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VOLUME THE
HI.
METHODS
OF
INTEGRAL
SECOND
TYPE,
INVARIANTS, DOUBLY By
Translation
of "Les
CELESTIAL
H.
M6thodes
MECHANICS
PERIODIC
ASYMPTOTIC
SOLUTIONS SOLUTIONS
Poincar6
Nouvelles
de la M6canique
C_leste.
Tome III. Invariants int_graux. - Solutions p_riodiques deuxi_me genre. Solutions doublement asymptotiques." Dover
NATIONAL For
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TT
F-452
'
(i,
P_,EQEDING
PAGE
BLANK
TABLE
NOT
OF
CONTENTS
VOLUME
III
CHAPTER INTEGRAL
FILMED.
XXII
INVARIANTS
Pa_m_ i Steady Motion Definition of
of a Fluid ............................................. Integral Invariants ....................................
Relationships Between the Invariants and the Integrals ............... Relative Invariants .................................................. Relationship Between the Invariants and the Variational Equation ..... Transformation of Other Relationships Change General
the Invariants Between the
..................................... Invariants and the Integrals .........
in Variables .................................................. Remarks ......................................................
CHAPTER FORMATION
4 7 9 15 19 27 32 34
XXIII
OF
INVARIANTS
43 Use of the Last Multiplier ........................................... Equations of Dynamics ................................................ Integral Invariants and the Characteristic Exponents ................. Use of Kepler Variables .............................................. Remarks on the Invariant Given in No. 256 ............................ Case of the Reduced Problem ..........................................
CHAPTER USE
OF
45 50 65 69 71
XXIV
INTEGRAL
INVARIANTS
73 Test Procedures ...................................................... Relationship to a Jacobi Theorem ..................................... Application to the Two-Body Problem .................................. Application to Asymptotic Solutions ..................................
CHAPTER INTEGRAL
INVARIANTS
AND
81 83 88
XXV ASYMPTOTIC
SOLUTIONS
Return to the Method of Bohlin ....................................... Relationship with Integral Invariants ................................
iii
91 115
Page
Another Discussion Method ............................................ Quadratic Invariants ................................................. Case of the Restricted Problem ....................................... CHAPTER POISSON
120 130 134
XXVI
STABILITY
Different Definitions of Stability ................................... Motion of a Liquid ................................................... Probabilitites ....................................................... Extension of the Preceding Results ................................... Application to the Restricted Problem ................................
142 143 153 156 159
Application
167
to
the
Three-Body
Problem
................................
CHAPTER THEORY
OF
XXVII CONSEQUENTS
Theory of Consequents ................................................ Invariant Curves ..................................................... Extension of the Preceding Results ................................... Application to Equations of Dynamics ................................. Application to the Restricted Problem ................................
CHAPTER PERIODIC Periodic Solutions Case in Which Time Application to the
XXVI II
SOLUTIONS
OF
THE
SECO_
TYPE
of the Second Type ................................ Does Not Enter Explicitly ......................... Equations of Dynamics .............................
Solutions of the Second Type for Equations of Dynamics ............... Theorems Considering the Maxima ...................................... Existence of Solutions of the Second Type ............................ Remarks ............................................... ] .............. Special Cases ........................................................
CHAPTER DIFFERENT
FORMS
OF
THE
176 179 188 190 197
228 232 241 245 246
XXIX
PRINCIPLE
OF
LEAST
ACTION
Different Forms of the Principle of Least Action ..................... Kinetic Focus ........................................................
iv
203 209 215
250 262
Page 268 270
Maupertuis Focus ..................................................... Application to Periodic Solutions .................................... Case of Stable Solutions ............................................. Unstable Solutions ................................................... CHAPTER FORMULATION Formulation of Solutions Effective Formulation of Discussion Discussion
OF
XXX
SOLUTIONS
of the Second the Solutions
OF
SECOND
TYPE
Type .......................... ...............................
to Equations
of No.
13
PROPERTIES of
the
Second
OF
Type
Stability and Instability Application to the Orbits
the
of
THE
SECOND
TYPE
of Least
Action
.......
............................................ of Darwin ..................................
PERIODIC Solutions
OF
Principle
CHAPTER
the
OF
Type
THE
SECOND
TYPE
................................
CHAPTER DOUBLY
329 340 348
XXXII
SOLUTIONS Second
322
XXXI
SOLUTIONS and
294 296 310 320
...................................
CHAPTER
Periodic
THE
........................................................... of Particular Cases .......................................
Application
Solutions
271 273
357
XXXIII
ASYMPTOTIC
SOLUTIONS
Different Methods of Geometric Representation ........................ Homoclinous Solutions ..................................... ........... Heteroclinous Solutions ..............................................
366 377 383
Comparison with No. 225 Examples of Heteroclinous
386 388
.............................................. Solutions ..................................
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NEW
METHODS
OF
CELESTIAL
VOLUME H.
MECHANICS
I II
POINCARE
ABSTRACT
Integral
invariants
are
introduced
using
the
steady motion of the fluid as an example. The usefulness of invariants in celestial mechanics is demonstrated. Various forms of the three-body problem are treated. Poisson stability is defined for the steady motion of a liquid, the general and restricted threebody problem. The theory of "consequents" is introduced in the discussion. The existence, stability, and properties of periodic solutions of the second type are treated. These are related to the principle of least action and the Darwin orbits. The concepts of kinetic focuses and Maupertius focuses are introduced in the discussion. Periodic solutions of the second type are treated. Homoclinous and heteroclinous doubly asymptotic solutions are discussed for the three-body problem.
CHAPTER INTEGRAL
Steady 233. of
In
integral
ticular Let velocity at time
order
to
clarify
invariants,
example us
from
consider
components t.
* Numbers
given
foreign
text.
in
it the an
of
the
XXII INVARIANTS
Motion the
of
origin
is useful
field
to
and start
importance with
of
a study
the of
idea
a par-
of physics.
arbitrary the molecule
margin
a Fluid
fluid,
and
which
indicate
has
let the
pagination
u, v,
w be
coordinates
in the
the x,
original
three y,
z
Wewill consider u, v, w as functions of t, x, y, z, and we will assumethat these functions are given. If u, v, w are independent of t and only depend on x, y, z, the motion of the fluid is said to be stead!. Wewill assumethat this condition is satisfied. The trajectory of an arbitrary molecule of the fluid a curve which is defined by the differential equation d__
dr _ d:
U
If
it
were
possible
to
is therefore (1)
g
integrate
these
equations,
one
would
obtain
/2
z = 71(6 xo, yo, zo), y --_,(t,,. xo, yo, Zo), z - ?3(t, Xo, Yo, Zo), such
that
their
x,
initial
If the position
the
figure
y
and
z would
values
x0,
be
Y0,
expressed
as
function
of
time
t and
z0.
initial position of a molecule of this same molecule at time
Let us consider fluid molecules F 0 at the initial instant of
placed, their group will form continuously deformed, and at consideration will form a new
a
were t.
the group time; when
known,
one
could
of which forms these molecules
deduce
a certain are dis-
a new figure which will move while being the time t the group of molecules under figure F.
We will assume that the movement of the fluid is continuous, i.e., u, v, w are continuous functions of x, y, z; there are therefore certain relationships between the figures F 0 and F which are obvious from the conditions of continuity.
will
be
If be
the figure a curve or
If the a simply
true
F 0 is a curve or a continuous a continuous surface.
figure F 0 is a simply connected volume.
If the figure F 0 is for the figure F.
In particular, is incompressible, able.
a curve
connected
or
let us examine i.e., where the
a closed
surface,
volume,
surface,
the
the
figure
figure
the
same
the case of liquids where volume of a mass of fluid
F
F will
will
hold
the fluid is invari-
Let us assumethat the figure F0 is a volume. At time t the mass of fluid which fills out this volume will occupy a different volume which will be nothing else than the figure F. The volume of the mass of fluid did not change; thus, F0 and F have the samevolume. Therefore, one can write (2) The first integral the volume F0. Wewill
is extended over the volume F and the other over /3
then say that the integral fffax
is
an
integral
by
the
It
invariant.
is known
that
the
condition
of
incompressibility
can
be
expressed
equation du dv d_ dx + _y + -37. =°"
The
two
equations
(2)
and
volume
Let us again consider of a mass of fluid
such
that
if
one
calls
p
(3) are
first
volume
F 0.
integral In
is
other
equivalent.
the case of is variable. the
density
f ff The
thus
ax
extended words,
(3)
of
a gas, Thus, the
i.e., the case where the the mass becomes invariable,
gas,
one
has
(4)
d,.=f f f podXo a.roa,.o. over
the
the
volume
F,
the
second
over
the
integral
fff is an
can
integral
invariant.
In this case, be written as
where
the motion
d(pu)
a_
is
d(po)
+--dT
steady,
the
equation
of
continuity
d(pw)
+
d_
=o.
(5)
The
conditions
234. second
The
us
hold
at
have
us
a a
for
assume
constant
of
(5)
remain
vortices
that
develop
us
Under
the
the
figure
that
the
In u,
these
Let
that
F
the
v,
this
that
satisfy
is the
that
are
of
along In
the
they
us
other
235.
words,
Due
presented
It
is
give
this
Let
that cases
simple
us
the
of
t
z
If
4
and
are
of
they their
could
it
is
with
a
The
same
compressible to
remains
or
forces
steady,
conditions.
It
is
not,
which it
is
not
useful
/4
the
integral
w dz).
it
has
is
an
the
same
to
value
along
the
curve
much
meaning
differential
functions
be
initial
can
be
the
of
= _
x,
x0,
examples
used in
of
by
which
which integral
invari-
generalizing it
is
not
of
the
their possible
form
dz = at, = -f
y,
I have
invariants.
equations
integrated, values
the
consideration
broader, to
invariant.
Invariants
question, the
invariants are
integral
Integral
the
one
these which
consider
of
lead
physical
given
curve.
subjected
motion
v dy+
integral
dx -_
Y,
closed
consider
shows,
nature
readily
for a
to
clear
definition
x,
us
F O.
Definition
to
provides
satisfied.
let
Helmholtz curve
a
only
certain here.
assumption,
is
whether
and
order w
F0
fluid,
conditions
assume
theorem
and
just ants.
Helmholtz
F.
(udz+ As
equivalent.
of
figure
temperature,
potential.
necessary to
assume
true
Let is
theory
and
example.
Let will
(4)
(i)
z.
one
would
Y0,
z0.
obtain
x,
y,
z
as
a
function
If we sent fine
the the
assume
that
the
coordinates of laws of motion
If these equations the point M at time
of
coordinates If group
x 0, Y0,
one
is represented
by
t and
M in space, point.
x,
y,
equations
are integrated once, one t, if its initial position
z0'
considers
of which
time
a moving point of this moving
z repre(i) de-
can find the position M 0, given by the
is known.
moving
points
which
obey
the
same
law
and
the
a figure F_ at the initial instant of time, the • different figure F at time t of these same points w!ll-form a will be a line, a surface, or a volume depending on whether the
group which figure
F 0 is Let
us
forms
a line, consider
a surface
or
a simple
a volume.
/5
integral (A dx + B dy + C dz),
(2)
where A, B, C are the known functions of x, y, and z. If F 0 is a line, it may happen that this integral (2) extended over all of the elements of the line F is a constant which is independent of time, and is consequently equal to the value of the elements of the line F 0. Let the
us
double
now
assume
that
this
F and
same
integral
F 0 are
A'
B'
C'
are
functions
integral has the same value the surface F, or over all Let triple
us now
assume
that
of
is
value
us
let
all
of
imagine
F and
(3)
dy),
and
z.
It may
extended over of the surface
F 0 are
volumes,
happen
all the F 0. and
let
that
this
elements
us
of
imagine
the
integral
a function for In
are
and
surfaces,
+
x, y,
which is of those
f f f
M
over
integral
dyd=+ B'a where
extended
F and
these
integral It
of x,
y,
z.
It
_l ax & d:;
is possible
(4)
that
it may
have
the
same
F 0.
different
cases,
we
say
that
that
the
simple
the
integrals
(2),
(3)
or
(4)
invariants.
occasionally
happens
integral
(2) will
only
have
the samevalue for the lines F and F0 if these two curves are closed, or the double integral (3) will only have the samevalue for the surfaces F and F0 if these two surfaces are closed. Wemay thus say that (2) is an integral to to
the the
closed closed
curves and surfaces.
that
(3) is
an
invarlant
with respect invariant with respect
integral
236. The geometric representation which we have employed plays no important role. We can thus lay it aside, and nothing prevents us from extending the preceding definitions to the case in which the number of variables is greater than three. Let us consider the following equations dxl
x_
where
XI,
one of
could t and
the
same
X2,
...,
their
the
.....
_
=at,
the
given
functions
them,
one
would
find
initial
terminology, and
dx_,
x,
X n are
integrate of
dx,
-
values
we
point
may
xp0,
point
..., M
the
x2,
...,
x_.
In order
system
of
x 0I, x0,
x n.
to
retain
values ...
x I, x2,
_.
Let us consider a group of points M 0 forming a subset F 0 and of corresponding points M forming another subset F (I).
Let
us
consider
an
integral
F are
of
the
continuous
order
subsets
(i) The
word
subset
is now
commonly
dxl,
...,
having
the
p
p
where _ is a function of Xl, x2, ..., Xn, and where of p differentials chosen among the n differentials dxl,
If
functions
group
F 0 and
of values
...,
x n as
Xn,
that n.
system
of Xl,
x I, x2,
...,
We shall assume dimensions where p _
M 0 the
x_,
call
(1)
d_
is
the
product
dxn.
employed,
so
that
I did
not
feel
it was necessary to recall the definition. Every continuous group of points (or system of values) is named this way: In threedimensional space, an arbitrary surface is a subset having two dimensions, and an arbitrary line is a subset having one dimension.
/--6
It
is
subsets
F It
two sets
possible and
may
to
F 0.
also
We happen
subsets F and are closed.
closed
give may
this thus
that
integral say
the
that
this
it
integral
same
is
an
has
F0, but only under the It is thus an integral
value
for
integral
the
condition invariant
the
two
invariant.
same
value
that with
these two subrespect to the
for
the
subsets.
Other
types
example,
let
lines.
It
of
us is
integral
assume
invariants
that
possible
p
that
=
may
1 and
the
that
the
same
may
also
be
value the
for
case
F
and
F0,
the
following
for
also
and
assumed.
F 0 may
be
For reduced
to
/___7
integral
f (A,dx,+ A, ax, +...+ has
be F
A, dx,,) = f _A,dx,
and
is
an
integral
invariant.
This
integral
IxBi dx_ + 2 X C,k dxi dxk,
where
B
and
C
I stated,
it
F
and
and
F0, The
are is
like
other
quantity
p
Let
us
A
of
that
the
will
called
Between
again
dr,
one
If the
could
integrate
integration
following
it,
were
all
the
dr.
of
performed,
may be
Xl,
x2,
...,
x n.
have
the
same
value
readily
order
of
Invariants
the
x, - _
If
may
the
consider
of
integral
examples
be
functions
this
similar
Relationships 237.
the
possible
the
and
As for
envisaged. integral
the
invariant.
Integrals
system dw,,
..... its
_
= at.
integral
the
result
(1) invariants
could
be
could
presented
be
formed.
in
form Yt = C,,
y, = c,, y,_ ,-= Cr,-t, "= t -_- Cn,
(2)
CI,
C2,
functions
...) of
C n are the
arbitrary
constants,
and
the
y's
and
z's
are
the
given
z for
the
new
variables,
x's.
Let us change instead of x's.
the
variables
by
taking
y's
and
Let us now consider an arbitrary integral invariant. Under the sign / (which will be repeated p times if the invariant is of the order p), this invariant must include a certain expression, the function of the /8 x's and of their differentials dx. After a change in the variables, this expression will become a function of the y's, z, and of their differentials dy and dz. Without changing the y's, in order to pass from one point of the figure F 0 to a corresponding point in the figure F, it is necessary to change z into z + t. Therefore_ when passing from an infinitely small arc of F 0 to the corresponding arc of F, the differentials dy and dz do not change (the quantity t which is added to z is, in effect, the same for the two ends of the arc). If one considers an infinitely small figure F 0 having an arbitrary number of dimensions F, the product of a number (equalling that
and the corresponding of the dimensions of
of differentials dy or dz will one figure to the other.
either
not
change
when
one
passes
In short, in order that an expression may be an integral it is necessary and sufficient that z is not contained in it; the dy's, and dz may be included in an arbitrary manner. Let discussed
us consider an expression in the preceding section
having
f
This
expression
represents
of xl, x2, ..., n differentials
Xn,
d_
is
an
We
would
like
to know
same
form
as
from
invariant, the y's,
that
which
_.Ad_o,
integral
a product
d.rl)
the
figure F 0 and F)
of
of
dx2,
whether
carrying out a change in variables expression (3) becomes fEB
the
order
p,
p differentials
. ..,
this as
(3)
a function
chosen
from
the
dxn.
is
an
indicated
d_',
A is
integral above,
invariant. we
find
that
By
we
B
is
tials
a
function chosen
of
the
from
y's
the
n
and
dy,, In
order
necessary only
that
and
depend Just
as
the
in
z,
_y,,
expression
sufficient
on
of
dm'
is
a
product
of
p
differen-
differentials
that
.",
dye-,,
dz. integral
(3)
be
an
all
of
the
B's
be
invariant,
it
is
independent
of
z
and
y's.
the
preceding
section,
let
us
again
consider
the
ex-
/9
pression
f
The
Bi's
and
After
the
the
Is
Ci. k
change
_x
B, dxf
are
in
+
functions
the
greater
symmetry
in r
the
xt-----yi, In
order
necessary
and
dependent
of
that
z,
and
that
depend
I have
(4)
be
an
of
the
on
238.
let
We to
us
are
the
now
led
closed
determine
the
to
Let
condition
by
A,dxj+
is
an
integral Let
and
our
us
invariant carry
integral
out will
with the
;
set
the
Xn=Z
B' i 's
following
,
integral
to
form
us
first
which
the
At dx2 +...+
respect
change
becomes
invariant, and
the
in
the
integral
assume simple
that
is be
in-
invariants p
=
i,
(i)
Andxn)
to
closed
variables
Bn-tdyn-t+
and
integral
lines. as
indicated
become BtJ/lq-B,d)',+..._
it
C'i.k'S
Invariants
attempt
subsets.
expression
y.
Relative
relative
x's.
t
n--I);
all
only
the
-t- "z_ C_.k dxidxk
(/----- T, 2.....
expression
(4)
this
notation,
.
sufficient
of
variables,
_/Z B_ dx?
For
_Ci.kdx_dxk,
B4dz)
t
above,
which I can write again, taking end of the preceding section
the
most
symmetrical
notation
from
B+d:;.
fx
the
(i')
This simple integral, extended over a closed, one-dimensional subset -i.e., over a closed line -- may be transformed by the Stokes theorem /i0 into a double integral extended over a non-closed, two-dimensional subset -- i.e., over a non-closed surface. We then have
f-'B,.d_;.: However, integral
lute
We In closed
all
be
can
J
fX' ¢,uB,._ abe) _._ \ dx_
the integral of the second member of invariant, and not only with respect therefore
order that lines it is
independent
Similarly,
and
conclude
(I) be an necessary
of
the
(2)
dxl / dxl dx'k.
(2) must be an absoto the closed subsets.
following:
integral invariant with and sufficient that the dBl
dBa
d_-;,
d.r;
respect to the binomials
z.
more
generally,
let
SA d_
(3)
be an integral expression of the order p, having the same which were discussed in the preceding sections. We would whether this is an integral invariant with respect to the of the order p.
form as those like to know closed subsets
Let us assume that this integral is extended over an arbitrary closed subset of the order p. A theorem similar to that of Stokes states that it may be transformed into an integral of the order p + i, extended over an arbitrary subset, which may be closed or not closed, of the order p + 1. The transformed integral may be written
fez
One ternately
i0
k 4-- _dA dxh. doJ.
always takes the sign + if p is even, and the if p is odd. [For additional details, refer
(_) signs to my
+ and - alreport on
the
residuals
to
my.report
de
l'Ecole
of
double
integrals
contained
in
condition
which
integral
invariant
of
that
be
(4) 239. let
an
us
with us
Let
M 0 be
its
that
respect
change
it
M
be
its
Volume
Edition
expression
is
to
closed
VIII),
of
the
and
Journal
and
Yl,
Y_,
the
make
z
.-.,
The
by
Bk'S
subsets p
the
be
+
is,
/1_!
is
i.
preceding
that
an
section,
an
integral
our
change
in
variables.
yn-s,
point
...,
Bk
of
yn-l,
will
writing
of
to
variables).
corresponding
appear,
order
invariant,
(i')
new
y_,
closed
the
(i)
(3)
lines.
form
F0
for
to
of
a relative
the
sufficient
respect
consider it
of
coordinates.
I will
p with
invariant
(with
the
and
integral
yl,
be
Centenary
necessary
order
to
a point
coordinates
Let
the
again
assume
Let
is
absolute
Let us
invariant
be
Special
Mathematica,
Polytechnique.]
The
and
the
(Acta
and
z+t
be in
F
functions
the
of
following
the
y's
and
of
z,
but
form
Bk(z).
If
that
the
is,
line
the
F 0 is
closed,
we
an
exact
depend it
not
must If
derivative
then
have
expression z [Bk(_
is
will
differential only
on
the
be
reduced
to
we
assume
that
of
Bk(Z)
which y's a
with
and
+
t) -- B,(_)]
I set z,
equal
but
also
(3)
dx',
to on
The
dV. t.
In
function order
V will
that
t =
o,
constant. t
is
infinitely
respect
to
small z,
and
expression
if
we
(3)
call may
B'k(Z) be
reduced
the to
11
The
expression x Bk.(z) dx_,
(4)
is then an exact differential which I set equal to dU. The function which is thus defined will depend on the y's and z, but it will no longer depend on t. I shall again make z appear by writing U(z). It then happens that dV
f + t)dx'_=jdU(
f
-37 =j-B'_(z
arbitrary
function
of
.. + t)=
U(z
is an
z of will
However, U(z) may be regarded as the derivative with another function W(z) which is also dependent on the then have
t.
d d--iW(.
On the other hand, may finally conclude
since that
The quantity +(t) designates be assumed to be zero without generality. One
then
+ t) = U(.
V must
V = W(z
/12
4-t)+f(t),
f(t)
we
U
be
-+ t).
reduced
+ t)--W(_:)+
an arbitrary essentially
respect to y's, and we
to
a constant
for
t = o,
?(t), function of restricting
t only, and may the conditions of
finds d B_.(.)= _
C k is
independent
of
z, so
that
f and the integral
of
12
first integral is an absolute
the
W(z)
expression
dW + f
(i')
may
be
reduced
to
XCkdx'k,
is that of an exact integral invariant.
240. In a similar way let us a higher order than the first.
+ Cx..
discuss Let us
differential,
a relative assume that
and
the
invariant
second
which
is
EA d_ is
this
invariant
The
which,
after
the
is
variables,
will
become
be
zero,
whatever
may
(i)
[B(z + t)-- B(z)] d_'= J
be
the
closed
subset
of
order
p
over
which
It
must
therefore
similar an
and
to
exact Let
now
limited
a
of
It
a
subset
integral
the
value
that
a
total
V
of
conditions"
differential
of
the
which
is
which first
v
subset of
p
-
i
p
dimensions,
dimensions
which
will
not
serve
closed
as
the
(i),
which
if itis
same
boundary
the
integral
equals
an
is
extended
calculated v, (i)
one
only
integral
for
of
over
other
will
obtain
depends
the
the
on
order
p
subset
similar the the
-
V,
subsets same
value
boundary
will
not
V',
V",
--
i.e.,
is
product
extended of
This tion
on will
p
-
over
the
v.
i
subset
(2)
is
v.
Let
its
changes by
v
and
while
clearly us
a
where
C
is
function
consider
its
_/7 do_'=
vB'(=
dm"
a
designates
function of
t,
of which
derivative
an the
arbitrary
y's,
depends
with
z and in
respect
t.
addito
t.
have
dt
As
(2)
subset
1 differentials,
integral the
the
be etc.,
j which
/13
order
it.
However,
having
certain"integrability
stating
consider
by for
The zero.
satisfy
those
differential.
us
boundary
We
it
extended.
are is
in
integral
fx must
change
last
expression
t
into
changing
z
t - h
=
shows, or
everywhere
when, into
this at z +
+ t; &,,'.
derivative the
same
does time,
one
not
change
transforms
when V
one (or
v)
h.
13
It
can
be
concluded
that
J----/Y D(z)
is
a
The
function
of
J has
the
following
form
D(z-4-t)dJ--fED(z)dJ,
x,
y,
z.
integral x D(z)d,o"
(3)
is of the order p - i, but it may be readily transformed into an integral of the order p. It is sufficient to apply the transformation which, in section No. 238, allowed us to change from the integral (3) to the integral (4), and which is the opposite of that by which, in the present section, we changed from the integral (i) to the integral (2).
to
The integral (3), the integral of the
extended order p
over
the
subset
v,
is
therefore
fEE(z)dt_' extended
over
the
subset
It
2.
It may
Under
is
zero
for
be
this
and
the
reduced
are
this
closed
to an
assumption,
integrals
However,
terminology employed for simple integrals, we (4) is an exact differential integral. And,
every
s= f
we
will
XE(z+t)d_'--
extended
equation
may
it is valid This
14
means
for that
an
arbitrary
subset;
integral
over
of
lesser
order.
have
f
__E(z)d_,',
the
also
fx[ B(.-+ o- E(z + t)] and
(4)
V.
By analogy with the may say that the integral in effect: i.
equal
subset
be written
= subset
V. as
E(.>] V.
follows
/14
v
.f_-[B(z)-is
an
absolute
We
therefore
Every tial
integral
241. of
the
so
that
In
order
The
same one
+
is
is an
the
be
the
However,
There
is
a
exact
in
If
invariant
of
of
the
p
so
order
that
which
is
if also
a relative
3,
then
to
us
Between
again
at
In into obtains
may
form the
order
xi + the
_i,
the
invariant p.
invariants, and
to
construct
very
quickly.
question one
be
is
wishes invariant
also
transform to
which
zero.
transformed,
one
is it
/15
illusory;
to
an
obtains
exact
again,
an
in-
differential
one
obtains
a
the
dx_ --
X2
of
dxn
variational
Chapter
equations,
and
we
disregard
the
linear
equations
dXk
dXk
(1)
equations
as
they
were
IV.
these
=
the
-- dr.
Xn
form
cl_,k
and
system
__.,.--
to
of
Invariants Equation
corresponding
beginning
system
in
transform
the
consider
XI We
differen-
zero.
dx,
defined
it
abandoned
invariant
Variational
Let
absolute
absolute
integral
is
this
wishes
be
which
The
p
exact
invariant
apply
procedure
order
an
successively.
to
lead
Relationship
242.
....
have
would
of
an
to to
invariant
but
how
applied
integral.
I,
one
seen
+
the
sum
from
would
the +
the
invariant.
continue p
the
would
an
integral, result
2,
which
differential
is
have
be
to
procedure
transformation
variant
also
p +
which
we
deduced
may
in
result:
integral
tempted
case
case
following
invariant
238,
be
order
this
the
No.
procedure
of
the
absolute
1 may
could
invariants
this
an
Section p
at
integral
and
d_'
invariant.
arrive
relative
integral
E(z)]
in squares
equations of
"+
dXk
the
(i) _i's.
we
change
One
x i
thus
(2)
15
There between of
is
the
a close
integrals
equations
relationship
which
of
(2) and
equations
may
be
the
readily integral
perceived invariants
(i).
Let F(h,
h,
..-,
i,,)=
¢onst.,
be an arbitrary integral of equations (2). This will be a homogeneous function of the _'s, which depends on the x's in an arbitrary manner. I can always assume that this function F is homogeneous of degree 1 with respect I would only have to a homogeneous Let
to the increase
function
us now
of
consider
degree the f
which
is We
an
integral
should
first
$'s. Because, if this F to a suitable power
F(dxt,
is
an
infinitesimal
infinitesimal of the first The
of
note
the
be
Under reduced
quantity
quantities order with
simple
d.c2,
invariant
F(dx,, of
of the respect
integral
(3) is
the case, to obtain
i.
following
that
were not in order
expression ...,
system
(i).
quantity
dx, .... the
(3)
dx,,),
under
the
sign
/16
f
, dxn)
first
order,
since
first order, and that to the quantities. therefore
dxl,
dx2
.._
dxn are
F is homogeneous
finite.
this assumption, let us first assume that the figure to an infinitesimal line, whose extremities have the
F 0 may following
coordinates
The quently
integral will
(3) may
be
reduced
to a single
element,
and
conse-
equal
Due to the fact that this expression is an integral of equations (2), it will remain constant and will have the same value for the line F 0 and
16
for
the
line
F.
If the line F0, and consequently the line F, are finite, we may divide the line F0 into infinitesimal parts. The integral (3), extended over one of these infinitesimal parts of F0, will equal the integral (3) extended over the corresponding infinitesimal part of F. The integral extended over the entire line F0 will equal the integral extended over the entire line F. Therefore, the integral
(3) is an integral
invariant. q.e.d.
Conversely, let us assumethat (3) is an integral the first order, and
invariant
of
F(_,,_,,.., b,)
will
be
and
In for
an
integral
reality, the line
F 0 is reduced coordinates
of
the
equations
the integral F, whatever
to
an
(3) must be the same for the line these lines may be, and particularly
infinitesimal
element
x_ As we
have
seen,
the
(2).
and
integral
whose
ends
have
the
FG if
following
x_+b.
(3) may
be
reduced
to
/17
F(_,,$,...., $.). Since
the It
integral
is
is an
therefore
an
Thus, an integral invariant of the first 243. higher
than Let
tions
Let
us
(2).
the
us
now
integral
of
this
expression
equations
these
two
be
constant.
q.e.d.
of equations (2) corresponds order of equations (I), and
to each integral vice versa.
to what
the
invariants
of
an
order
correspond. two
particular
arbitrary
solutions
of
the
equa-
Let
{h, _, _;, h _;...... ..... _,
be
(4) must
(2).
determine
first
consider
invariant,
(4)
(5)
solutions.
17
The following
which
depend
No matter duced to In
on
functions may exist
the
xi's
what the constants other
,
the
eL'S,
and
the
two chosen solutions, which are independent
words,
the
function
d_
T!
s
at
the
same
these functions of time.
F will
dXk
_i
be
dX1,.
an
time.
may
integral
of
be
the
re-
system
dXk
dt - d_T _' + _
_' +"" + _
_"' (6)
d_ = -_ dX_., _"÷ _-i which
the
and
Let us F has
that
and
_i's
the
_i's
formulate the form
Aik'S
It may
the
are
then
a more
definite
of
stated
that J =rE
is an
integral
invariant
mal
Let us assume that parallelogram whose
t
O
=
xl,
gram
dX_,. _,,, ,, _; + "" + -d-_
satisfy.
functions be
_
of
the
the
hypothesis,
x's
the
and
let
us
assume
alone.
double
integral
_t_,l.rid_k
equations
(I).
the figure F 0 may be reduced to an infinitesicorners have the following coordinates at
x_ + _l, xi + _,,
xt + ¢t-4- $i.
The figure F will also be similar to whose corners will have the following
an infinitesimal coordinates at
parallelot = t
x_, _ + b, _i + [_, _i + _i+ _. have
and the and
18
The integral precisely
J the
will be following
reduced value
-- since, under the hypothesis, system (6) -- it will have the FO.
to
a
single
element
which
this expression is an same value for the two
will
integral figures
of F
/18
Let
us
now
assume
that
F and
F 0 are
two
finite
surfaces.
Let
us divide F 0 into infinitesimal parallelograms, to each of which an elementary parallelogram of F will correspond. The value of J is therefore the same for each element of F 0 and for the corresponding element of F. It is therefore the same even for the entire surface F 0 and
for
the
entire
surface
The
integral
J is
The
converse
of
preceding
F.
therefore
this
may
an
be
integral
proven
in
invariant, the
same
q.e.d.
way
as
in
the
section.
244.
The
theorem
is
obviously
general,
and
may
be
applied
to
invariants the third
of a high order than two. Let us present it for those of order. Let us consider three special solutions of the equal (2), _i, _i, _i" These three solutions must satisfy the system T!
tions
d_,k
_
dXk
7i = _
777_, ._" (7)
d_;+ X_ aXe, , d_,_
'_
dXk
-RF = _'=J_ If
the
system
(7)
includes
an
,
_"
integral
of
the
form
I
ZAi.k.t
(8)
_k _;, _ ],
1 where
the
A's
are
functions
of
the
x's,
the
triple
integral (9)
XAiktdxid_kdxt will
be
an
integral
invariant
of
Transformation 245.
With
the
invariants
variational equation, one may make it possible to transform If
one
knew
a
certain
the
of thus
equations
the
(i),
of
vice
versa.
Invariants
reduced
to
the
readily find several these invariants.
number
and
integral
integrals procedures
invariants
of
of
the
which
the
equa-
tions
19
d J-i
dt -- X_, one could tions
deduce
from
each
of
them
an
(1)
integral
of
the
variational
equa-
d_,. %,_dX, d--7=Z., ,-t_ $_" By
combining
integral the
these
of
different
equations
integrals,
(2),
from
(2)
we
which
one
the
case
will
may
obtain
deduce
a
new
a new
invariant
of
equations(l).
Let
us
commence
by
studying
of
first-order
invariants.
Let
be
a
certain
will
be
number
functions
Now
of of
integrals
the
q
integral The
(dx_.), f
pend the
invariants
functions
of
under
the
the
depend
on
the
on the xi's differentials
xi's
in
an
and
must
be
homogeneous
first
(i).
These
integrals
F, (dxi)
their
same
equations
order
.....
of
of
these
t
the
...,
dxi's.
However,
with
They
will
respect
deto
dxn,
first
order.
Then
will of
2O
be the
integrals first
order
of
the
with
equations respect
(i).
Fq(dx_)
manner.
dxi
rq(dx_),
differentials
arbitrary
and
f
f
sign
dxl,
they
equations
F,(dx,) .....
Ft (dxi),
will
of alone.
let
fFt be
xi's
tO
(2)
and
the
_i's.
will
be
homogeneous
and
/2O
-V
¸
Now
let O(_,, _,, ...,_p; F;,F,, ...,Fq)= O[_k, Ft],
be
a function
of
an arbitrary with respect
the
manner, to the
_'s but F's.
and
of
which
the
F's,
which
is homogeneous
depends and
of
on the
the
@'s
first
in
order
Then O['>,,FI(_)] will be
be
a new
integral
a homogeneous
of
function
the
equations
and
of
the
(2).
first
In
order
addition, with
this
respect
will
to
the
$i's. It
thus
results
that
J /O[¢k,
is an
integral The
same
invariants
by
For
invariant result
of
could
changing
the
the be
Fddx_)]
first
order
of
readily
achieved
invariables
of No.
the when
equations
(i).
transforming
the
237.
example,
and
will
be
integral
246.
The
invariants. same
calculation
may
be
applied
to
invarlants
of
a higher
order. Let
be
the
p integrals
of
the
equations
(i),
and
/21
let
21
...j
be the q integral functions
invariants
of the xi's
of the second
and the products
order.
The F's will be
of the differentials
dxi dx k.
They will products.
be homogeneous
and of the first
order with respect
to these
Then
will be integrals
of the system
(6).
If O[%,F_] is
an
arbitrary
function
of the first order with
of
the
_'s
respect
and
of
the
F's,
which
is
homogeneous
to the F's, the expression
will be an integral of the equations (6). It will be homogeneous addition, and of the first order with respect to the determinants
As a result,
in
the double integral o['P_,Ft(dx_dxk)]
will
be
the
247. means
cedure
an integral
of
example,
invariants possible to
second
order
of
equations
(1).
of the same order, we thus have other invariants of the same order.
of the same order obtain new invariants
are
known, the of a different
same proorder.
I
assume, which and homogeneous
is func-
let fF,(dxi)
be two integral invariants the most general case, that tions of the differentials
22
the
Knowing several invariants of combining them to obtain
When several makes it For
invariant
,
of F 1
dx i.
the and
f e,(dx,), first order. F 2 are linear
/22
v
The
expressions
will
be
and
these
homogeneous will
In
the
and
be
same
of
the
integrals
first
of
order
equations
be
integrals
As
a
respect
to
the
$i's,
way,
F,($_), will
with (2).
of
the
F,(_')
equations
(6).
result,
(i0) will
be
an
integral
Since
As
of
F 1 and
a
F2
result,
the
are
system
linear
expression
(6).
with
(i0),
respect
which
to
the
changes
$i's,
sign
we
when
will
one
have
exchanges
!
the
_i's We
and may
the T
the
_
's,
does
conclude
function
that
of
coefficients
integral
therefore Now
the
depend
invariant
deduced
two
order and to
change
this
when
one
expression
changes (I0)
is
_i a
into
linear
_i
+
_i'
and
determinants
on
the
of
from
x's
alone,
but
not
on
the
$'s
the
equations
and
the
integral the
are n
second
order
expression
of
(i)
may
(i0).
let
and F2
the
this
fFj
be
not
's.
An be
_i
thus
homogeneous
and
the
invariants second
linear
and
differentials
is
(dx,.),
of of
equations
the
homogeneous dxi,
:F2(dxidxk)
the
second
(I); order.
functions, second
with
the I
first shall
the respect
first to
is
of
the
assume
that
with
respect
the
n(n-l) z
first F I
pro-
ducts
23
dxt dx_.
The
/23
functions
F,(_,),F,(_-- h_D will
be
integrals
The
of
the
system
(6).
expression
. i t + F,(_i)F,($i_;__$k_ )
will
be
an
On
integral
the
linear
and
other
of
the
hand,
it
homogeneous
system
(7).
may
readily
with
be
respect
_
An
integral
Now
invariant
of
two
the
&
to
verified the
invariants
We
which
I
Then
the
it
will
be
_
third
order
may
thus
of
can
deduce
can
write
F,(dz_dx.),
the
from
as
f
be
deduced
from
it.
second
it
follows,
two
that
is,
F,(dx, dz.)
order
of
integrals
for
purposes
equations
of
(1).
equations
of
(6)
--
brevity.
expression
+ F,(_U)F,(_"T) + F,(_"T) F,(_') F,(_t)F,(_'_") + Ft(_'_')F,(_') -4F,(,_') F,(_'_') -iF,(_'_') F,(_"_)
24
that
determinants
let
f be
(ii)
(12)
will
be
an
equations
integral
to
In
the
of
the
equations
addition,
this
will
be
respect to the determinants the corresponding quantities I shall linear
with An
from
continue respect
integral
It
by
adding
the
following
a linear
and
homogeneous
that
products of
F I and
the
F 2 are
function
quantities
with _i and
homogeneous
/24
and
dxidx k.
the
fourth
order
could
thus
be
deduced
(12).
should
be
noted
zero
when
we
to
obtained
formed, with,, four,,,of _i , _i ' _i "
assume
the
invariant
expression
equal
to to
system
(7)
that
this
invariant
does
not
become
exactly
set F,= F,.
Expression
(12),
divided
by
2, may
be
then
reduced
_, (_')F, (_'_") ÷ F, (,_') F, (_'_ ') + F, (_')F, An invariant invariant of the
of the fourth second order.
be obtained by the the order 2p would number)
order may always be An invariant of the
same procedure. be obtained from
In
general,
More generally, it (2p being an
deduced from an sixth order would an invariant of arbitrary even
let fF,,
be
two
arbitrary
p,
and
the
invariants
second
is
of
the
of
fF, equations
order
with
respect
to
the
products
(i);
the
first
is
of
the
order
q.
I shall assume that F I and F 2 are first with respect to the products
second
(_T).
•
248.
the
to
linear and homogeneous of p differentials dx,
functions, and the
of q differentials.
Let
_?',
_?' .....
_F+_'
25
be p + q solutions of equations (2). the system of differential equations
a_,_j dt = ,_dXk$_ _
of by
(i,k=,,_
These
....
solutions
,n;S=,
_" ....
Then
the
(13)
when each product determinant formed
, _P'.
In the same way, let F'2 represent the quantity when each product of q differentials is replaced by determinant formed by means of the q solutions
_F+,,,
satisfy
, _, "", p+q).
Then let F'I be the quantity which F 1 becomes p differentials is replaced by the corresponding means of the p solutions
_",
will
_F+,_....
/25
which F 2 becomes the corresponding
, gpw,.
product F_F;
will
be
an
Under
integral this
of
system
assumption,
(13). let
us make
_'_, _" .... undergo
an
arbitrary
permutation.
the
p +
q letters
, _Y+q' The
product
F'1F' 2 will
become
F;F; and
this
will
still
be
an
integral
of
system
(13).
We shall give this product the sign +, if the permutation under consideration belongs to the alternate group -- i.e., if it may be reduced to an even number of permutations between two letters. On the other hand, we shall assign the product permutation does not belong to the alternate group be reduced to an odd number of permutations between In any
case,
the
expression +F_FI
26
the - sign, if the -- i.e., if it may two letters.
(14)
will
be an integral
of system (13).
Wehave (p + q)! possible permutations; we will therefore obtain (p + q)! expressions similar to (14). However, we shall only have
which will be different. This is due to the fact that expression (14) does not change when the p letters which are included in F" 1 are only interchanged among them, and, on the other hand, when the q letters which are included in F'2 are only interchanged amongthem. Let us now take the sum of all the expressions (14). Weshall have an integral of system (13). However, this integral will be linear and homogeneous with respect to determinants of the order p + q, which can be formed with the letters
An
invariant
of
the
order
p +
q of
equations
(i) may
thus
be
/26
de-
duced. If p = q and if F I is identical to F2, the invariant thus obtained be equal to zero if p is odd. However, this will no longer be the if p is even, as I explained at the end of the preceding section.
will case
Other
let
249. us now
Based trace
Relationships the
Between the Integrals
Invariants
and
on the knowledge of a certain number of invariants, the manner in which we may deduce one or several inte-
grals. I shall
first
assume
that
we
know
two
invariants
of
the
tn_-h order
and
where ratio
M and M' _-will
M'
are be
an
functions
of
the
x's.
integral
of
equations
It may
be
stated
that
the
(i).
27
Let
us
consider
the
variational
_P', be n arbitrary equations. These n tions, which as system s. Let _i (k).
A be
solutions
equations
_p' .....
which
are
determinant
linearly
independent
of
these
a system of differential equa(6) and (7), which I shall designate
formed
by means
of
the
n2's
letters
Then M_
will
let
_"_
solutions will satisfy is similar to systems
the
(2) and
be
integrals
of
system
S.
_I'_
and
The
same
will
also
hold
for
the
ratio
M' M and, must
two
since be an
this ratio only depends on integral of equations (i).
The
same
Let
us perform
integral
result
may the
invariants
be
obtained
change will
the
x's,
in another
in variables
and
not
on
the
_'s,
it
manner.
as was
done
/2__17 in No.
237.
Our
become
fMS
d3, dy, . . . dy._,
dz,
dy, dy_ ...
dz
and fM'J
dyn-,
J designates the Jacobian or the working determinant of ables xl, x2, ..., xn with respect to the new variables Yn-l, z. According
to No.
237,
MJ yl_
and
this
is an (i).
also
integral
holds of
for
the
equations
and
M'J
must
y2,
-..,
Y,_-I,
ratio (I),
M' _--. this
only
Since ratio
depend
every is
an
the old variYl, Y2, ---,
on
function integral
of the of
Yi'S
equations
q.e.d.
28
250.
This
For
procedure
example,
may
be
the
also
p
in
several
...,
/Fv(dx,)
linear
invariants
./F,(d.r,),
of
the
first
order.
the
Mi's
It of
Let
us
assume
that
the
differentials
we
have FI = 312F_ + 313F_+...+
and
ways.
let
fF,(d.ri),
be
varied
depend
may
be
equations
Let
only
stated
on
that
the
x's,
the
Mi's
MpFp,
and , if
not
on
p _n
+
].,will
be
dx.
integrals
(i).
Aik
be
the
coefficient
of
dx k
in
F i.
We
must
then
have
ALk = M: A_.k + M_ A3.k-_-...+ MpA p._. Let
us
perform
invariants
then
the
change
in
7l:',(dx',), If
we
also
variables
as
was
done
in
No.
Our
237.
become
:F;(dx',),
...,
f
F'Adx'O.
/2S
set F_ = E A_k dx_.,
we
must
have A',.k = .M: A'_k + M: A'3k+...+
We the
shall
Mi's,
then
provided
have
n
that
p =< n
linear +
equations,
on
The
z.
integrals 251.
same
to
No.
is
237,
therefore
of" equations Now
the
an
integral.
which
only
on
we
may
obtain
!
Aik
true
s depend for
the
Mi's,
that
the
y's,
and
is,
the
Mi's
not are
(I).
let F(x,, x2, ...,
be
from
i. !
According
.M_ A_,_.
It
is
apparent
[\'_telF dx,
xn)
that
-+ _dF
dx, +.
dF ) . " +--_x_xnaxe,
29
will
be
an
integral
One
may
Let
us
then
invariant formulate
consider
an
No.
In 237;
order our
The
B's
and
order. question:
invariant
dxl+A_dx2+.
D
of
C's
must
this, let become
depend
us
on
the
sign the (1)?
make
the
y's,
dU is an exact following form
the
first
order
+Amdxn)
that the term under the f be the relationship between the integrals of equations
If this expression therefore have the
must
i
first
following
integral
to determine invariant will
the
the
the
f(A and let us assume tial. What will differential and
of
but
is an integral
exact
change
in
not
differential,
on
of
differenthis exact
variables
of
z.
the
function
U
U ----U0 +=Ut; U 0 and
U 1 are
integrals
of
equation
(i).
We
will
will
have
then
/29
have
dU --_
If we
return
to
the
old
dU
variables dU
therefore
results
integral
of
U is 252.
30
an
equations
integral We
could
dU
x,+...+
dU
_
Xt÷..+
(i). Ul = o,
and
we
_,
x..
that
dU Xt+
an
xi,
x,+ _
d_-q is
U|.
a_J
7i- = _ It
=
of cite
equations numerous
a_J
"
If
_'_ this
X,,
expression
is zero,
we
have
U = U0, (1). examples
of
this
type.
I shall
only
present
one
example.
Let
us
Let
&
Let
us
invariant
be
the
_'
Let
invariant
of
the
of
chan_e
in
the
the
first
order
quadratic
variables
having
form
according
the
form
_.
to
No.
237,
and
our
become
be
J be to
an
discriminant
make
will
Let
respect
consider
the
the
the
discriminant
Jacobian
x' 's.
of
or
We
will
the
the
quadratic
working
form
determinant
_'.
of
the
x's
with
B' 's and
the
C' 's)
have ._' _ _J_,
The
quantity
integral
of
Now
After
let
the
A'
will
equations an
be
(like
the
an
(i).
invariant
change
obviously
in
the
of
the
tn_-h order
variables
according
be
to
No.
237,
it
becomes
/3O
MJdr_dx;...dz,], and
MJ
must
I
may
be
an
conclude
integral
from
of
this
equations
(1).
that
A' 51_ji' i.e., A M_
must
be
an
integral
of
equations
(i).
31
Change
in
Variables
253. When the variables x i are changed in an arbitrary manner, without affecting the variable t which represents time, it is only necessary to apply the customary rules for the variable change for single or multiple definite integrals to the integral invariants. This is the procedure we have already followed several times. However, encountered. cannot lead Let
when the variable t is changed, greater It would even appear a priori that this to any result.
us
consider
the
system d_
l
dt = _ Let
us
introduce
a new
difficulty is transformation
d,r,
= x_
variable
dX n
......
(i)
x.,"
t 1 defined
by
the
relationship
dt dt I -- Z; Z is
the
given
System
function
(i) must
of
Xl,
x2_
...,
become dr,
dz',
dr,,,
dr, = ?._. - zx, ..... Let
us
coordinates
assume of
that
a certain
If the motion representing time,
of at
x n.
the
initial
point
values
M 0 in
(2)
1C"
space
0 x?,= x2, having
0
i
g
B
_
the
Xn represent
n dimensions.
this point is defined by equations (i), with the time t = _ this point will move to M.
t
On the other hand, if the motion is defined by equations (2), with ti representing time, at the time t I _ T the point M 0 will move to M'. Let different
us now consider points M 0.
If the motion equations (i), at
the
32
If the motion figure F 0 will
a
and the the time
figure
F 0 occupied
at
deformation of this t = T it will become
is defined by equations become a new figure _
(2), which
the
figure a new at is
time
zero
by
are defined figure F.
the time different
by
tI = x from F.
/31
Not
only
will
_
be
different
from
F, but
longer coincide with one of the positions is different from the time t = T. It
thus
appears
of the problem, duced from the invariants
of
Let
that
we
have
profoundly
us make
by
changed
and we must not expect that invariants of (I). However, order
in general
occupied
F
the
it will at
a time
given
no which
quantities
the invariants of (2) may this is what occurs for
be
de-
n.
the
change
in variables
of No.
237.
System
(i) will
become
at and
system
a_,
ay,
0
0
dr,
0
must
a_
0
(1')
I
(2) dt, = dy,
We
&n-,
then
assume
that
d.r._,
0
d_
(2 ' )
0
Z is expressed
as
functions
of
the
y's
and
of
Z.
Let
us
then
set
with integration being performed with assumed to be constants), and starting may
depend
on
System
the
have
Then
which
(2) becomes &'
dy,
0
will
y's are origin
y's.
dti
and
respect to z (the with an arbitrary
the
same
&__,
0
form
as
0
(2")
dz, I
(i').
/32
let M_Idzl...i_n,
be
an
changed
invariant according
of
the
order
to No.
n of
237,
equations
(i).
When
the
variables
are
it becomes
MJdy, dy,...ayn_,d_;
33
J is be a
the Jacobian function of And
of the x's the y's.
with
respect
to
the y's
and
z; MJ must
then
_ls&,&,...dy__,_, will
be
an
invariant
of
equations /MJdYtdy, T
(2"); dy__,dz
...
will
be
an
invariant
of
equations
(2'),
will
be
an
invartant
of
equations
(2).
General 253'.
Let
us
consider
and
finally
Remarks
a system
of
differential
equations
dxi = X_dt, and
their
variational
the
Let us assume first order
(1)
equations
that
equations
(i)
include
an
integral
invariant
of
E Ai dxF Expression On
with
Y,Ai_ i will the
E being
other
an
be
hand,
arbitrary
an
integral
these
equations
infinitesimal
Let z,. = _dt)
34
of
equations
(2).
(2) will
have
constant.
the
solution
/33
V-
be
an
arbitrary
solution
of
equations
(i).
If
c is
a very
small
con-
stant x_= ?_(t + _)=
will
still
be
a
solution
of _i=
will
be
a solution
It
thus
of
+ _ d-F
equations
(1),
?i(t+¢)--?i(t)=
equations
results
_(t)
and
dxi _ --dS-= _Xt
(2).
that 2A,,_i = s_ A_.X_
must
be
a constant.
Therefore, Let of
the
us now
second
EAiX i is an
integral
assume
equations
that
of
equations
(i)
include
(i). an
integral
invariant
order
f f
E Aik dxi dxk.
Then 2 hi,(i,_i--
will be
be
an
deduced Let
integral by
us
of
changing
equations the
"_k,o v,
(2)
of
equations
is permissible,
because
_i's
into
_=
_Xl,
and
(2'),
which
may
_'i "
set
with _ being a constant. solution of (2').
This
_
=
eX i is
a
Then "2,',;k( $,,Xk-- X_-$k) will
be
an
integral
of
(2).
This
shows
that
35
is
an
integral
This order
n
invariant
procedure -
i,
may
sometimes
may
be
Let
one
now
the
it
knows
first
order
possible an
illusory,
to
us
makes
when be
equal
of
to
invariant
because
the
of
equations
obtain
an
of
order
the
invarlant
(i).
invariant n.
which
is
of The thus
the
/34
procedure obtained
zero.
envisage
an
invariant
having
the
following
form
Z(Ai+tBl)dzt, where
Ai
having
and
this
Bi
are
form
functions
of
the
We
x's.
shall
encounter
invariants
below.
Then Z(Ai+tBl)_t
will
be
an
must
be
a
integral
of
equations
(2).
As
a
result,
E(A_+tBi)Xl
For
constant.
purposes
of
brevity,
let
us
'P = ZAiX_.; and
the
set
q', = $BiXI,
expression "b + tap t
must
be
a
constant,
which
entails d'P )-7 +
the
dq't t o-7 +
condition
,l,, = o,
or
a,
_x_xlXt +q',+
same
36
The Xi's , the Ai's holds true for
, and
the
(3)
tX, a,b
z.. _-x7 Xi = o.
Bi's
are
functions
of
the
x's.
The
+, The
identity
(3)
'h,
can
A__X,.,
only
_Xt.
occur
E
if we
also
have
identically
d, l,I Xl = o 2_i
and _d,l' xi. + 'h = o.
The
first
relationship
253".
shows
us
that
_I is
an
integral
of
equations
(I).
Let q_
--=
COIISI.
be an integral of equations (2). The function _ must be of a specific form, a whole and homogeneous polynomial with respect to the _i's, where the coefficients depend on the xi's in an arbitrary manner. Let
m be
the
degree
of
this
polynomial.
The
(where _' is nothing else than _, where the $i's differentials dx i) will be an integral invariant Under specific Let equations
this form us
assumption,
let
I be
an
arbitrary
expression
were replaced of equations invariant
by the (i).
of
the
237,
and
_.
make
the
(I) will
change
in variables
according
to No.
the
become dr; _ TIT -o,
d; _- =,,
and, if one employs n i and _ to designate the variational equations of (I') will be
(i')
the variations reduced to
of Yi
and
z,
d_, d_ dt -- _ --=o.
37
/35
With
these
new
variables,
_ will
have
the
specific
form
_0, which
is whol_
homogeneous, and has the degree m with respect to the ni's and _. The coefficients may be arbitrary functions of the Yi'S. However, according to the theory presented in No. 237, since we are dealing with
an
integral
The xi's relationships
invariant,
these
cannot
are functions of the y's and of z, and between the variations may be deduced
_i _ X
The_'sare
coefficients
therefore
linear
clz¢
functions
One then passes from the (4), whose determinant
Let I of _.
I 0 be the invariant We will have
of
form _ to is J. of
the
on
z.
following
(4)
d_ i
the
n's
and
terminant of these linear equations (4) is nothing bian of the x's with respect to y and to z. I have J.
tution
depend
_0,
which
the
form
of
_, and
the
de-
else than the Jacocalled the Jacobian
_0 by
corresponds
linear
to
the
substi-
invariant
I = loJP with
p being
the
degree
of
the
invariant.
However, I 0 is a function of the coefficients of _0 and, consequently, a function of the y's, which is independent of z. It is therefore an integral of equations (i). Let we have
and
M be
the
last
multiplier
of
an
(i).
38
(i),
in
such
a way
that
that 31 dx t dz, ...
is
equations
integral
invariant
We have seen Therefore,
in No.
of 252
the
order
that
dxn n.
MJ will
be
an
integral
of
equations
/36
Iu(MJ)p-=
I_Ip
will be an integral of equations (i). therefore corresponds to each invariant Now
let
C be
respect to the variables _. If
C O is
a covariant
coefficients
the
of of
the
_,
corresponding
An integral of these of the form _.
form
and
the
covariant
_, having degree
of
the
q with
_0, we
equations
degree
p with
respect
will
to
the
have
C =-- CoJP.
The and they those of
coefficients are therefore
Therefore,
is an
C_[p is
integral
C, where We
the
of C O are independent
an
integral
invarlant _i's
therefore
functions of z.
of
of
been
have
a method
replaced of
invariants. the so-called
The particular case absolute invariants
attention.
If
will form
be an a new The
order.
integral integral
same For
C,
for
example,
example,
may
(i), by
(2);
true
of for
therefore,
where
C'
is none
other
than
forming
an
applied
/37
dx i. a great
number
of
integral
absolute
covariant
to
integral
of
of
of equations (i). One may therefore without knowing the last multiplier be
_0,
in which p is zero (i.e., the case or covariants) merits particular is
invariant invariant
procedure
equations
equations
have
of the coefficients The same holds
invariants
M.
of higher
let EA,-_dxid_rk
be
an
integral
invariant
of
the
second
order.
The
bilinear
form
39
which is integral
an integral invariant.
of
equations
(2)
and
(2')
is connected
with
Every invariant or covariant of this form, multiplied by appropriate power of M, will be an integral of equations (2), and will consequently produce a new integral invariant.
this
one (2')
In the same way, if one has a system of integral invariants, a system of forms which are similar to _ may be deduced from it, which will be integrals of equations (2), (2'). An integral of equations (i) will correspond to every invariant of this system of forms. An integral invariant of equations (i) will correspond to every covariant of this system of forms. For
example,
let
F and
F 1 be
two
quadratic
forms
with
respect
the _'s. They become F' and _i when the _i's are replaced ferentials dx i. Let us assume that F and F I are integrals that, consequently,
are
integral Let
us
invariants consider
the
of
by of
to
the dif(2) and
(i).
form
/38 F--_F,
where % is an unknown. When stating that the discriminant of this form is zero, we shall obtain an algebraic equation of degree n in %, for which the n roots will obviously be absolute invariants of the system of forms F, F I. These will therefore be integrals of equations (i).
and
However, this is not all. F and F I can be written in
Let %1, the form
12,
''',
In be
these
roots,
F ----l,h_+itA|+...+XnA_, F,= AT+ A|+...+ At, with AI, A2, ..., A n being the purely algebraic operations. The
40
quantities
AI,
A2,
linear
...,
forms
A n may
be
which
regarded
may
as
be
determined
the
covariants
by
of zero degree of the F, FI system, so that
are the integral invariants of equations (i), if A_ designates what Ai becomeswhen the _i's are replaced by the differentials dxi. However, there would be an exception if the equation for X had multiple roots. For example, if XI were equal to %2, it could no longer be stated that
are integral
invariants,
is an integral
but only that
invariant.
Nowlet f
be
two
integral
EAikdxidxh,
invariants
f
of
ZBikdxidxk
the
second
The
order.
two
bilinear
/39
forms
¢,, = z B,k(_A_,-- _*-_') will
let
be
The most n = 2m. Let
and
integrals
let
equation
us
(2) and
interesting
consider
us make for
of
its
X of
the
(2').
case
is
in which
n
is even;
therefore,
form
determinant
degree
that
n =
equal 2m.
to
However,
0.
We the
shall first
have term
an
algebraic
in this
41
equation tion of
is a perfect order m. The
will
integrals
be Now
_ and
square, m roots
of
_I
so
equations
can
that
(i),
for
the
in
the
form
be written
,I, = Z
it may
ti(PiQ_--
+, = 2(P,Q;-
the
be covariants (2') to which
There would multiple roots. If we
to
an equa-
reason
as
above.
QiP_) polynomials polynomials,
with respect where the
to the _m'S
expressions I',Q_--QIP_,
wii1 (2),
same
reduced
Q,P;)
and the Pi's and the Qi's are 2m linear _'s. The P_'s and the Q_'sare the same have been replaced by the _''s . Then
be
of the be
have,
the system integral
an
for
P2Q_--Q_P_
....
,
PmQ_-Q,_P_
0, 01, invariants
exception
to
and
this
consequently will correspond.
if
the
equation
integrals
for
of
X had
example, it = t,
it
could
no
longer
be
stated P,Q;-
are
integrals
of
(2),
(2'),
that
the
PiQ,,
P,Q;-
but
only
two
that
P, Qi -- Pi Q, + P,Q; is
42
an
integral
of
(2),
(2').
expressions
P;Q, their P;Q,
sum
/4O
CHAPTER XXIII. FORMATION OF INVARIANTS
Use 254. readily
There
when
is
the
an
last
of
the
integral
Last
Multiplier
invariant
multiplier
of
the
which
may
differential
be
formed
very
equations
is
/41 known.
Let dxl X,
be
our
differential
Let that
we
us
that
have,
there
function It may
M
is
then
dt,
(1)
is a function
M
of Xl,
X2_
...,
order
n
x n,
so
identically d(MX,) dxi
This
dx_ _-=
equations.
assume
also
dx2 =-XV .....
be
d(_X,) dx2
called
the
stated
d(MX.) dxn
last
that
_0.
multiplier.
the
integral
of
the
J =f_ld_,dx_...dxn is an
integral
integrated; integration
the
integral
invariant.
expressing constants
J will
Let
x I, x2,
assume
...,
xn
that as
equations
functions
of
(i)
have
t and
of
n
been
of
the
become J =fM .I
The quantity with respect
us
Adatd_... d_,,
A is the Jacobian or to the _'s. We will
the functional then have
determinant
x's /42
dJ f dM dt =.] --d[- d_l d_2 "" d_"
43
However, dM t_ = M d._ + ', dM d--7-_5 -_; dM
On
the
other
_ "--
hand,
I may only write the first deduced from it by changing Therefore,
A +
dt
dA _
dM
dxt d.v, -_=1' c-_t''"'
line _i
of this to _2,
must
be
xi-+dt_ with
respect
to
the
a's.
the xi's with respect to x i + Xidt's with respect
This
dx,! . d=t
determinant; _3, .-., an.
the
Jacobian
of
the
others
may
be
the
=x,.+Xidt will
be
the
product
of
the
the _'s -- i.e., of _ and the to the xi' % which I shall call
Jacobian
of
Jacobian of the D. I may write
dA A+dt-_=a.D. However, the Jacobian D may be readily formed. The elements principal diagonal are finite, and that belonging to the _ to the _ column may be written
The other elements line and to the _
are infinitely small; that belonging column (i < > k) may be written
to
of the line and
the
dXt d;ck
will
It thus have
results
that,
neglecting
"
terms
on
the
order
of dt 2, we
/
dXi D = r+ dt'_ z.a ,' from
44
which
it follows
that
/43
d-_
dX_
_ One
may
conclude
which
we
,T_,"
that
dM dt
from
- _
dM
finally
dX,.
E
d(MX_)
have
dJ dt
_0,
q.e.d.
Equations 255.
In
the
gral invariants learned how to equations, gral
and
in
invariants The
case
may form
the
integral
preceding
integral
is
of
the
I obtained
second
which a
will
second
dynamics,
we
3,
Vol.
order
integral and
which
order
may
and
integral
be
A
be
written
number
56 the how
of
inte-
on, we variational to
deduce
inte-
d_+_Z,+...+
d_.r.).
of
the
little
which
I may
as
follows
it
greatest farther
I may
deduce
167)
is
as
follows
const. from
which
j,=fz
p.
deduced
is
follow.
invariant may
great
learned
I,
_i _, + _ _, - _i _, + ....
is
as
follows
Importance on
(still
it
is
for p.
the
167,
Vol.
I),
write
=
The
a
From Sections integrals of
chapter
(equations
invariant
statements
of
formed. number of
J,=fcd_,dz,+ It
Dynamics
them.
_ _,The
equations
be readily a certain
from
first
of
of
CONSt.
from
of
the
fourth
dxtdyidxkdyk.
45
The
summation
combinations In
where
of
the
the
indicated the
same
by
the
indices
way,
summation
the
is
sign
i and
7
may
be
extended
over
the
n(n
- i) 2
/44
k.
integral
extended
over
n(n
-
i)
(n - 2)
combinations
of
6 the
three
indices
i,
k and
i, will
still
be
an
invariant,
and
so
on.
We thus obtain n integral invariants if we have n pairs of conjugate variables. One of these invariants Jl will be of the second order; another J2 will be of the fourth order; another J3 will be of the sixth order,
...,
and
However, different. the second an
it
last
is not
Jn will
be
necessary
of
to
the
assume
order that
2n. these
invariants
are
At the end of No. 247, I stated that from an invariant order, one can always deduce an invariant of the fourth
invariant
Jn which from the
the
of
the
sixth
order,
I have just defined first of them.
These
invariants
may
be
and
are
so
none
on.
The
other
than
considered
in
invariants those
another
which
way.
At
Jl,
J2,
may
the
be
all
of order, --., deduced
beginning
of page 169, Volume I, I demonstrated the manner in which one could deduce the Poisson theorem from the integral (3) on page 157, or -- which amounts
to
the
same
thing
--
from
the
Following the same procedure with a theorem similar to that of Poisson.
integral
the
invariant
invariant
J2,
Let
be
four
integrals
of
the
equations
of
dynamics.
Let
be
the
Jacobian
of
these
four
integrals
'z'i,
46
yl,
Xk,
with
yk.
respect
to
J1-
one
would
obtain
The
expression Y'-,_Ik,
where i,k,
the will A
the
summation still
similar
invariants
is
be
an
extended
over
all
combinations
of
the
indices
integral.
theorem
would
J3,
"'',
Jq,
be
obtained
by
commencing
with
any
of
Jn"
of
However, according to the statements which I have just made, the theorems is different from that of Poisson in reality.
be
However, attributed
It It
could be is known
from among all to the last of
obtained that the
of these them
invariants,
great
importance
by the procedure given in the preceding equations of dynamics have unity as the
none
may
section. last
multiplier. 256. coordinates on
page On
I shall now assume that the x's designate of n points in space, and I shall employ
169 page
of Vol. 170,
the the
rectangular notation given
i.
we
obtained
the
following
integral
of
the
variational
equations NfyT_
The
corresponding
,_7 dV
integral
invariant
\', [ v ,It
In
the
same
way,
the
to the
written
dV dx).
+...
+ &,,,)
integral
_'r_l
The
be
invariant
( &,, + &,, corresponds
may
i --_ const.
invariant
_ (xli
d)" _.i --.)"
li dx2i
--
x2i
rlyli
+
y2i
dxli)
47
_J
corresponds
to
the
integral z,(x,i_,,i-)q,_i--
However,
none
of
these
invariants
may be immediately deduced of gravity, and area. This does not hold function V is homogeneous In No. variational
_,,i_,. + y._,_,,-) -- const.
from
is the
of
56, we learned equations have
removing
the
It may the same
indices,
be stated procedure
we
mi
which
we
invariant
obtain
The
which
second
the
has
integral
48
an
the
degree
-i,
a very
integral
the
-4- const.,
E #' \ m
dx
_
be
order
invariant
special
dxdVdx) '
nature
since
it
depends
on
time.
be
readily
written
\ :znz
of
of
+c°nst'
--#Ydx)-+(P--2)t/EO_Y
may
/4_k
dV _) + consl. d,r
\ m
integral
f and is t_erefore seen that
_ki
when
more generally that if V is homogeneous leads to the following integral
J--/E(2xd)
an
dxhi
of
they
center
have
E(_txvj--py_)=(a--p)t
from
since
energy,
that if V is homogeneous the integral
E(2x,,+y_)=3tZ(y__
p,
interest, of
for the following, which occurs with respect to the x's.
\
or,
great
integrals
an
._,,),
exact
differential.
It may
is
none
other
The gral the
the
invariant
taken values
arc
chapter. as
As
J
energy
is
of
constant,
the
first
is
the
figure
When
this
I
explained
a result,
which
figure
is
in
the
we
The
we
therefore
of
a
to
time.
not
curve
is
integral
single
_" --py
is
value
This it
in
to
the
this
if
figure
is is
one
if
inte-
C1 be arc.
F 0
in
the
preceding
become
F,
C O
and
C 1 do
not
4_7
Co).
the
two
is
reduced
variations
ends
of
to are
the
arc
an
arc
proportional
correspond
true
if a
the
arc
relative
the
curve
invariant,
of
as
is I
desig-
chapter. that
the
may
the be
integral.
constant
arc
of
the
added
under
For
example,
curve the we
f
is
closed,
sign
without
may
an
add
_),
coefficient.
integrals f
relative
(which these
constant.
also
assumes
of
dx)
F
the
therefore
differential
value
arbitrary
Thus,
also
an
Let C O and ends of this
_)t(CI--
.,y
_(zdy+y
are
therefore
called
dx)-}-(p--
_._..e,:_....
energy
preceding
exact
changing
the
integral
However, arbitrary
an
C.
chapter.
however,
constant,
for
the
when
deformed;
particular,
closed.
with
is
call
have
constant
here)
The
nated
have
shall
integral
is
In
it
curve. the two
preceding
.f:,:
a
I
order;
deformed
J =fE(2x ,/
to
which
along an arc of an arbitrary for the energy constant at
This
change,
than
Z y dx,
fear
dy
invariants.
49
We may The
saw
in No.
always be invariant
238
deduced of the
that
an
absolute
invariant
from a relative invariant second order which is thus
of
the
second
order
of the first order. obtained is none other
than Jr_=/V dx dy, which
we
studied
This
is
above.
the
case
in which
the
expression
which appears under the f sign, becomes an exact differential. This is the case in which p = -2, which would occur if the attraction, instead of following Newton's law, follov_ed the inverse of the cube of the distance. We then have X(_x_ The
quantity
respect
Z2xy
to
is
time,
and
therefore
--2ydx)=
E2xy.
a polynomial dx
Z2xy=
the to
expression time.
Zmx 2
Jacobi
reached
However,
in
of
the
first
degree
with
since
is
a polynomial
this
result
d = _ Emx_,
E_mx-_
of
at
the
the
second
beginning
degree
of
with
his
respect
Vorlesungen.*
general, X(2xd:--pydx)
is not In the
an
exact
the
differential.
special
following
case
of Newtonian
fX(2xdy+y
Integral 257. invariants *
50
attraction,
Invariants
dx)--3t(C,--
and
invariant
takes
Note:
English
Co).
Characteristic
It may be asked whether there in addition to those which we
Translator's
our
form
title
is
are other have just Lectures.
Exponents algebraic formed.
integral
/48
Either Chapters
IV
invariants and
the
the
method
and
V,
to
least
with
Let
may
respect
us
set
an
method
As
be
we
which
have
of
the
applied
motion
more
to
the
integrals
of
be
or
employed. the
could
equations it
Bruns,
be to
procedures the
However, at
may
correspond same
applied
of
to
I
seen,
employed the
integral
variational these
in
equations,
equations
as
are
themselves.
advantageous
to
modify
these
procedures,
form.
arbitrary
system
of
differential
equations
d'_ - Xt, dt
and
their
variational
/49
equations (2)
d._ dXl dt ----'X
Let order
us
first
having
try
the
to
determine
which
the
expression
differentials
dx,
These
_k.
the
integral
invariants
of
first
form (BL dzt
in
(1)
and
invariants
+ B, dx, +...+
under where
the the
sign
B's
correspond
B_ dz.),
to
f
is
(3)
linear
are
algebraic
the
linear
with
respect
functions
integrals
to
of
of
the
the
x's.
equations
(2). What which the
are
are
a
us
assume
periodic
will
be
period the
conditions with
under
respect
to
which the
equations
_'s
and
(2)
algebraic
have
integrals
with
respect
to
x's? Let
to
the
linear
the T.
that
solution known One
following
values of
functions
can
then
are
period of
derive
assigned
T. t,
The
which
the
to
the
x's
which
coefficients will
general
be
of
periodic
solution
of
correspond
equations and
have
equations
(2) the (2)
form
_ = X_Ak_,'+_,_ .
The
_i,k'S
ponents, We
are and
can
in
periodic the
then
Ak'S solve
functions are the
integration linear
of
t,
the
(4)
_k'S
are
characteristic
ex-
constants. equations
(4)
with
respect
to
the
51
unknowns
Ake_k t, and
we
will
obtain Ake _d =
are
periodic
functions
El_tBt,k
The
0i,k'S
of
the
There will therefore be n relationships _'s, and there will be no others.
(5)
t. of
the
form
(5) between
If equations (i) and (2) include q different integrals which are linear with respect to the _'s and algebraic with respect to the x's, some of these q integrals may cease to be different when the x's are replaced by of equations What
the values (i).
may
then
be
corresponding
to
one
of
the
periodic
solutions
done?
/5O
Let
1Ii=
be of
these q linear the x's, which
Bli_,-_-.B_i_t-+...+Bni_=
eonst.
(i=r,2,...,q)
integrals, where the B's will will correspond to q integral
be algebraic functions invariants of the form
(3). They are different between them having the
-- i.e., following
there form
are
no
identical
relationships
_, IT,+ O,IT,+. + 3_"_ = o, where
the
coefficients
8 are
constants.
Neither
(6) does
the
following
form
Occur
+, lit +
with
the Is
having
_'s
it the
being
integrals
then possible that following form
'_t lit +...
of
+ +y llq =
equations
there
may
?t lit-i-?Ill2 +...+
(6')
o,
(i). be
a relationship
?yII V_-=_ o,
between
them
(6")
with the @'s being arbitrary functions of the x's alone. According No. 250, if the same relationships hold, the ratios of the functions must be the integrals of equations (I).
52
to
Wewill
therefore have ?, ",
with
the
_'s
being
integrals,
'?_. +2 and
_ ?_ _+I
consequently
,.., If, -I- "+++,_ TI2+. •. -i- '5,I IT,,,_ o, which
is An
between
contrary
to
identity the
our
hypothesis.
relationship
of
the
form
(6")
cannot
therefore
exist
Hi's.
However, if the values corresponding to one special solution, whether it is periodic or not, are assigned to the x's, it could happen that the first term of (6) vanishes identically. It could happen even if equation (6) which is not identically satisfied whatever may be the /51 )
x's, would hold when functions of t, that special
shall
the x's are replaced by the appropriately is, by those functions which correspond
to
chosen a
solution.
Every special designate as Under The
this case
solution under which a singular solution.
assumption,
in which
the
two
cases
this
may
phenomenon
be
periodic
solutions
in which
they
not
consider
a singular
is
produced,
I
presented. of
equations
(I)
are
all
singular; Or,
the
case
258.
Let
which
it
us
are
_,Bk.,+ from
singular.
solution
_,B_., +...+
$.
Let
us
set
_qBk.q = B+,
follows B,_+
Since
all
relationship
(6) was
B,_, +...+
not
B,,_,, = _,I;_+
identically
_,H, +...+
verified,
_qHq.
we
do not
have
iden-
tically
R,= D, .....
B.=o.
(7)
However, since relationship (6) must be verified by the solution S, these relationships (7) (which are algebraic, according to our hypotheses) must be satisfied for the values of the x's which correspond to the
53
solution S. Nowlet us set dBl
_= x, _
and thus
dBl
dBt
+x, 7_, +'"+
x_ _-_x.'
dB_
aB:.+
B_=Xt-_t+X2dx
The
solution
S must
_
obviously
• • "+
Xn
satisfy
dB}
d--_x_ .....
the
relationships
(7') and
the
and
so
B_=o
(i=,,=
....
,n),
B_=o
(i= ,,_, ...,n)
relationships (7")
on.
We shall therefore successively form the relationships (7), (7'), (7"), etc., and we shall stop when we have arrived at a system of relationships which will only be the result of those which will have been previously formed. Relationships (7), (7'), (7"), etc., will be algebraic to our hypotheses, and all of them together will form what called in No. ii a system of invariant relationships. Therefore, if a system of differential periodic solution, it will permit a system tionships. It
is probable
that
the
three-body
braic invariant relationships except am still not able to prove this. Under solutions.
this For
equations permits a singular, of algebraic invariant rela-
problem
those
permits
which
assumption, let us assume that each of them, we must have J3,B,.., + _¢ B,'.t +...+
according I have
we
are
no
other
already
have
several
_q B,-.q = o.
alge-
known.
I
singular
(8)
Only the constants B will not be the same for two different singular solutions. It is therefore not apparent that these two singular solutions must satisfy one and the same system of invariant relationships.
54
However,
this
is what
takes
place,
as we
shall
prove.
/52
In order to formulate our ideas, let us assumethat q = 4; the line of reasoning would be the samein the case of q > 4. Let us consider the n relationships ° Let
us
form
determinants be zero. If which clude
this
must only
Table by means
is not
the
be satisfied the x's and
the
If they are relationships
The
M's We
the
formed
are will
of
therefore
(i
,,_,
..
T of
the
4 n coefficients
of
the
four
case,
we
shall
columns
obtain
B.
in
one
are
the
equal we may
If all zero,
three 4
let
us
first
order
of
Table
or more
relationships
M, III+ M211t÷ M3H_+
M_H_=
T.
o.
(18)
of the assumes
coefficient of Sk to be identically
form that
(6"), which is all of the M's
of
the
form
minors the
of
minors
the of
first the
order
second
4 for
_f2 , 3 and
4 for
of Table
T are
identically
one. minors obtained by taking by cancelling the lines 1 and
143.
become 51'_ I11-I_I_Hz -I-_I_Hs = o.
This relationship the first member assumed
of
zero.
for M' I , 2 and
have
the must
have
Let M' I, M'2 , _f3 be three of these arbitrary columns in the table and
It will
of
table
to zero, let us consider three deduce the following from them
shall therefore have a relationship to our hypothesis, unless one only
identically
All
this
This relationship (18) must be identical, because the is one of the determinants of Table T, which I assume zero. We opposed
(17)
n).
by all the singular solutions, which will inwhich will not include the indeterminate B !s.
identically (17), and
minors
---
must be identical, because is one of the minors of the
to be
identically
(19) the coefficient first order of
of Sk in T which I
zero.
55
This would still only assumesthat all cally zero.
be a relationship of the form (6"), unless one of the minors of the second order _ are identi-
If this is the case, it will Bi, IIt-which
is
still
a relationship
becomeidentically
BaHt=o, of
the
form
(6").
It may therefore only be the case that all of the determinants of Table T vanish identically. We shall therefore have at least one relationship (and, consequently, a system of invariant relationships) which must be satisfied by all the singular solutions of equations (i).
tions
It may be immediately concluded (i) cannot be singular.
But this solutions. We
have
is not
Just
all;
defined
we may
the
expand
singular
integrals H i of equations (2) which and which correspond to q invariants of equations (I). In lar
the
solutions
same
way,
with
we
may
respect
equations the _''s.
with
(2) and
These integrals respect to the
of
our
of
the
a definite
I"12, ..., (2')
solutions
definition
solutions
q arbitrary
equations
all
with
are linear with (linear and of
provide
to I[I,
of by
that
of
of
equa-
singular
respect
to
/54
q
respect to the _'s the first order)
definition
of
the
singu-
integrals
Iiq obtained
must be homogeneous _'s and with respect
and of to the
by
replacing
the
the same order, _"s. They will
_'s
both be
whole polynomials with respect to these variables, but they will not be necessarily linear with respect to the _'s. They may therefore correspond to integral invariants of a higher order, or to integral invariants of the first order, but which are not linear.
not
In addition, these satisfy identically
I may then tionship (6) is solution.
56
integrals must a relationship
be of
different the form
-- i.e., (6), (6')
they must or (6").
state that a special solution S is singular if satisfied for the values of x which correspond
a relato this
We
shall
then
have lli:
The
quantity
Ak
of factors _, and the Bk. i s We
shall
is
a monomial
_2, are
..., _n' algebraic
first
set,
formed
no
shall
changes
Every fies
one
at
the
as
was
will one
The
and
role
products
all
periodic
this
would
In
is
me
reasoning
to
suitable
power,
pursued
above.
one
the
q
integrals
H i
invariant
relationships.
envisages
the
Bn+|.il_l+
We
satis-
integrals
in
the
as _,
to
only _i's
delve
be
may
products the
of
which
either the
reasons
singular
will
without
--
be
in
any will
the
form
for
solutions
/55
these
relationships.
repeated, Bk. i
to
solutions
invariant
--
into
respect
singular
quantities
the
be
with
these
algebraic
the or
B_+,.it$_+...+B2,,.it_n.
solutions, and
of
the
integrals
values
I
the
which
--
in
(i) to will
corresponding
afield
that the
(2)
include
still to
_'s be
that
and
be they
and
the
my
play
$i's,
t$ i.
my
belief
the
subject;
it
is
that
case
of
may
the
singular. equations
when
a We in
the
periodic
respect x's
to
are
all
system to
of
the
shall No.
integrals
this
that
according
of
with
return that
unlikely
satisfy --
q different
different
very
be
algebraic
shall
assume
problem
necessary
numbering
a non-singular
I
provisionally
three-body
would
order
and
from
shall
observing
of
notation
respect
far
only
solutions
equations with
too
meantime,
correct,
section
employ
q
of
the
preceding
the
if
need
of
cannot
take
relationships,
these
system
wish
invariant
If
of
respect
same,
proof
and
not
periodic
linear
number
_qBiq,
algebraic
singular
above
solutions
proposition
again
line
valid
the
same
_i
later.
the
a
certain
problem.
This
of
in
I do
three-body
of
the be
the
of
259.
this
of still
presented
the
to
a
above,
B2.t_l_...+Bn.i_a+
coefficients
same
to
Bt.i_l@
The
the
the
with
still
proof
changes.
done
in
system
definition
satisfy
of
form Iii=
integrals,
product
conclusion.
solution
are
the
_iBi.1+ _Bi.2+...+
made
same
same
results
following
The
be
the
singular and
These the
need
arrive
by
E_' $_' "''' _n raised functions of the x's.
Bi: and
Z B_,/A_.,
257.
which
are
the
x's,
replaced
by
solution.
57
By stating that these q integrals are constants, and by replacing the x's by the values corresponding to a periodic solution in the equations thus obtained, one will obtain q equations of form (5), but where the exponent _k will be zero. These q equations must therefore be included among equations (5). Therefore, in order that equations (i) include q different respect to the x's_ it ponents _k be zero for Let
us
now
try
to
integral invariants which are linear with is necessary that q of the characteristic exevery non-singular periodic solution. determine
7_ZAidx,+
the
integral
2B,_dz,_lz,=
These invariants will correspond to (2) which are quadratic with respect F($i)=
f
invariants
of
the
form
(_.
(7)
the integrals to the _'s.
of equations The integral
(i) and
const.
will correspond to the invariant (7); this integral must be quadratic with respect to the _'s and algebraic with respect to the x's. In this equation, let us replace the x's by the values corresponding to a nonsingular periodic solution. We shall have
V*(_,)-_ ¢o._t., where F* is a quadratic polynomial to the _'s, whose coefficients are
(8)
which is homogeneous periodic functions
It must be possible to deduce all equations (5) in the following manner.
equations
of
with of t.
the
form
respect
(8)
from
When dealing with a problem of dynamics -- in particular, in the case of the three-body problem -- we have seen that the characteristic exponents are pairwise equal and have the opposite sign. We can therefore group equations (5) by pairs. Let us set
When multiplying equations (5') and (5") by each other, we an equation of the form (8), and all equations of the form linear
combinations If we
therefore
of
the
assume
equations that
thus
equations
obtained. (i) have
of the equations of dynamics, and that they contain gate variables, we shall have p pairs of equations and (5"). Consequently, for each periodic solution,
58
will obtain (8) must be
the
canonical
form
p pairs of conjusimilar to (5') we shall have p
/56
equations
of
Let
us
the choose
combinations for all of the other nomial to the only
form
(8) which
one
are
equation
instance, periodic
linearly
from
these
defined
for
values
We
now
determine
a way that or even of problem,
the the
Let
us
of x which
attempting
now
try
i.e.,
those
having
where
F is
a linear
to the
equations
and
the
F* of
to
determine form
linear
function
at
the
the
be
present
invariants
a double
of
with respect which are
a periodic may
for poly-
made
/57
solution. in
such
algebraic functions of the x's, x's. I shall simply pose this it
the
of
selection
are the
solve
to
of
time.
the
second
order
--
integral
products
dxidx k
(the
of this linear function are naturally functions of the invariants of the second order will have the following
retain
their
of the second degree functions of the x's
correspond
whether
coefficients of known functions
without
p
F*($i). Let us follow the same procedure solutions. We shall then have a certain
F*($ i) which is homogeneous and _'s, whose coefficients will be
must
independent.
coefficients
x's). These significance.
Let us select equations (i) and (2) once again (we the numbering given in No. 257), and let us form
shall always in addition the
equations d?,.
They will may write
lead us to equations as follows
_,_ dXi
which
(2a)
_i.
are
similar
to
(5),
and
Ai.e_ t = z _)Oi_. only
order
According to the preceding chapter, the invariants will then correspond to those of the integrals of
and
which
algebraic Let
are
from
linear
with
equations
with
respect
(5) because
respect
to
to
the
x's.
F($_$_--
$k$_)
I
(5a)
They
(2a),
differ
which
the
the
letters
are
accented.
of the second (i), (2) and
determinants
59
be
one
of
these
integrals.
If
corresponding to a periodic having the form
the
F* will
and whose We the
In
now
(9), the
divided
a linear
coefficients
have
form
be one let us
be
into
like
us choose obtained.
periodic
solutions.
are only solution.
now
these coefficients of the x's. Let According
us now to No.
of equations nents become If,
6O
for
the
manner
of
respect
in which
return 29,
equations
(5a)
(5).
be
of
formed.
may
be
Let
B;,.e--=','= .X_,_0h,
(5a")
then
of
the form
the
and of
have
linear of
example,
these
of
all equations procedure for
of the form all other
a relationship
of the determinants. The be functions of the x's which
x's
corresponding
to a periodic
the
selection
be
or
two
exponents
made
so
that
the
known
functions
of
the
first
order.
(4),
or more
may
even
invariavts
equations
when
by (5"), (5a") by (5'), and having the form (9). Each
all other equations of the form those which thus may be formed.
functions
(5), is modified equal. nine
relationships
dynamics,
whether
to
all
t.
may
algebraic
the
of
determinants
(5a')
values
are
the
A;,.._,,' = z_,;0.,,
shall
determine
values
equation
solution,
one equation from among Let us follow the same
for
the
an
periodic
equations
We
to
functions
term will be a linear function of this linear function will
defined
We must
periodic
will give us one, linear combinations
Let (9) thus
by
obtain
(9)
pairs. Let us multiply (5a') We shall obtain an equation
pair of equations (9) will only be
whose first coefficients
with
to a given
of equations
pairs
of these subtract.
be
determined
case
replaced
will
6_6',) = const.,
function
will
relative
are we
/58 F'(_iLI.--
where
x's
solution,
and
consequently
characteristic
equal
zero,
we
that expo-
may
write
the corresponding equations (5) in the following
/59
form (i0)
P_.-= X,'_,O,h-, The quantity Pk designates constants for coefficients.
a whole
polynomial
with
respect
to
t, having
These polynomials are of the degree q - 1 at most. In order to specify this more precisely, the number of polynomials is q. The first may be reduced to a constant, the second is of degree one at most, the third is of degree two at most, and so on, and finally the last is of degree q - 1 at most. In
the
its maximum a derivative one, and so
case and of on.
in
which
In every case, the groups. In each group, stant, and each of them In
order
the
degree
of
this
last
polynomial
is equal to q - i, the polynomial before the last, the q - 2nd one the derivative
that
q polynomials may be the first polynomial is the derivative of
there
may
be
p linear
reaches
the last is of the q - ist
divided into several may be reduced to a conthe following.
integral
invariants,
it
is
not sufficient that p of the characteristic exponents are zero. It necessary that p of the polynomials Pk be reduced to constants (or, which is the same thing, that these polynomials be at least divided into
p
groups).
From cance
is
of In
the
point
equations No.
important.
216
(i0)
we
This
of view
of
where
defined
study,
Pk may
an
invariant
our
not
integral
has
the
what be
is
then
reduced
invariant
to
the
signifi-
a constant?
whose
role
with dx.
respect
is
very
form
t."+ tf[:,. where F and F I are functions x's, and linear with respect A having
similar the
invariant
following
which are algebraic to the differentials
corresponds
to
an
integral
to
of equations
the
(2)
form
where F and F I are functions x's, and linear with respect
which are algebraic to the _'s.
with
respect
to
the
61
pond
In to
this integral, if I replace the x's a periodic solution, we shall have F'.
where whose
the
values
now determined (ii) starting
Let us consider two constant, and the second derivative of the second. written
the
0i's
and
the
which
corres/60
const.,
(ii)
F* and F_ are functions which are linear with coefficients are periodic functions of t.
We have tionships of
where
tF_=
by
the manner in which with equations (i0).
we
respect
may
to the
obtain
all
_'s,
rela-
polynomials Pk, the first being reduced to a being of the first degree; the first is the The corresponding equations (I0) may be
,_,= z$,ot,
(I0')
A, + A, t= z_iO_,
(i0")
e'i's are
periodic
in
t.
We
may
thus
deduce
Z$iO_--tZ_,Oi = const., which
is
a relationship
of
the
form
(ii).
We should note that equation (i0'), raised to the square, provides us with a relationship of form (8), and that a relationship of form (9) may be deduced from equations (i0') and (I0"), that is, (E6_O.I)(E$}O'_)--(E$}Oi)(E$iO'_') 260. Let us apply this let us determine what may be of the several types studied That is:
tials
The first dx;
type:
linear
= const.
procedure to the three-body problem, and the maximum number of integral invariants, in the preceding section, for this problem.
invariants
with
respect
to
the
differen-
The second type: invariants where the function under the sign f is the square root of a second-degree polynomial with respect to the differentials of the x's; The with
62
third
respect
type:
invariants
to products
of
the
of
the
second
differentials
order, dxidXk;
which
are
linear
The fourth type: invariants having the form considered at the end of the preceding section -- i.e., having the form
These of
different
integrals
of
The
first
The
second
The
third
_i _ 'k _
_k _' i
The
types
equations type:
of
linear
type:
invariants
(2) and
integrals
quadratic linear
type:
correspond
(2a),
that with
integrals
with
with
different
to
_'s;
types
is: respect
integrals
to
/6__!
respect
respect
to
to the
the
_'s;
determinants
• '
fourth
integrals
type:
having
the
form
F -F-tF,, where
F and We
may
periodic
F 1 are
linear
assume
that
solutions
of
with it
the
respect
to
is extremely three-body
the
probable
problem
In the three-body problem, the number number of characteristic exponents is
the
_'s.
is
that
none
of
the
singular.
of degrees of freedom twelve. According to
is the
six;
ideas presented in No. 78, there are six, and six alone, which vanish; the six others are equal pairwise, and have the opposite sign. There are therefore six equations of form (I0) and six polynomials P., of which four are of degree zero and two are of degree one. Or, _ere are three pairs of equations tions having the form (i0'), Let each
type
us
therefore
there
will
having the form (5'), (5"), four equaand two equations having the form (i0").
determine
how
many
independent
I shall state more precisely what I mean. variants of the first type as independent
./'lq, or
n
invariants
of
the
second f_t,',:
or n
invariants
of
invariants
of
be.
the
third
.f,::
....
I do not
regard
n
in-
, .fF.,
type f,/t+'_:
...,
7f']_
type
63
or n invariants
of the fourth type fF,,
when there the form
is an
/62
fF=, ...,
identical
(Ft
relationship
-{:}+tF_),
between
'PlFI-_ 'P2F2-_"._.- -*-'PnFn
where
¢I,
¢2,
"'',
_n are
integrals
It is apparent that we cannot the first type, i.e., no more than already known.
FI,
F2,
...,
Fn having
o,
of equations have more the number
(11.
than four invariants of equations (I0')
of
We cannot have more than thirteen invarlants of the second type, of which three will come from the three pairs of equations having the form (5') and (5"I, and the six others will be obtained by means of the squares of the four equations (i0') and of their products by pairs. These last ten exist in actuality. However, they are not independent of the four invariants of the first type, since they may be deduced by the procedure variants.
which (5')
given
in No.
245.
We
may
therefore
have
three
We cannot have more than eleven Invariants of the three will come from the three pairs of equations (5"I. Six will be obtained by combining the four
by pairs; two will be with the corresponding
obtained equation
by combining (i0').
the
two
new
third type, of having the form equations (i0')
equations
(10"I
Seven of these invariants are known. One is the invariant No. 255; the six others are those which may be deduced from the equations (10'I, but they may not be regarded as independent of four invariants of the first type, since they may be deduced by procedure given in No. 247. We
type,
therefore
have
Finally, we may not i.e., no more than One
64
may
of
these
four
new
have more the number
Invariants
invariants
of
the
third
than two invariants of of equations (i0").
is known,
that
of No.
In-
256;
Jl of four the the
type.
the
we may
fourth
still
have a new invariant. It is probable that these new invariants, the possibility of which was not excluded in the preceding discussion, do not exist. However, /63 in order to prove this, we must resort to other procedures -- for example, procedures similar to those of the method advancedby Bruns. Use
in
261. The invarlant still another form. Let
us
set
of
of
Kepler
the
an arbitrary
fourth
system
a._, ,Iv dt Let curve
us
consider
the
type
of
in No.
canonical
256
may
be written
equations
4>',__ _:v.
-- dyi'
TFF -
following
(i)
d._t
integral
taken
along
an
arbitrary
the
curve
arc
J -- f(_, Let
Variables
us
assume
that
we
are
dy, + x_ _,
writing
+...
the
+ x,, dy,,).
equations
of
arc
along
which integration is performed, expressing the x's and the y's as a function of the parameter _, and that the values of this parameter which correspond to the ends of the arc are s 0 and _i. The integral will
J
equal
dx J d_.
Let us in the
assume that we are preceding chapter,
F 0 for
t = 0.
Then such
the x's,
as
dF F, dx'
dF dy'
We
shall
have
the ...,
y's
considering our curve which varies with time
and
will
., =.:L_d
be
_7/
the
functions
functions
dx-4-
of
of
arc and
the
_ and
llke the figure F may be reduced to
x's of
and
the
y's,
t.
f[XX dtdxjd¢_
or
65
when
integrating
by
parts
,1<_:+drr.<,_<m,_ _j 1<_"- [_: <m ;i.7 j " i6__A
However,
and
therefore d--7=
F -- Z x dx J _=:t." (2)
If we assume that F is homogeneous to the x's, it will become
and
has
the
degree
p with
that
the
equation
respect
dF Ex _ =pF.
may
Let C be be written
the
energy
constant,
so
of
energy
F-----C.
Let C O and C I be the a 0 and al; it will become
values
of
dJ at -('-P)(c,-
this
constant
which
correspond
to
(3)
c_).
Therefore, strictly speaking, J is not an invariant. However, its derivative, with respect to time, is constant and -- to use the expression defined in the preceding section, it is an invariant of the fourth type. 262. Let
Let us
us now
divide
the
assume pairs
and let us use xl, Yi to the first class, and let Jugated variables of the
66
that of
F presents
conjugated
another
type
of homogeneity.
variables
into
two
classes,
designate the pairs of conjugated variables of us use x'i, Y'i to designate the pairs of consecond class.
I to
shall
assume
the
xi's
, to
us
then
set
that
the
F
is
(x'i) 2 's,
dF Let
homogeneous and
to
I_(
_/[:<:
of
the
d_F,
the
order
(y'i)2's,
p with
so
that
we
respect have
,dF\
@) + " :C:'@'-/_')]
/65
or +-E
from
which
it
follows
, (x _ _d,,,,_y, d=,kl d_ lJ d_,
that
<,
+,<,:,)] f[
-+-
'(
E.'_ dd_t
-+- _ Z
<_'."'__,,<_'='_1 d_ dt ] J da
,v'--da dt
or
f[ or,
integrating
by
, _
_v
, _F)]
parts,
dF
or
or
, (..
I Z
z,
dF, +y,
dJ = [F _pF dt
_=='
finally
/2 = (,_p)(c,-
co),
dt
which
shows
263. problem,
that
Let and
J
us let
is
apply us
still
the
determine
an
Invariant
of
the
preceding
statements
the
in
change
the
fourth
to
the
±nvarlant
type.
three-body of
No.
256
67
with
the
different
In No.
ii,
we
variables used
the
will
as
be
in
an
[_(Ldl
following
as variables
,qL,
_G,
_0,
_'L',
_'G',
_'0',
l,
g,
O,
I',
g',
0'.
F is homogeneous of degree first series. Therefore,
f
chosen.
+ Gdg
-2
with
respect
+ OdO)+ _'(L'd/'+
to
the
G'dg' + 0',!0')]
variables
,- 3t(C,--
of
the
Co)
invarlant.
The same No. 12,
homogeneity
remains
A, X,
II, h,
Z, _,
if
A', X',
the
H', h',
following
variables
are
chosen,
Z', _'.
Therefore, f(Ad_+
will
be
an
If
the
the function to the _'s, As
H dh + Z d_+A'd'A'+
are
an
following
are
chosen
as variables
A.
A',
_,
_',
p,
p',
X,
_',
_,
_',
q,
q',
F wlll be homogeneous to the n's, to the p's,
3t(Ct--
Co)
(see
No.
of degree -2 with and to the q's.
12)
respect
to
the
A's,
a result,
_ +p dq--
q dp)+
St(Or--
Co)
invarlant.
The sign E indicates that the accented must be added to each X_d_ =_d_
68
Z'.:_')+
invarlant.
fX(aAdt+_d_--_ is
lt'dh'+
term whlch is deduced term. Thus, we have + ['dn'.
when
the
letters
/66
If
we
finally
shall
we
see
select
the
be
an
A,
A';
vi,
)_,
)";
=i,
H _
invariant
of
Remarks 264.
of
In
No.
the coordinates namics take the
the
on
256,
E(':,'_l'ri
fourth
the
we
We
considered
of n points in following form
an
and
we
have
in
this
special
then
137
that
of
the
cases
Co)
Given the
in No.
case
and
256
in which
in which
the
the
x's
designate
equations
of
dy-
dV
fourth
p with
respect
to the
x's.
case
,¢y +py
merit
-_- 6g((]t-
= d.-_'
degree
invariant Two
and
/67
of
seen
eid'¢i)]
space,
is homogeneous
J =f_(2:v
is
131
type.
d_ :v
V
--
Invariant
"_ _ where
Nos.
that
['_,-\
will
variables
d,v) + (p -- _.)t(C_--
Co)
type. particular
attention.
Let
us
assume
that
have J = _,/'s
(x ely --y d_v)
,J
and
J is In
points readily In
an
invariant
particular, which attract
of
the
first
type.
this is what occurs when one assumes several material each other in direct ratio to distance. This may be
verified. this
case,
we
have x=:Acos_t+Bsint
and
69
9' = --m),A
sin), t-4- m), B cos), t,
where % is an absolute constant, while A stants which are different for different It then becomes
and h pairs
are integration conof conjugated variables.
#_c= cos). t dA -4-sin), t dB, _'," :: -- m_. sln).t dA + m), cos), t dB, from
whlch
it
follows
that .7:dy --y
which
shows
dx --=m),(A
dB- - B dA),
that
J =_XJEm(AdB--BdA) is an invariant, since time has constants and their differentials Now
let
us
material points their distance. The
set
p = -2.
attract
invariant
Thls
each
J then
the
quantity
under
is
other
the
in
and
case
that
only
which
holds
ratio
to
inverse
the
integration
when
the
several
cube
of
becomes
J -= 2fz(x
Here,
disappeared, enter.
dy +y
the
dx) -- 4 t(C,-.-
slgn
f is
the
Co).
exact
differential
of
the
expression S = _xy, so
that
gration
if arc
the are
values
of
designated
S corresponding by
S O and
to
the
two
ends
of
the
inte-
S I, it becomes
J = (2 S, -- 4 C, t) + (= So -- 4 Cot). In particular, if we assume that one of the ends of the integration arc corresponds to a special situation of the system, where the n material points are at rest and are located at a very great distance from each other, the mutual forces will be very small, so that the velocities of these material points wlll remain very small for a very long period of time, and the distances will remain very large. As a result, C O will be zero, as well as SO, both for all values of t and for t = 0, and we wlll
70
/68
still
have J = aSi-We
will
therefore
4Ctt.
have S = _Ct + B,
where
B is
a new
constant,
and
C is the
Xxy= or
or,
constant,
or
else
xCt+B,
dx
performing
integration,
E
where
energy
A is This
Vorlestmgen
a is
third the
however,
X_
which
Dynamik
Case 265. considering
2
=Ct_+Bt+A,
constant. result
Uber
nl
Jacobi
(Lectures
of
the
obtained on
at
the
beginning
of
his
Dynamics).
Reduced
/69
Problem
We may reconsider the question which we discussed in No. 260, the problems pertaining to the three-body problem, which are, somewhat
simplified.
I shall first consider what I have designated as the restricted problem, i.e., the problem discussed in No. 9 where two masses describe concentric circumferences, while the third, infinitesimal mass moves in the plane of these two circumferences.
the No.
There are form (5'), 259).
then two degrees of freedom. (5"), one equation (i0') and
Therefore, we can have which is already known, two is known, two invariants of one invariant of the fourth
There is one pair one equation (i0")
having (see
at best an invariant of the first type, invariants of the second type, of which one the third type, of which one is known, and type, which is already known.
71
three
We can bodies
also consider the plane moving in one plane.
_roblem
--
i.e.,
the
problem
of
Finally, we may assume that the number of degrees of freedom has been reduced by the procedure given in No. 16. Let us make this assumption in the case of the general problem. We shall then arrive at what I have designated as the general reduced problem. Let us assume that this is true in the case of the plane problem; we will then arrive at what I have designated as the plane reduced problem. A resum_ different
of
cases
the is
discussion
given
in
which
the
would
following
be
followed
in
these
table.
/70
Problems
Re-
Plane
General
stricted
Number of degrees of freedom ...............
4
Number (5 _')
of pairs (5'), ..................
4
2
3
3
4
1
2
1
4 13 ii 2
1 4 4 i
Number
of
equations
(i0')
Number
of
equations
(i0")
2
of possibl,
First type .......... Second type ......... Third type .......... Fourth type ......... Maximum number of new invariants:
possibl_
First type .......... Second type ......... Third type .......... Fourth type .........
72
Reduced General
6
2
Maximum number invariants:
Reduced Plane
I
0 3 4 i
0 2 2 0
0 3 3 0
V
GHAPTER USE
OF
INTEGRAL
Test
series
266. In Volume which formally
XXIV INVARIANTS
Procedures
II we discussed different satisfy the equations of
procedures for the three-body
/71
finding problem.
Since these series may be of great practical importance and since they are only attained at the price of long and difficult computations, every method which one may find to verify these computations may be very valuable. The consideration of integral invarlants provides us with one method which is of interest.
Let
us
call
planets (as we the first must of
the
first
ponents in
of
series
those
and
their in
xi
(i =
i, 2,
of
the
Sun).
In
x i and of two
On
These
following
manner.
the
4,
5,
6)
the
other
and
coordinates
hand,
quantities
we
introduced
of
six
us
call
Yi may
and XV, instead
Yl which I have just defined, in planets we employed twelve other
addition,
let
x i and
of Chapters XIV these chapters,
two
in of
be
Yi
the
com-
developed
particular, the twelve
order to variables
define
the
arguments
r
while
the
No. ii and as we have always done since, Sun, the second to the center of gravity
momentum.
Let us recall the results obtained in No. 155. In
variables positions
3,
stated in Section be related to the
t
setting
six
other
integration
constants Ao, _%, _,o
and we found that following way. The
the
equations
,_,'_, _;o,
of motion
x',o,
could
be
satisfied
in
the
/72
quantities
73
k.2
may
be
developed
in
powers
periodic with respect on the two integration
of _
and
to the w's constants
of
the
x'iO's
and the _ '% A 0 and Ah.
.
and
The constants n i and {i may be developed in 0v x'i s, and depend on A 0 and N 0 in addition.
the
The
_-i's and
the _'i's
are
six
integration
Each
term
depends
in
powers
of
is addition
_ and
of
constants.
Finally,
is an exact differential when the twelve variables A, l, o and T are replaced by their expansions, and when the w's and the _ s are regarded as six independent variables and the quantities AO, A_, _i 0 are regarded as
constants
in
these
Our quantities pressed readily by
expansions.
x i and yi which I have Just means of the twelve variables
defined A, l,
may be exo and _.
It may be concluded that x i and Yi may be developed in series in powers of _ and of the _iO's, as well as according to the cosines and the sines of multiples of the w's and the _ 's . In addition, each coefficient depends on A and N O . The
will six
be
expression
an
exact
independent
differential, variables
and
if
the
w's
and
A 0, N O , x_ 0 are
the
_ 's
regarded
are as
regarded
as
constants.
We need barely point out that the series thus obtained are not convergent. They are only of value with respect to formal calculations, which gives them, however, a certain practical utility as I explained in Chapter VIII. Nevertheless,
if we
substitute
these
expansions
for
the
xi's
and
the Yi'S in the expression of an integral invariant, the result of this substitution must, from the formal point of view, satisfy the conditions which must be satisfied by an integral invariant. This provides me with the verification
74
procedure
to which
I wish
to
draw
attention.
We
267.
saw
above
f
is
an
integral In
that
a change
In
dy
E('zx
dz)--
3t(ci--
(i)
Co)
invariant.
order
perform
/73
that
order
we
of
may
make
variables
to have
use
of
which
greater
this
is
symmetry
Ao= _,,
invariant,
similar
^_ = _, ;
in
to
the
we
that
are
given
notation,
let
going in No. us
to 237.
set
_i ° = _i+,.
We have seen that we may develop the x's and the y's in series depending on the w's, the w' s , the A0, A'0, and the x_0's -- i.e., with our new notation, the wi's and the Ei's (i 3 i, 2, 3, 4, 5, 6). For the
new
variables
differential
we
may
equations
then
take
of motion
the
will
Ei's
take
and
the
the
wi's,
and
then
form
(2) d_i
_.
d'_'t
0
[just
as
in
No.
237,
=
lzi
equations
(1)
become,
dy i _ dz 0 as we
have The
dt
after
the
change
in
variables,
- dt
i
seen].
ni's
are
functions
of
the
$i's
alone.
However, it is more advantageous to select other variables. Due to the fact that the six ni's are only functions of the six _i's, nothing prevents us from taking the ni's and the wi's as variables, instead of the
_i's
An
and
the
integral
wi's, so
that
invariant
the
differential
dn_
dwi
0
/7-I
of
the
a =f(zA_dn_+ where
A and
B are
functions
of
the
equations
become (3)
= dr.
first
order
will
take
the
form
EBldwi), ni's
and
the
wi's.
75
x.J
I may assumethat figure F is a curve arc for which the equations, which
are
variable
with
where the variables and of a parameter tirely traversed.
time,
have
nt=fd_,
t):
these
which
stipulations,
it follows
f
However,
we
,,,, =.f;(_,,
o);
it
d_
(dA,dni
y
dBi
finally
dt
J is an
dwl
d*-
n
t
d2
cvi
\l
+ A_ -dTd_ + n,. -3/-il_ 7"
have
"f[( If
write
that
d_ nl dt d_
which
then
o).
z ,% -_- + ,.:B, -3:,/d_,,
dA _. dt -- y nk dBi d-7" = xnk
from
form
t),
,,'i =f;(_,
I may
J = from
following
n i and w i are expressed as functions of time t a which varies from a 0 to a I when the arc F is enThe equation of the arc F 0 will then be hi=f,(,,,
With
the
--
--
Vt
dni
absolute
el"- w i dni "dt-_ ----_
o;
follows
dA i d-_ ' dBt _t_' '
that
E_tnk
'"
d_v k q-
integral
Bi
)
+
dwiZk,',;;
invariant,
'",1
_?i_kj.
we
must
therefore
have
dDt ]_k
nk
_
(4) ---_ O_
dA_
x_,,k _G Let
us
now
determine
what
B's are periodic functions of oped in trigonometric series. Let
76
us
first
consider
= - D,.
occurs
in
the w's
equation
and
(4),
(5) the may
and
case be,
when
the
A's
consequently,
let us
set
and
the
devel-
/74
Bi _= E[7) cos(re,w,+...+ where
the
b's
Equation
and
the
(4)
becomes
E(m,
which
may
only
b"s
depend
the
I_v,_l....+
m6 ws)] '
ni's.
m_n6)[--bsin(mtwl+...+ m6wt) -I- b'cos( rnt _vz -4-..-.-4- m_ we)]
= o,
if I._[ll_
or
on
b'sin(m
/75
nt+...+
hold
m6w_)-+-
I -_-.
,
.._-
D_[6 /'J.6 _
0 t
(6)
if
However, pendent tion
the
m's
variables (6)
are
integer
between
therefore
which
entails
the lnl
This its
means
that
known
term
independent Let
the --
of us
i.e.,
now
----
B i
_,
for
Equation
pass
to
may V(mt
Let such
that
then
have
us ml,
first m2,
In
the
contains
_
• . , _
DI 6
_
the
n's
are
may
our
inde-
hold.
Equa-
O.
expansion
is
a
function
of
of
B i may
the
ni's
be
reduced
alone,
to
and
is
of
then
brevity,
be
n. +...-+-
consider ...,
second neither
a
Let
us
set
of
written a sin_o -+- ct' cos o_) -_ -- Bi.
a
whlch
term not
-_-
zero
, . .--_
B i does term
instead
m6ns)(--
m 0 are
term,
(5).
Z(acost0+a'sin_),
D_ I l/_t
term
D_3
equation
purposes
(5)
relationship
w's.
Ai= writing
and
linear
following
trigonometric
the
constants, no
for
,rn6n6
not cosw,
is at
dependent the
same
on
the
time.
w's,
We
l,e.,
shall
<>0.
depend nor
on a
the
term
w's. for
This slnm.
second As
a
77
result,
the only
we
have
Therefore, known term on
the
A i does not depend of its trigonometric
equation
In general, every first order, where the to the x's and the y's must
and may be reduced to a term which depends
ni's.
However,
w's,
on the w's, expansion,
have
the
(5) may
then
be
reduced
/76
to
linear absolute, integral invariant of the term under the sign f is algebraic with respect and, consequently, periodic with respect to the
following
form
EAidnt,
where the Ai's depend only for the absolute invariants differentiating of gravity.
the
However,
the
on the which
integrals
relative
of
ni's. In reality, we know and which area,
energy
this is what occurs are obtained by
or motion
of
the
center
invariant
J =-fr.(_x dy ._-yd.r) deserves
more
attention.
We
have
seen
that
J --St(C,--Co) (where C O and C I are the values of ends of the arc F 0) is an integral
the energy invariant.
constant We shall
at the two therefore have (7)
dt--dJ=
If we
3(Ct
dni
+
Bidw_),
(7) becomes
•
78
Co) = 3fdC.
set
J =fx(A_
equation
-
dAi
i_i)+
dlviEknk
dBt
]
dC -_,
V
because
the
energy
Equations equations
constant
(4)
and
C is
(5) must
only
a
function
therefore
be
of
the
replaced
ni's.
by
the
following
/77 dBt
xkn, _
=o,
dAt
dC
x,n_ _ The
A's
and
the
B's
must
If we treat equations equations (4) and (5), we
We
= 3_ be
--B_.
periodic
(5') functions
(4') and (5') Just find the following:
i.
The
Bi's
are
independent
of
the
w's;
2.
The
Ai's
are
independent
of
the
w's;
3.
And
that
finally
the In
the
of same
the
w's.
as we
treated
obtain
X('_x dy + y dx) = XAidn_'+ where
(4')
Ai's
other
depend words,
only
on
the
3"_ dc _a'_n_
d*vi,
ni's.
expressions
or
(8) do not depend on the w's and are only functions of either the _'s or the n's, depending on whether everything is expressed as a function of E's and the w's, or as a function of the n's and the w's. In
the
same
way,
we
shall
have
79
the
dz, _l:_,_ = 3_,5 ,lc_" Ei _Xi_t,v_---k+)'i(l_vk/ AS I in powers
have already of _ and of
equations ties.
(9) may
in
stated, the xi's , the the _i 0 's. Expressions
therefore
All of the expansion powers of _ and of the
also
be
(9) y_'s, _8)
developed
in
terms of expressions x'i0 's, must therefore
and C are developed and the two terms powers
of
these
(8), which are be independent
in
quanti-
expanded of the
WVS.
On must
the
equal We
other the
thus
268.
hand,
each
expansion
corresponding
have
numerous
I have
stated
term
of
term the
procedures
of
the
second
for
first
member
of
(9)
member.
verifying
our
computations.
that
x i d_i
is an exact the w's are We
differential, regarded as
then
if the _i's are regraded independent variables.
dC since
regarded
and
dz) = 37"z
de
's
the d-_i as
constants,
obtain
E(_x dZ +y
or,
as
depend
only
on
the
_'s,
they
must
be
consequently
constants f
E(2xdy
from
which
it
follows
that
from
which
we
finally
have
+ydx)=3E
dC
(lO) f_x
80
us briefly
return
= 3X N "<m dn i to
the
tui
--
Z xf.
notation
given
in No.
162.
In
this
section,
Just
as
in
No.
152,
we
chose
the
follo_rlng
as
variables (ii)
t
and
we
A, ),,,
A ', _'j,
_i, ,_,-,
set
On Yi, are several
the
other
hand,
conjugate times, the
the variables
variables. expression
As
(ii),
a result,
Just
as
the
Just
as
I have
variables
xi,
explained
•vi dye- - A d)., -- ._,'ag,', -- _ _id';i = dU is an
exact
readily of
the We
differential.
formed, xi's
which
and
then
the
I should
may
be
add
consequently
the
regarded
dC
,
Relationship It
is known
Dynamik, Jacobi attraction, the
Jacobi
The
as a known
that
to at
demonstrated mean value
,
/79
be
function
theorem
equations
a Jacobi
Theorem
the beginning
of his Vorlesun_en
fiber
the fact that, in the case of Newtonian of the kinetic energy equals, with the
is
directly
of motion dx
(12)
in Chapter XV is applied, one equation (12) furnishes us with
exception of a constant factor, the mean value of assuming that the coordinates may be expressed by series having the same form as those which we are This
U may
have
Just as when the procedure outlined is led to formulate the function S, and the desired verification in a new form.
ments.
function
Yi'S.
E
269.
that
may
related be
the potential energy, the trigonometric presently studying.
to the
preceding
state-
written
l
ml--a_=_ir
from
which
it
follows
that
81
v
Then
the
-V
represents
kinetic On
degree
the -i,
the
potential
energy
other we
hand,
shall
due
to
equation
having
us
--V=
E
take
multiplied
the
fact
may
note
y_m'---; = ;l
making
(9) from
No. by
217 n k.
and We
let
us
shall
add
them,
after
have
dC
that
= n k) , we
may
a comparison
dx
dwk
--
conclude
_X_di-
with
dt
that
+yi
tlt ]=
the energy
dC
shows
/80
.__._!'_
x _k _ which
of
be written
respectively
E
When
V is homogeneous
dxL _..r_ -2-i"
dx
dwk dt
that
d rt dyl xy_ _?i + _:xt_IY = C.
E n_
(since
and
dyt xl= zxl -3y,
therefore
equations them
dV _
dr, If we
energy,
have
,
Let
total
energy.
E
The
C the
energy,
that
C must
equation,
we
find
degree
2 _ with
that
2
= _ c,
be homogeneous
of
respect
to
nk's , which could be seen directly. The mean value of a function which I shall designate by the notation [U], will be zero if U is derivative of a periodic function. We shall therefore have
82
the U, the
[f'_Tf Jx,+ and,
connecting
from
which
this
with
the
xi _I dt J = °
energy
equation,
[Vxl _I= dt J we
is
the
Jacobi
If
the
partial derivatives dx similar results
derivatives_,
_C,
2
theorem. dx d-_k are would
"zi (-"i &' and
obtain
have
[--vj This
we
considered
be
instead
obtained.
We
of
would
the
total
_
obtain
d:_/_
consequently
gy'l
[
,_c
[ EiYi _"d_vx.]1 = --3 7[n--k" dc Application
to
the
Two-Body
Problem
270. In particular, the preceding considerations may be applied to the two-body problem. Let us consider a planet and the Sun, and let us refer the planet to axes having fixed directions and passing through the Sun. Consequently, let us consider the relative motion of the
planet
Y2,
Y3
Let
Let
with
respect
x I, x2, be
the
_, n,
x3
three _ be
be
to the
the
three
components the
Sun.
three
coordinates
of angular coordinates
to particular axes, i.e.: The major axis line to the minor axis, and a perpendicular We
shall
of
the
planet;
let
Yl,
momentum. of
the
of
the line
planet
with
respect
orbit, to the
a parallel orbital plane.
have
83
•rt -, hi _ t- I_'l rt + h"__, x_ where
the
h's
tionships orthogonal.
are
which
In the
constants
which
Indicate
same
: h;,_-_ h'_q-,-h'_.
that
way,
we
are
the
shall
connected
by
transformation
the
of
well-known
coordinates
relais
have
Yi = t_hi d_ 3"e ' _ h_ dn _ + _hl d_ 2-t' where
_ Is
the
mass
of
the
planet.
It is now evident that _ is zero, and that g and q are of one single argument w, which is the mean anomaly, and of which are the major axis a and the eccentricity e. In addition, or more generally, three angles. Thus,
the
x's
the of
and
h's are the functions of three arbitrary functions
the
y's
are
functions
functions two constants,
the three Euler angles, ml, _2, _3 of these
of w,
a,
and
n as
e,
and
of
the
_IS.
If we motion, we
designate C as the shall then have
energy
constant
the mean
#
+
and,
in
addition,
the
dy,
] =
3 d'--n
dx,_
!:( +' 2"ri
"-(_-_
•
must
be
with
Some of the statements no new verification.
84
dw
expressions E(ar.
independent
yt
"q-
Yi
de
]
dxi
"_
of w. were
apparent
beforehand,
and
provide
us
In actuality, coefficients
the dXi's are linear functions of the xi's dmk depend on the m's and are such that
As a result,
we may write the following identity ot I dx,
where
the
tions
of
a's the
are _'s.
dx_
arbitrary In
same
a result,
we
and
way,
=,_ay, + =2 _
As
we
shall
+=3 _dZ, =
setting
k =
must be we have I, 2,
::r,.
.:_::q'_
reducible to three similar
3 -- we
may
¢V3 _%'1--
Let
is not
us now
a new
us determine x's include
func-
have
Y,
y,
y3
investigate
the
3"2
X3
[
r, 3"2 ?f _i )'sI" ",,_ a constant which is independent relationships which one obtains
of
I =
const.
X2 =_ COflSt.
result;
\ Let The
given
.Y2 '25"3 -=-= COIISt,
,Y2 :_'1 ---YI
this
the
write
ylx3--.)'3X
However,
are
have
(
by
k, Oi s
the
"_1
This expression w, and -- since
ott
dx_
constants
the
_
whose
these
are
the
area
equations.
expression
-35 + y_ da / "
the manner in which the x's and the y's depend on a. a as a factor, and the y's include_-_, because we have dxi {Ixl yi = _ -dF = 11n _j.
85
/83
_J
We
therefore
have d:vt _ ,Tt. da a '
Our
expression
therefore
dy__ 2 = r_ + y l dn da a n da
becomes Ext'y_
It may be readily law of Kepler, we
verified have
A....n da]
that
it
l_q-a
from
which
it
follows
is
S _
....
We
not
must
obtained
now
the
third
the
_'3,
COIISL_
3 da -t .....
IL
still
to
that 2 dn
We have
According
zero.
examine
a new the
=:0. (_
procedure
two
for
the
proof.
only
vary
expressions
"ZXi _l¥ "L-yi dw ] ,),,re_l_dy, -_-Y_ 'ilJ dri'_} = t';.
Z( e and
w remain
i,e.,
the
therefore
to
be
direction choose
varied. of
We can
the
major
particular
axes
x,=_
--=a _
x,=_-_
The
functions
includes
J all
the
We may
are
therefore
deduce
moved,
86
becomes
the
and
may
set
e+EJp-,(Pe)
the
/84
We may
C e)_inp'v]" -p--j
The from
orbit.
-_
index
to
+%
p
under
with
the
the
sign
exception
Z of
following
-- _anEJp_l(pe)sinp_v
y_= W
the
functions. values
yt=
Expression
of
a (I--2Z--_[EJP-'(P .
Bessel
integer
therefore
axis
_an
l_e--_ZJ_)-,(pe)cosp
following,
if
the
w. common
factor
_a2n
is
re-
O.
3eVJl,_.tpcospw--9,,xJi,
--'_(I
For
purposes
Jp_l
(pe) We
t cosp,v P sinpw P
- e_)XJp-I
of
brevity,
Xjj,_t Pc°sp_v+[xJp
X Jp_,p
I have
sinpw
written
must
then
Jp-I
=: lV'.
everywhere,
instead
of
have
However,
C
m
-- e2)[Z tp_. t cospw]_
•
dC 3--. dn
W---
where
-4-(|
.Isinpw]i
designates
therefore
the
,,t ;,:,
.......
mass
of
l_.|
the
(g 3 _
Sun
t,q:p
plus
that
of
we
have
a
a
series
the
planet.
series
of
We
have
C
-=4 --
I)_.$F?_
_ 2
and
dC :3_l,i ..... However,
we
__ _! l'man a=
__ t,a,_t.
since
find
When ships A which
identifying
between
the
study are
of will
271.
in be
Numerous
provided.
For
motion
as
having
taken
we
examples
of
after
just
into
account
done
/85
us
to
functions
of J
relationships
and
their
first
time.
these
having in
the
terms
of
could
relation-
J.
Bessel
this
one
terms,
leads the
included
example,
--
E
which
have
masses
similar functions
expression
similar,
derivatives
disturbed
the Bessel
apply
particular treated
preceding the the
degree same
applications the
case
section zero
of --
with
principles
could
be
Keplerian i.e., respect to
the
after to
the
entire
87
group of terms to interesting
tional
of degree results.
i.
There
doubt
that
Using the same procedure, we could also study equations which we discussed in Chapter X.
integral
this
would
the secular In place of
lead
variathe
invariant
we would have the advantage of defined in Nos. 261, 262, 263. We
shall
put
these
272. take
Let
the
us
apply
coordinates
employing
questions
Application
us
is no
similar
principles
x i and
which
we
aside.
to Asymptotic
these
invariants
Solutions to
asymptotic
solutions.
Let
the
dxi yi = ml_-_ as
the
variables.
Let
us
consider
the
invariant
J =fz(_zc_+ydx). We know that if C is the energy constant, and C I and C O are the values of this constant at the two ends of the integration line, we shall have J--3t(C,--
If we following
consider form:
The
a system xi's
where
the
coefficients
of
and Alea_
are
Co)=
t,
eonst.
(i)
asymptotic
the
yi's
A,e=,t_
periodic
. . ,_
in
solutions,
will
be
it will
developed
in
have
powers
the of
_k¢=Lt_
t + h,
where
Ai, A_, ..., Ak, h are
k + If
energy
i arbitrary these values equation, the
constants. of the xi's first member A le_tl
88
Al°qt
and the Yi'S are substituted in the is always developed in powers of ..._
Ake_t
where
the
pendent If
t, it will
the
(i), we
and,
coefficients
of
values
shall
of
are
periodic
also the
be
in
t + h.
independent
xi's
and
Yi'S
of are
Since
it must
A I, A2,
...,
substituted
Ak
in
be
inde-
and
h.
equation
have
consequently, J = const. In J,
the
expression
under
Ale_t, t, The k +
coefficients are 1 differentials
must
therefore
sign
Aae_t,t,
periodic
dAt,
We
the
in
dA_,
.,
.+
f,
dAk,
developed
in powers
of
Ake"+-tLt'l
t + h;
...,
is
it
depends
linearly
on
the
dh.
have
(2) E 2x at+ +9" dh
The
first
terms
of
equations
= eonst.
(2) are
developed
in
powers
of
the
Aie_it's. All terms of this development must be zero, except for the known term. One thus obtains a multitude of relationships between /87 the coefficients of the development of the xi's in powers of the Aie_it's. By first
way
term,
of
an
and
example,
I shall
I shall
X i ::
where
X i and We
may
Z i are
confine
myself
to
considering
the
write
periodic
X i q-
in
Z i ,% e _t
t + h.
deduce y,. _ +m[X_.-+-Ae_'(Z}-!+ +_Z+)]
where
X_
and
Neglecting
Z'i
designate
all
terms
the
derivatives
in e 2_t,
etc. , we
of
X i and
then
Zi.
have
89
((r ,,.x 7/;( We
therefore
(]_"l-: _-y ,lA /
provides
X
Zi .
and
The
us
with
the
+'2¢_XZ-'_
first
the an
90
us
with
first,
since
equation
is
principle.
X'Z)--o,
relationship
between
the
coefficients
relationship
E
furnishes
_z) _ x'z].
have
Zm(_XZ'
which
z,,,_'.[,,_x(z'+
dy -l-y '2x
another
one
--
it
obtained
when
which
which, is is
d:v ) =: const. [l_
in
combined an
reality, with
immediate
would the
first
consequence
not
differ
relationship of
the
from -energy
CHAPTER
INTEGRAL
INVARIANTS
Return
273. the
Before
results
to
sum
my
point
up
of
the
We
saw
in
AND
to
proceeding
given
the
any
Chapters
results
a
and
to
Bohlin
I must
XIX
I wish
SOLUTIONS
of
further,
supplement
XX.
compare
I would and
some first
which
of
like
will
serve
as
departure. in
Chapter
Vll
that
if
djz:._Xi dt has
ASYMPTOTIC
Method
VII,
which
XXV
periodic
a
(i
system
(i)
_=,,_,...,n)
solution
and
if
we
set Xi _
the
(2)
_i's
may
be
developed
in
Xl
increasing
powers
of (3)
Al£7_tj
where
the
coefficients
are
gration
constants;
the
periodic
solution
(2).
The vergent
series under
There having
the
is
always certain an
are
satisfy
the
between _-q
where cient line
the y 5.
is
coefficients whole, When
B
are
positive,
writing
this
or
of
(i)
which
we
case
where
the _- X_
whole,
t
t.
characteristic
equations
in
Ane_t,t
functions
the
conditions,
form
. ..,
periodic
_i's
exception
following
._.zc%tj
The
formally.
have we
exponents
Ai's
are
exponents
of
They
discussed have
a
in
intethe
are
con-
No.
105.
relationship
_
,
-- zz = o
positive,
or
negative.
relationship,
(4)
(See I
zero,
and
Volume
assumed
that
I,
the page the
/89
coeffi338, unit
of
91
time
was If
chosen there
so is
that
the
period
a relationship
be developed in powers of quantities (3) and of t.
the
of
the
having
solution
the
quantities
form
(3),
(2) equalled
(4),
but
the
_'s
in powers
2_).
cannot of
these
This is precisely what occurs if the equations (i) have the canonical form of the equations of dynamics. In actuality, in this case two of the exponents are zero, and the others are equal in pairs and
have
the
opposite
sign.
In the case of equations of dynamics [or, more generally, when there is a relationship having the form (4)], we were still able to obtain a result. It is sufficient to give special values to the integration constants A, so as to cancel the values of these constants corresponding to a zero exponent, and one of the two corresponding to each pair of equal exponents having opposite signs. [More generally, the constant A corresponding to one of the exponents included in the relationship having the form (4) would be cancelled, so that there would no longer be a relationship having this form between the exponents corresponding to the constants A which are not zero.] For
example,
if =z-- - =_,
we
would
=i_ -- _s....
,
=*,-' -----=,, (n even) ,
make AI----A2----o_
A3 ----o,
A_=o,
...,
An-I
_
o.
The _'s may then be developed in powers of those quantities (3) which are not zero. However, we shall no longer have the general solution of equations (i), but a special solution depending on the number of arbitrary constants which is less than n (i.e., n - 1 in the general 2 case
of
the
equations
of
dynamics).
We have thus arrived at the asymptotic solutions: We have done this by cancelling a certain number of constants A, not only those which we have set equal to zero for the reason which I have just given, but also those which we had to cancel in order to satisfy the convergence conditions given in No. 105.
For _'s
92
the
in powers
time of
being, _ or
of
I shall ¢_.
not
deal
with
the
development
of
the
/90
In
Chapter
basically
only
is
reduced
partial
to
studied
obtaining
suitable
cases
a
that
are
which
in
the
and
that
S
this
case
in
which
following
which
satisfies S
is
a
the
the
function
S::So
_- v'TS,
us
Bohlin_
are
those
limiting
case be
which
since
the
an
equation
form
the
S may
M.
which
mean
is
problem
is
approximately
between to
as
has
there
interest
that
by
method,
function
designated
saw
derived
Jacobi
coefficients
greatest
have
we
method the
function
whole
of
I
section,
¢_,
the
having
which
the of
Only
for
relationship
this
I
application
derivatives.
larly
to
XIX an
a
with particulinear
motions.
The
which
are
similar
(No.
207).
In
developed
in
powers
of
form
-_- aS,-_-...
dSp da',
is
periodic
with
the
period
2_
with
y_, (employing
the
notation
However, variables
In
which
the
which
section
are
in
results
the
discussed
No.
206,
periodic
with
ya,
simplified
in
No.
respect
n
to
Y_, and
which
In
I
No.
regarded
210,
as
we
indicated
be
I defined
the
by
+
functions
the
Ya,
i
Only
the
retain new
equations
the
...,
change
in
y., of
Y_=:Yl--
canonical have
the
210.
variables
periodic
_,
Y}=3"l
d_
equations
performing
and
solutions.
following
xl_x_+_,
The
above).
209
generalizations
set
to
..., y,
section
may
was
respect
form
the
following _
= _ = y,
(/>i)
dg
with
the tnvariant
new
variables
xJ,
Yt
I.
/91
relationships
= o,
93
which, with respect to the new as generalizations of periodic
with
respect
our
Without canonical
If zero
to the
equations, may be regarded just as is the case for
ones.
limitin_ the conditions of _enerality, we may assume equations imply the following invariant relationships
this
for
old
canonical solutions,
is
the
the
case,
we
derivatives
saw
dS_,
in No.
and
210
a double
that
Yl
zero
= 0 is
for
the
that
a simple
derivatives
dyl
dSp
(i > i).
dyi Thus S, the expansion have
or rather S - S O , may be developed in powers will begin with a term of the second degree.
•
%"
S=:$0+)_2y_
where
the
E's
are
series
in powers
of
/_.
In
functions
of Y2,
YS,
Unfortunately, The
function
- i arbitrary
whereas
n would In
to the tion
order change
given
in
depending
addition, "'', this
be
94
if
we
define,
it may
is not
Y2, be
(5)
YS,
''',
Yn
seen
that
the
sufficient
is defined
required
for the
in variables, section
by
for
our
equation
Just
the
and
are
E's
developed
are
periodic
purposes. (5),
in
solution
in greater
given
in No.
i.e.,
I
as
complete
study --
I
and
on
$
depends
only
constants
to pursue this
_
Y, and We shall
Yn"
S, which
on n
3
_--_3y,+-_Y,+""
of
the
=i_y
indicated
detail,
206.
if we
i ....
of
If we
we
the
problem.
shall
employ
resort the
nota-
set
;,
section,
, ,.,
the
variables
x_,
Ul, Vl, and the functions T and V, the to v I and to the zi's will be periodic Volume II, p. 361).
derivatives of V with functions of the zi's
Let us examine in greater detail the equations which beginning of page 363 (Vol. II) and which are written
the
respect (see
appear
/92
at
y, --O(_,,y_,y3,...,y,), x_ --_._ (v,, Y2,y_....,y,,) Regarding equations
Y2, Y3, (always
"''' Yn Just as
as constants, let us consider in the indicated section)
When we vary v I, the point (x I, Yl) will describe to study. Let us assume that we vary x'I, instead
constants
x'2, x'3, ...,
corresponding We
to
assumed
X'n, and
different above
we
shall
values
that
the
are The
like
following
a generalization
following
point
obtain
infinity
invariant
of
curves
relationships
hold
_0
of periodic
will
an
a curve which I of varying the
of X'l.
X! _ X 1 -=Yl
which
following
x, = C,(_,)
y, = o(v,),
wish
the
correspond
solutions.
to
these
relationships
_I :--Yl _ O
i.e., the origin of the in the vicinity of this Let S defined
coordinates. point.
¢_
into
like
us assign to x_ the value corresponding by equation (5), and we shall have x,= _V_y,.
The
I would
corresponding -_,
we
would
curve obtain
3E3y_+
passes a
second
to
to
study
the
our
special
curves
function
....
through curve
the passing
origin.
By through
changing the
origin.
We therefore have two curves crossing at the origin. The center curves may pass near the origin, without reaching it and without intersecting each other, so that all of our curves together will look like (in terms of their general form in the immediate vicinity of the origin) the figure formed by a series of hyperbolas having the same asymptotes
95
/93
and formed by their
asymptotes.
274. In order to study these curves and their corresponding functions S in greater detail, let us limit ourselves to the case in which there are only two degrees of freedom. Let us assumethat the change in variables of No. 208 was performed in such a way that is a periodic solution,
which amounts to stating
that for
we have dF
Let of
us
degree
develop
0 would
F in only
dF
dF
increasing
depend
dF d ,¢l it will
be
reduced
a constant, Let
we
us
to
may
try
find
that the
and
of xl,
x 2,
since
we
F Is
only
and
must
Yl.
The
term
have
--0
a constant.
assume
to
powers
on Y2,
Since
this
terms
term
of
of
the
degree
first
defined
up
zero
zero.
degree.
is
to
Since
dF dF = =- __ o d:v_l ayl there
wlll
be
no
other
terms
of
the
first
degree
except
for
a term
for
x2• Let
us
now
set
a., _z_, It
can
be
seen
that
y,= Eyl, F may
be
x_=_x_,
divided
by
y_=_;. ¢2
and
that,
if
one
sets
F = _,F', the
equations
retain
the dx_.
canonical dF'
a-7 = 2-//
96
form dy_
_r
and dF'
....
d_'
become
(l)
In
addition,
F' will
be F'=
F' can be coefficients
developed, on are periodic
developed _;-_
in powers
_F', ,- _F;
the other functions
+...
H,
A,
B and
We shall
C are
apply
periodic
a method
hand, in of Y'2-
Let write
us
xi,
remove
Yi'
I would
F,
our
accents
like
to
of similar
to
x_, _1. finally
that
of
play
the
have become ' F' F'. of x'i, Yi' ' i" that
/94
The
yV2.
which
state
form
_ Cy', l
the parameter e will in Chapter XIX.
F.I instead
first
is
the
powers of x' 1, We shall have
functions which
equations. In this method, that the parameter _ played
e in
;
F_ = Ilx_ + A.x'_2 + _Bx',y'l where
of
I may
useless,
always
Bolin
to
same
and
our
role
let
us
assume
II=i. If
this
were
not
the
case,
I would
choose
the
following
for
new
variables
_fdz, x_ = llx_,
The
canonical
form
of
Y_-- J
the
II '
equations
would
not
be
when
Y2
changed,
since
x_ dy_ --x_dy_ = o is an
exact
In
differential.
addition,
2_. I may always stant equals 2_. 2_ will function to
Y2* increases
by
a
constant
increases
by
select the Then every
be a periodic F will not be
unit of time in such a way that this conperiodic function of Y2 having the period 2_ . The form of the function o f Y *2 wlth " p erlod " changed;
only
the
first
term
Hx 2 will
be
reduced
x* 2 . Let
us
I may
therefore now
state
assume that
we
that may
H -- i. assume
A=C=o, Let us have
form
our
canonical
equations
B= (I)
const. assuming
that
e = 0,
and
we
/95
97
dy,
dx,
d-T =--I;
dt
dr, --
dyi--2(Bx'
t-Cy,)
dx, 71r-7= -- 2( ,t x, -_ n'YO
dx 2 and
an
equation
for
d-_-which
I
x2 + hx_-_ The
linear
they
will
the
equation
of
energy
2 I}zlyl -:-Oyli ---_ const.
d, rl
d j# I
_.)3; =: .2(.\xi
equations have
_,
¢,
constants,
It
by
dy i -- -- 2(B.z'l + Cy,), having
the
¢1, and
may
I may
Under
¢I a
readily equal
are
coefficients.
a
general
In
virtue
of
No.
29,
solution
functions
of
Y2;
_
and
6
are
integration
constants.
seen to
.._Byi)
,yi-_-iP?l+w,_l,
periodic
b
set
for
W?+iVlt_
are
and
be
this
periodic
following
Xi::
which
replace
equations
are
where
may
that
b
=
-a
and
that
_I
-- ¢_i
is
a
constant,
i.
assumption,
let
us
make
a
new
change
in
variables,
setting
=, = =', ? +Y,'k; #
y, = xl _,-,'-.x;,I,,,
,
,'
xl=xi_lIz',_+aI(x,yl+Ly',i;
where
H,
form that
of
K, the
L
are
functions
equations
is
of not
t
Y2,
y,=y.j,
chosen
changed.
in
For
such
this
t
a way
that
purpose,
it
the is
canonical
sufficient
.'_',dy, - _', dr; _-x, dr, -- _, d.r; be
an
exact
It creased
may by
differential.
be the
seen
that
dr, [x?(_,?'
The
98
quantities
Xldy
I -- x_dy%
equals
an
exact
differential
in-
amount
@', 41,
...
??D +y'?('h'b'-'_'_)
designate
the
-_ .-._'_'+o-_' ,:,,.,_--9'7_)];
derivatives
of
_,
@i,
-..
with
L
J
v
respect
to Y2.
In it
order
that
is sufficient
which
the
to
canonical
form
of
the
equations
is
not
/96
changed,
set
It may be seen that H, K, it follows that the form
L are periodic of the function
functions F will
of Y2, from not be changed
either. However,
if we
set
e =
0, our
equations
must
have
the
solution
may
assume
r
x t =- _e+ay_;
from
which
it
follows
that
a "2
Without
limiting
the
conditions
I1 =l,
from
which
it
follows
A =
_ince
we
Fo=
We
shall Let
follow us
this
perform
of
generality,
C _o,
have
B=
we
that
const.
removed
the
accents)
xa+_Bxtyt.
procedure a change
from
this
in variables,
point
on.
setting
Yl log--- = _v.
:r,yj = u,
Xt
Since d(x, X t dyt
is
an
exact We
then
differential,
U
the
dv
,_-
.r, )
--
canonical
form
will
not
be
changed.
thave _,=
The
--
function
F may
e_;
then
y,=
be
e-_¢_.
developed
in
powers
of
99
We
have F0_xz+
Let
us write
F in
the
_Btt.
following
form
F(,r_, u; y_, ,,,) and
let
us
define
a function
S by
\ dy,' where
C is
a
constant.
Let S
":So
C-:
shall
us
the
develop
us
S and
I'-:S,+s_S_
_-"-,
Co+_C,-_'s_C_-4
-....
alSo dS_ 2j_ + _ l_ dJ
Let
/97
equation
7/_ ; y2, v ) =: C,
In order to determine S0, $I, have the following equations
As I have already function by _. In the In the third equation,
Jacobi
82,
...,
C in
by
powers
of
a recurrence
c
method,
we
:Co,
dS,.
,I>
Ct,
dS2
dS_
't, -t- C_.
(2)
done previously, I shall designate second equation (2), I assume that I assume that S O and S I are known,
every known S O is known. and so on.
set Sv :_oY_+,%v
with
the
condition =_+aB_o=
Since Co is arbitrary, arbitrarily. Nevertheless, Following is the reason for Let
us
assume
that
the two constants it is important this.
it has dS0
may
i00
be
developed
in powers
Co.
been dSt
of
shown
a o and _o may be that we do not set
that dSi,
chosen 80 = O.
f.j
We
may
conclude
(if
80 is
not
zero)
_V
+ _-JV
since the quantity under the This conclusion could not be that ¢_uln
this
may
be
e _ i)-,
•
that
+'"+
radical reached
it
us
now
consider
includes
same
holds
true
for
_p J_-
may be reduced to if B0 were zero.
reached,
the
depends
due
The
coefficients
second
to
the
equations
on v and _'
_
A are
_
A
m.n
on
Y2,
_nav+'ln.y,
.4-
constants
80 It
presence
The
second
equation
St=
the
(2)
_'0'_+
then
gives
(2).
and
for e = 0. is important of
the
/98
radical
has
function
following
form
may
depend
positive, E, I have
us
the
on
a0
negative, shown the
and
on
or zero term in
80. which
following
A mn emv + tn.r,
13,_'+ z.., -_ +--q--ff-_-
condition _t+ 2B_,=
Except for this condition, I shall therefore assume that _|
A,0+ C,.
the
_
_1
_
constants
al,
sum
up,
the
derivatives
81 and
C I are
arbitrary.
O.
I shall determine S 2 by the third equation that this equation has exactly the same form as treated in the same manner, and so on.
To
The the
Aoo.
which
The indices m and n may take all whole, values. When removing it from the sign these two indices are zero.
with
the
F. Let
which
conclusion
C "v 1
dS d-_2 and
dS _v may
(2). Due to the fact the second, it will be
be
developed
in powers
of
If one
compares
this
analysis
with
that
given
in No.
125,
it may
i01
be seen that there is an exact analogy between them. However, instead of having only imaginary exponentials e±iY,,
we
here
have
real
e±_,,
....
e_lYn;
exponentials e _g"
275. Once the function Jacobi method, we may arrive given in No. 127. The The
80.
S has been determined, by applying the at series which are similar to those
function S depends energy constant
on v,
C---
is
a function
the
of
s 0 and
Co+
on Y2,
and
As a solution of our following equations
differential
dS
"=
_ ;
may
are
two
new
S may be following
developed
have
four
in powers
By pursuing we may see that
of
x_,
102
from
of u,
reasoning
y2--(n,t+w,),
then
have
--"
a_o'
a;_-%'
constants. n2,
which
depend
on
s 0 and
if
B0,
I set
dSo u = _ = _o; dSo n2t÷w2-d_o --_'"
which
E, depending
a line
1_ t =
-4-
developed in powers of E and, as the first approximation
equations
we
dC
integration
dSo x, = _ = _o; dSo n_t--r-m_= d_o =y2; We
1£2l
;TEo'
'_" = -
It may be first seen that n I and be developed in powers of E.
In addition, E = 0, I have the
/99
s 0 and
dS "
17J I _
dC
m2
equations,
(IS II I l "4-
"' = - TiE; and
constants
zC_-q-...
canonical
dS --"
ml
two
80.
x, = 4r.,'
where
the
on
we s0,
may 80,
exactly v--(n_t+w=)
obtain nl like
x2,
t +
u, Y2
ml,
that
n2
given
and
v
t + w 2. in No.
127,
may
be
developed
in
powers
of
E1
The
same
will
I would in
powers
hold
like
to
C ±i{a_/+lT;_)
true
add
j
for
that
C±Injt-_).
¢_u, x I,
all
of
and
these
Yl.
quantities
may
be
developed
of ,
and
S -- S O may
If
we
be
set
developed
for
the
in
time
y_ --(nit the
two
powers
of
/loo
being ÷
m.,) =
z_,
v --(n21
q-m2)
equations dS
will
take
the
dS
form z: = _+2,
where
_2
and
_3
may
be
E,
[and,
for
example,
in
e_i(n't+r_'),
_0_
we
In cient
order
to
us
which We
to
apply
Let VII,
formulas
saw
now I
for
in
e±iy
the
theorem
compare
reiterated Chapter
of
_oe
-_n,t-t_,_,
z2,
z3
have
prove
the
(3)
powers
Vl_oe("_t+_J_
'
similar
z3 = _q_,
developed
_l_c" = 4_c".'+=, and
= z3,
2,
(
,+ -7 z_ + -_
and
the at VII
in
results the
30
obtained
in
_!I
presented No.
beginning
that,
_
+
""
)
B_oe-V].
postulate given
+
1.2
the
of
above,
to
equations with
this
vicinity
that
it
is
suffi-
(3). given
in
Chapter
chapter. of
the
periodic
solu-
tion xl =yt
= x_ = o,
103
the
variables
Xl,
Yl,
x2,
Y2
e_i_'qt+_,
may
_,
be
developed
A e'_] t ,
A 'e-n]
where A, A' are integration constants, stants, depending only on the period of characteristic exponents. We powers
The
two
have of
just
results
seen
clearly
that
these
are
in
same
t,
et
then
write,
for
We
It
is for
this
must
may
o
be
first
conand the
developed
set
. constants which may be /i01 may be reduced to n_ and
=
C_J_t. cInJ--n_It,
the second factor in powers of e, s0, B0. factor will then be developed in powers of reason
in
example,
_t_t
and may then develop addition, the second
of
t ;
variables
In addition, n I and n 2 are constants, but developed in powers of e, s 0 and B0, and which n_ for _ _ s 0 = B0 = 0. We may
powers
n_ and n'2 are absolute the periodic solution
agreement.
A = 4 0e½,
in
that
we
saw
in
Chapter
VII
the
time
In t. t and
its
powers emerge from the exponential and trigonometric signs, which could have led to a certain amount of difficulty in certain cases. The preceding analysis shows that this difficulty was entirely artificial. If I shall
I now wish to compare consider the curves
out
result
y, = 0(,,,),
with
those
given
assign a constant value developed in powers of e±Itt,t
form
When n2t + _2 is varied, which I described at the
only
In conclusion, valid from the
104
XIX,
z, = ;,(¢_)
whose definition I presented at the end of No. 273. the equations for these curves, I need only take the x I and Yl and and x I may be
in Chapter
to a0,
B0,
nl
In order to expressions t + _i.
obtain of
Then
Yl
have
the
+_3_ I .
it may end of
be seen that No. 273.
the
curves
I should point out that all of these results are formal point of view. The series only converge in
the case of asymptotic solutions, by
for which one obtains the equations
setting
I mean
by
this,
setting
CYst='-or even
CF e
setting _0_0,
I mean
by
this,
ID'2
_-
--
setting
d77< ,= o, where
A and 276.
A'
designate
Let
us proceed
degrees of freedom. different manners.
of the
_¢ _
The
the
= .,,,
finite
to
the
constants.
case
preceding
in which
results
may
In order to explain this, it is sufficient freedom. It may happen that we may wish to vicinity
of
a system
of
invariant
which play the role of a generalization the sense of that given in No. 209. It may a true
also
periodic
happen
that
we
wish
there be
are
more
generalized
202
in
two
two
to assume three degrees study our equations in
relationships
of
the
to study
periodic
them
in
solutions,
the
vicinity
in
of
solution
In the first case, there are four invariant relationships linear relationship between the mean motions, a relationship have represented in the following form, employing the change of No.
than
and one which we in variables
if necessary _l=O.
two
In the second case, linear relationships
sented
in the
following
there are five invarlant between the mean motions,
relationships and which we have repre-
form #11_0
l
_=0.
105
Weshall begin with the first F = _'F'; The equations powers of c,
x,=.x_,
remain canonical in the following
case, and we shall set y,= _,
equations, form F'= F_+¢F_+
We
then
x,= _'x;, and
xa=_'z;
F' may
be
developed
in
....
have F% = h,z% + A,='_+ A =',, + 2 S z;y; + C/,',
or,
removing
the
accents
which
have
Fo_h_x_+h3_3 The functions h2, h3, A, with the period 2_ with
become
useless,
we
have
/10__!
_-An_+_l]_2yiq_Cy_.
B, C depend only respect to these
I am going to perform the Everything which I have stated point of view.
change remains
on Y2 and Y3, two variables.
in variables valid, but
and
are
periodic
of No. 274 again. only from the formal
In order that I may apply the principles of formal calculation, it is necessary that there be a parameter with respect to the powers of which the expansions may be performed. This will be the parameter _. F and, consequently, h2, h3, A, B, C may be developed in whole powers of _. I should add that, for _ = 0, B and C may be reduced to 0 and that h2, h3, A may be reduced to constants which I designate as h_,
h_
and
Let
us
A O. try
to integrate
the
WF = --h_, I shall
try
to perform
integration Y_---Y'_,
following -_?F = -
equations h3.
in such
a manner
that
Y3--Y'3
are periodic functions having the period yh and y_ which must themselves have the
27 of the following
The quantities n 2 and n 3 are constants which of p; _2 and _3 are integration constants.
106
(i)
may
be
two new form
variables
developed
in powers
Equations (I) then take the form
We
shall
<6")-
dy3
(2)
k3.
set hi :: D,° -_ ,_hCl)+ ,:h_2_-) -, l i ..Yi __:yo
1
_,
..Yifl
.-1
t- lJ-_,v_ :_)d-" ....
ni -"no +- lJ../,.i)+ _:n_, "2' t-...
and
we
shall
periodic
assume
that
functions
stants,
as
functions
we of
of
have y_
Y2
the
-(k)'s n
are
and
o_ Y3
(the
seen),
and
y_,
and
constants, h0's i
that
may
be
reduced
finally
except
the
that the y_k)'s are 0, the Yi s, which may be
for
(k)'s hi to
are
con-
periodic reduced
to
y_. In powers us
to
equations of p, and
determine
These
(2), we we shall the
shall equate the have a series of
y_k)'s
equations
may
and
be
.l,,I l ) ll_.
0
..,l #(._
n_k)'s
-l-
ll;I
.'.4!) r!Y2 "--7
-,--
.o #_ 3
"-
Ji
d ,,_o)
-7
,r.._,l _.Y_ _
• -i-
-I Jl /_ 1 --
, -r
.it2) , 2
I -:, " <:.r, <6:,
I shall designate every known that the y._0)'s and l assumed the We
o,s,
Yi
then
the
that
method.
dvO --" _. ---t-
d_O -+
/_.1 3
_ "-)-
))f2) "'3
Ji --_
d_.,o _J_ --_7"
<:y, <(r,
=
_
_
,rb --)
)
(3)
function by I. In the second equation, the n_(0) 's are known; in the third equa.L
y
, the
n
, and
the
n
1)'s,
and
so
on.
have ,.)',_
so
}--
<:::, <(r,
.h
tlon
by a recurrence
having similar which will enable
written
ci,<¢ i) 0
" ; " ! ",--
equations equations
equations
"'
--v'"
(3) may
be
'6"2' )
.0>_<_,y]+
reduced ,,_ '_")
_-
:lYe')'d ,4y?'
to
-,- ,,?, = ,
(3 ' )
,,_,) _
I07
f
to which
we
must
add
the
following
equations
' dy;
3 ,Iy_
which may be deduced from the second equation (3') are from the first equation (2). All of these equations take, for example, the
us it We
shall
the be
periodic
procedure
able
to
in Y'2
Having
which
satisfy and
thus
(2),
Just
may be integrated in the first equation (3'). The
contains (like., all the other functions shall set n_lJ equal to the mean value
ploying
(3")
'
:'-
we
our
_) of
as
equations
same manner. Let function _ which
is periodic in Y'2 this function, and
have
already
equation
by
a function
applied
as
functions
several y_l)
and by
times
which
Y'3' era-
we
is
Y'3"
determined
Y2
and
Y3
of
Y'2 and
Y'3, I may
set
x; = _, dy,
dy_
dy,
It
is
apparent
dy_
that
_;dy;+ _;dy;- =, dy,--x,dy,, which is canonical x_,
seen
x_,
zero, form Y2, '
an exact differential the equations is not
for
new
variables,
I may
i shall Let
us
that
the
therefore
make now
not
--n,x_--
n3x_=
coefficients
of
assume
this
that
assumption
integrate
the
and, consequently, that changed when one takes
instead
The form of the function F is that we have the identity
which shows constants.
108
Y_
is of
of x2,
changed
h, x2+
x_
h 2 and
from
equations
and
either,
Y2,
Y3.
but
it may
be
h_x_,
of x_
h 3 are
this
x3,
the
may
be
constants.
point
on.
reduced
to
/_05
dx!
dt
or,
which
is
the
dy_
same
dx,
dxl
+ h,
,@, h,_+h,_= us
try
to
satisfy
The
a
is
a
constant,
equations
Let
us
z
and
= --
Cy,),
(4)
dr, 2(h.,+By,).
these Xt
where
Byl),
thing
h,
Let
-- a(Az,÷
--_-=
=2(BzI+CY*)'
equations
_
eat
_'_
s
are
Yl
by
_
eat
setting
$_
periodic
functions
of
Y2
and
Y3.
/i06
become
develop
A,
B,
C
d;
dr
ds
ds
in
-- 'z(Bz
powers
+ Cs),
(4')
of
_
A _ .\o_-t_A_
in
the
following
form
_ ....
B =: Bo-t-I_B,+..., C == Co _- IxCl t-.... We In
the
should same
point
way,
let
out
that
us
develop
A 0 is h 2
hi= The
coefficients
other of
hand,
_in
of
let
the
us
these
the
S
In
We true
order
to
write
need
the
only
for Under
present
_
known
z,
s
a_l_+
that
B 0 =
C 0 =
0.
quantities.
and
a
in
On
the
increasing
powers
that
A in
AI, of
B
A 3, and in
....
!-....
equations
of
assumption,
_a_tl+
.%-i-Sl_-_S_It
the
expansions
this
are
unknowns
_,,_
expansion
recall
the
and
h 3
form a-:
shall
constant
h_ L'l_h_+....
expansions
develop
following
a and
in the
form
A 5,
...
a
more
are
symmetrical
zero.
form,
The
same
I
holds
C.
equations
(4'),
I
shall
equate
the
109
coefficients having similar powers of _. I shall employ (4' p) to designate the two equations obtained by equating, on the one hand, the coefficients of __7_ in the first equation (4') and, on the other hand, the coefficients
of _
in the second equation (4').
The equations (4' 0) and (4' i) will
determine al, sO and Zl;
The equations (4' i) and (4' 2) will
determine a2, sI and z2;
The equations (4' 2) and (4' 3) will
determine a3, s2 and z3;
and so on. I meanby this that equations (4' p) will determine sP and Zp+1 up to a constant, that they will determine aP, and will complete the determination of Sp_1 and zo_, which are determined by equations (4'
p-l), If
up we
to
a
recall
constant. that B0_lh=C0--C,=o,
it
may
be
seen
that
equations
(4'
0)
d: z h_ _ +/'_ h_ ds0
_
and
equations
(4'
i)
may
be
t h_ _,lzi
equations
(4' (/za
2) •o
may
dz, 77_
be
written
=o, (4'
O)
(4'
i)
_ h3o clso
,-F;_=°;
written
+
dst
and
may
be
h_ _clz_ --a_zl=-dsl
--
(Zi
2Ciso,
$0
written
dg3
(4 ' 2) i h_-_ydsl +n3-_).3--a_so--atsi= • o ds_
[the Y3'
ii0
letters which
are
_
designate zero
in
the equations
_zAozl+9.
known (4'
periodic 2),
but
Atzt+aBaso
+q_
functions which
in
I have
Y2
and
written
/107
v i
nevertheless
because
Equations proceed to two terms.
now the
they
will
appear
in
the
following
equations].
(4' 0) indicate that z I and s O are constants. equations (4' i), and let us equate the mean We have --a,._= --
a15
0
Let us values of
--_s0[C,],
=
2Ao_1,
which determines al, s O and z I . For al, we obtain two equal values having the opposite sign. Equations (4' i) then determine, up to constant terms, z 2 and Sl, which are periodic functions of Y2 and Y3" We can therefore assume that the following are known
/108
=2--[=_land s,--[sL].
of
Let us turn to equations (4' 2) and let us equate the two terms. We shall obtain two equations, from
obtain
a2, If
will
[z 2] and
the
mean
provide
This
these
two
of
the
z 3 and
functions
of may
Since we have two solutions.
have
be
values
procedure
values may
[Sl].
us with
of periodic
the mean which we
Y2
solutions.
terms
s2,
up
and
Y3.
then
found Let
two
be
two
to
equal,
constant
equations
terms,
in
(4'
the
2)
form
continued.
values
for
a :a,
_ ::
a ---- -- et,
z :'= +,
The
are
al,
o,
general
s=:
the
equations
(4')
will
71,
s ::: I,w1
solution
of
equations
(4) will
be
x, -- Ac'_'? -_ Be -.t_, yl
We
may
always
-: A Cae?t
assume .o'h-
We
will
-_- Be-att_l.
then
see,
as was
,_, 'F = ,-
the
case
in No.
x, = x', ? + y; ,,+,
and
if
H2,
K2,
274,
y, = x', ?, +/,
.r+:: .c_ -_- 1[_._.',-"_- 9.k_._,f',
-+. 1,2 y, ,
y, --=y_,
_:, ":: .r'j-t-
-_
Y:_ =)/3,
L2,
H3,
ll3:r't 2 -I-- _. 1(3 :_",/,
K3,
L 3 are
that
the
L:_y't _ ,
suitably
chosen
if we
set
q,,,
periodic
functions
Iii
N
of Y2
and The
to
Y3,
the
form
of
a constant, We
may
canonical F will
and
not
A and
always
form be
The rest of and 275, and The
three
the the
us
in
277. assume
x i and
s0,
J0
periodic
powers
Let that
and
B0,
The
of
_,
A
having
be
B will
: G
_
be be
changed. reduced
o.
of
e_i(
of
e ±i(nJ+='),
nl,
s0, _'0,
in powers
_i
and
and
the
of
_,/_,
and
n 2 may
/109
of
of
themselves
be
B 0.
to the second generalization study the equations in the
method, vicinity
and let of a true
form X'I
----- dvl "_--)'l
--'--,Y'I
---- O.
set F = _'F',
:r; = ¢x'l,
it
follows
s
y,=ty';,
x_ = _'x',, from.which
but
not
to O.
developed
constants
_,
us proceed we wish to
solution
shall
either,
reduced
:: coils{.,
Yi may
XI,=
We
be
will
computation may be performed as was done in Nos. following conclusion will finally be reached.
-_oeC,,t+_,),_e-_n,t+=,). developed
equations
set
variables
constants
the
changed
C will
Ii]
274
of
ys=ty,,
x,=,x,,
¢
y_ = y'3,
that F = F_+ ,F; + ....
The
equations
remain
canonical
equations,
and
we
have
Fo-'- h.; + +(_',, y',, G, y;), where + is a homogeneous quadratic efficients of _ and h are periodic
and
However, we shall
we shall remove simply write
the
form in functions accents
_i, _I, Xt2, _2" of Y3 = Y'3"
which
have
become
The
co-
useless,
F0 = hxa + _(x,, y,, x,,y,). Just
as
that
h may Let
112
in No.
us
be
274
and
reduced
now
276, to
consider
it may
be
a constant. the
equations
shown
that
we
may
always
assume
/
/
dy,
d--7=
-- h,
dxl --
dz,
d'l'
at
dO
=
dyt
d---;=-@q,' They
are
linear
will
have
the
The of
A' s
and
have
dyt
,
_
-d[ --
= --
dO
'
-_,
d4'
--_ ---'a-_,"
periodic
coefficients.
Their
general
solution
form
are
integration
may
be
constants,
and
the
_'s
are
periodic
functions
/ii____O0
Y3" It
readily
shown
that
expression
%i?k.,-- W.,?k.t+ ?i.,?x.._-%_?k.,
is
zero,
except
in
the
two i=T,
In
these
I may
set Let
It
two
may
cases,
equal us
then
to
now
set
be
seen
this
following k = 2;
cases i=3,
expression
k = 4.
may
be
reduced
to
a
constant,
which
i.
that z_ dy, --y,
dz,
+ z, dy,
-- y_ dz,
= _', ay', - y; _',+ _; a+,':--/, a_; + +ay_,, where
_
whose
coefficients
If
is
we
a homogeneous
then
are
quadratic periodic
form functions
of
respect
to
dl,
Yl,
_2,
3/2
Y3"
set z, = x_ -- -', 2
the
with
y_ = y;,
expression
113
\
\
will tions
be
an
exact
is not The
differential,
and
the
canonical
F is
not
changed,
form
of
the
equa-
changed.
form
of
the
function
only
F 0 may
be
reduced
to
where
h, We
A
and
shall
B are then
constants.
set i
xty'i
= ut,
]ogx_
= _vl,
]o_ Y'_
and The
the calculation may following conclusion The
xi's
constants
s0,
and
the
_0 and
be performed as was will be reached.
Yi'S
I/_e
The exponents E, s0, 80 and
nl, _0.
may
be
developed
B'0, of e_I_,t+_.J, and
n 2 and
'_,t__,,
n_ may
done
in
Nos.
in powers
of
275
E of
and
276.
Jill
three
of
_e-(_,t_
themselves
_,_
be
developed
in powers
of
This generalization may be directly applied when there are n degrees of freedom. The first case, which is the case given in the preceding section, corresponds to that in which there are n + 1 invariant relationships and one single linear relationship between the mean motions. This is what we discussed in Chapter XIX. The second case, which is what we are discussing in the present section, corresponds to that in which there are 2n - I invariant relationships describing a true periodic solution, and where there are n - 1 linear relationships between the mean motions. This is the case of asymptotic solutions which we discussed in Chapter VII.
iant
However, there are intermediate cases in which we have n - q invarrelationships and q linear relationships between the mean motions.
Then the xi's and the Yi'S may be powers of q real exponentials and
114
developed in positive or negative of n - q imaginary exponentials.
/ J v J
Relationship 278.
Let
us
assume
with
that
dri
the
(IF
_lt = @,' have
a periodic
solution
Integral
with
canonical
dxi
dF
-,If
,Ix_
the
that
h _i
is and
multiples
Let
an _i
of
be +
constant. developed
the to
the
preceding
x i
and
Yi
A's
are
and
In
the
addition, t +
exponents
a)
depend,
We
know
_'s
h,
that
of
the
period,
sines
by
and
A2ea,¢,
A_e-_,
are
in
such
cosines
a way
of
the
in
there
they
may
in
and
be
written
of
2n
powers
which
I
/112
shall
-
call
t
arbitrary
integration in
powers
coefficients
addition,
T.
of
These
on
the
integral
constants.
The
of
AIAII,
•..,
xi
and
of
coefficients
energy
A2AV2,
Yi
are
(just
constant
ex-
periodic
like
the
C.
invariant
x d_ dy, from we
which must
it
follows
that,
We
have
could
increased the
if
6
and
(2) y
are
two
integration
constants,
have
_\_0-
is
in 2
_,t
developed
an
periodic
A_,-t e-_"--'t
period
is
this
.......
expansion
having
near
developed
pairs, A_e
1
are
be
Ale_, t,
be
the
of
be
statements_
will
conjugate
themselves
functions
T
which
form:
_ may
(I)
n)
form
series
solutions
A,,-t e _,,-,e, The
Let
in
......
ponents
...,
yi_:+,(t+h),
According
which
(i=I,2,
h).
consider
following
quantities
may
__2_ (t T
us
solution. the
integration
equations
following
xi=?,(t+h), where
Invariants
write by
following
68,
this and
-,ty-
equation that
as
d_ /
dy
in
_o..,t.
another
a result
for
form. xi,
Let Yi,
us
Ai e_it,
assume "'',
that
B
we
increases:
115
=
\
_.ri, On
the
other
as
a result
hand, we
let
have
us
the
_i,
assume
_Aie _t,t, that
following
equation
may
_'yi,
We
second of the
increased for
xi,
by
6'y,
Yi'
and
that
"''
....
be written z ( _¢ _'y,--
The function
y is
increases _':ci,
Our
....
number is a constant. integration constants
obviously
(3)
_y, _tc,) --=o_,st. By this I mean that multiplied by 6$6'y.
it
is
a
have LXe_t = e=l(3:k + t_=).
On
the
other
hand,
we
have
d.ri ,_ ctxl _, dxi _AKe _t +Z---dci-- _,Ake ' - _, c, _:cl=-_Gou +_lh on + _/ t_l( Akj.j,t) = d( A'_e-=_') dx 7(d
It can
where
thus
El'
be
51.i'
seen
qi'
•_ dx .__ td(AKA]I)
that
_x i and
ql.i
are
6Aeat's and 6A'e-at's. In the Aeat's and the A'e-at's of _
(t + h).
The
,
6y i have
the
following
with
respect
to
linear addition, and the
expressions
It is sufficient to change be then seen that equation
.
of
they sines _'xi,
may and
form
6C,
_h,
be developed the cosines
_'Yi
may
_ into _' in those (3) may be written
be
and
to
the
in powers of of the multiples
readily
obtained,
of dx i and _Yi' in the following
It may form
D -4-Et + Ft-' = const., from
which
it follows
that
F = '2(_,.,._',._-are of the
116
developed
in powers 2n the multiples of _-
of
the
(t + h),
Ae at, and
_',._,.;) A'e-_t's they
are
and
the
bilinear
sines with
and
cosines
respect
to
._,JJ
_'Ae'_q The
first
term
must
be
_'A'e
has already provided must be satisfied by Thus,
with
D must
respect
to
be
the
?YC,
independent E
which which
at,
_
["
8'h. of
following
and
we
shall
have
--o_
us with certain the expansions
independent
t,
of
t.
verification of the xi's
It will
relationships and the yi's.
therefore
be
linear
determinants
t
L't,t. ,3'A). -- _'Ak L'_%, (4) A'_(SA_8'h--8'A._Sh) (_.(" _,'h --- 8' 8Oh)
(or with by
respect
to similar
interchanging The
A k with
coefficients
A'k,
will
depend
which
The time must disappear. can only happen if each or
addition
6A'e -at,
A new
series
279.
Among
or
on
be
will
A'e -_t
in
determinants or
determined
Aj with
developed
_
from
the
former
j).
in
powers
of
the
AkA_'s
, and
C. The exponentials must therefore disappear, factor Ae at is multiplied by a factor
6'A'e -_t.
of verification
relationships
may
thus
be
deduced
from
this.
real. tive. call sign, is
the
ak exponents,
Among the real However, since _k
from
between
I shall
positive Let
imaginary We
not
if
us now
it
limit
then
are
imaginary,
and
others
are
exponents the
which
conditions
are of
equal
generality
and by
have
opposite
assuming
that
_k
real.
cancel
exponent, will
two
is
some
exponents, some are positive, and others are negaI may arbitrarily choose an exponent which I may
or have
the
coefficients
to a positive the
following, :_._.=o,
Ak
which
correspond
to
an
exponent. if
_k
is real
A_o
117
\
and
if _k
In
is imaginary
addition,
I shall
set C - ("0,
where
C
0
periodic
the
series
AVV ._
value
s
the
under
we
be
t, h and
a certain
number
the
of
which
of
_'s
t and
of
will
corresponds
represent
VII.
They
exponents,
2n equalities
Between
invariant
and
in Chapter
to negative
have
functions
eliminate
constant
convergent,
studied
correspond
as
energy
to
the
consideration.
therefore
Yi'S
the
then
which
, which shall
of
will
solutions
We and
the
solution
Our totic
is
which
these
these
2n
relationships
the
include
as
arbitrary
will
express
constants
h
and
equalities, between
the
we
asymph and constants.
the _.
xi's
!115
If we
shall
have
Yi'S.
If a group of values of the x.'s and the Yi'S is regarded as reprei senting a point in space having 2n dimensions, these invariant relationships will represent a certain subset V of this space. This is what I shall designate as the asymptotic subset.
Let
us
reconsider
the
integral
invariant _dxi(O"
and let us subset V. the xi's satisfies
and
extend the integration In other words, let us
and the Yi'S, our invariant
I may
state
It
sufficient
is
this
is
that
apparent,
118
which
it
follows
over a portion of assume that every
which form a part relationships. the
integral
for
me
because
to
that
of
invariant
demonstrate
we A_
from
i
have o,
C = Co,
the
integration
will the
this asymptotic system of values
be
fact
zero. that
region,
of
i /
J J_ /
_Ak=o, _'Ak= which set
shows
that
all
of
the
_C --o, o,
_'C-----o,
expressions
(4)
are
We
cancelled.
can
also
c = co, A_),
A_=o •_h----A_=o
We
shall
have
quently, apply. The
for
No.
an
260),
a new
asymptotic
procedure
followed type,
obtained
a new
which
real
(for
imaginary
series
subset
we
arbitrary i.e.,
(for
of
followed the
ak).
asymptotic
to which
bilinear
having
_k),
for
the
the
invariant
solutions
same
and,
conclusions
invariant
(2)
(invariant
could
of
conse-
will
the
be
third
/116
form
ff n,t ,ax,, where of
B is
the
sign
a
function
of
differentials
the
dxi,
Xi's
dx k may
the
replaced
respect to Invariant
f B
is a
function
differentials It may
must
be
of
Yi'S
and
by
dy i
where or
one
dy k
or
under
two the
expression
will still be linear with still apply to a quadratic No. 260) having the form
the
and
E. The
where
(5)
be
be
linear
of
dxi,
the
B xi's
dx k may
seen
that
the
with
respect
be
the quantities (invariant of
d k, and
(6) the
replaced
Yi'S, by
and
dYi,
where
one
dy k under
or the
two
of
sign
E.
expression
to
the
expressions
t A_.A) 8Ak_Ak _A_,_A_, i
(4). This would the second type,
(4')
A),._C_h 8Ak _C,
119
\ \ N
and
to
A'k,
those
Aj
and
which
may
be
deduced
from
subset,
the
when
interchanging
A k and
A'j.
For every asymptotic (6), must be cancelled.
Another
in a
them
Discussion
280. This same different form.
study
may
For
shall
assume
example,
we
invariant
be
(5),
like
the
invariant
Method
pursued
farther,
that
are
we
while
dealing
presenting
with
it
a problem
of
dynamics, that the xi's are the coordinates of different points of matter of the system, and that the conjugate variables Yi are the tomponents of their momentum. We plan to study the integral invariants which are algebraic with respect to the xi's and to the Yi'S, and to determine and which
whether one is written
may
exist
in
addition
to
the
one
which
is
/117
known,
f f zdx,dyl We
have
seen
that,
the
Yi'S
may
and
consider
these
in
be
the
vicinity
developed
expansions
even By
in powers
of
equating
the
2n equations, t+h. We
which
C.
we
In
xi's
a periodic
in powers
again,
the energy constant corresponding so that the expansions will not but
of
but
of
we
shall
the
Aeat's, ....
assume
that
We
the
solve
they
yi's with
to
will
depend
these
respect
going
value
of
is zero, the Aeat's,
on
t + h.
expansions,
to the
xi's
are
the
to the periodic solution only proceed in powers of
addition, and
the
shall
solution,
we
Aeat's,
obtain
C and
have Az,e_d = _h', A _. e -_l,t
_
f_.,
C = ,I,_ (7) _ot-t-_lo--=-T--(t+h)=O.
We of
C and
uniform periodic
120
should of
the
point
out
AkKk'S.
functions solution.
of
that
a0,
It may
be
like seen
Ok, may that
the xi's and the Yi'S in In addition, the xi's and
be
fk, the the
developed _k,
@,
in
cosO,
vicinity Yi'S may
of be
powers sinO
are
the developed
to
in powers of the fk's, the f,k 's ' and 0 and according to the sines and cosines of the multiples of @. On the other hand, the expression (3) which would
corresponds correspond
developed
to the invarlant (2), or the to another bilinear invarlant
in powers
of
the
fk,
_k,
#'s
0.[_, _'% In
addition,
when
we
replace
as we
i.
In
the
exponential
form;
2.
In
the
form
cosine
of
3. Outside shall see.
of of
It must
enter
i.
In
is necessary the following
not order
the the
that
the
be
billnear
it
does
of
f'k, _,
of
or
0 by
t.
The
sine
of
these
not
with
respect
to /i18
"' £helr
time
the
three
enter
in
values
t may
be
multiples
exponential and trigonometric second degree and more). in any
expressions which form (5), must be
08.
fk,
this expression must be independent in three different ways:
and
similar of the
(7), introduced
of
(t + h);
expressions
(and,
ways. the
exponential
and sufficient that the expression be quantities which are similar to (4)
linear
form, it with
respect
to
aA_,fk--o A_f',, AJ;( _A-;'A - _J',_'A.), (8) (_,t, ;/o -- 8'q, _o),
where
the
coefficients
may
be
of
¢.
it @,
2. In order that t does is necessary and sufficient but only on its variations
developed
in powers
of
the
fk's,
not enter that our _O, 6'@.
in the trigonometric expression does not
_k'S,
form, depend
and
on
121
We must
o
outside the out that we
now
determine
exponential have
and
the
condition
trigonometric
_/h= e_a(_&.
under
which
expressions.
t does not enter We should point
__A_t _k),
_J_.-: _ ",'( _Ai.- A'_t _,.,,,.), ,; :' .-: _C,
(9)
_0 :-_ _% -_-t _o.
We may distinguish five types of terms in our expression, depending on whether they contain as a factor a quantity (8) included in the first, second, third, fourth, or fifth line of the table (8). Under we
shall
this see
assumption,
that
the
if we
five
replace
types
of
_fk'
terms
... by
include
as
their
values
a factor,
(9),
respec-/119
tively,
A%A)
(_A,
_'Aj
--
_Aj
_'Ak)
+ A%A) t [A_:({=k _' A/+ AkA_A/A) tt(_eik _'¢_]
+ A_t(_A,
be
Let with
seen us
terms
that
first of
It may
make
the
be
the
the
can
terms
type
that
_'ak),
+ A_,.A_tt(_ak o'ao -
_'a_ _ao),
t(_C _'"o -- _'C _xo).'
time
second
stated
_*tj
_'_o--_'A_.-_ao)
(_C _' _o-- _'C _o)+
It may
_'a, _As ) -- AI( _aj _'A, -- _' "t _A j,)] "-
enter for
or of
the
in
the
second
t 2 disappear. the
fourth
coefficient
power. They
may
only
begin
type.
of
t,(_=,_' as- _=s_',,,) must
be
zero.
in
actuality_
constants the
are
of
vanish
All
There
the
_k'
with terms
would
n - ] exponents
122
to
arbitrary,
exception
_'ej's
due
be si"
we
and
the in
the
fact
may
in
the
exception t 2 cancel,
an We
exception could
no
that
assume same of
the that
way
_'%
with
displacements
all
6ei's
the
it may
be
in
vanish,
assumed
the
with
that
all
the
.
the
if there longer
virtual
exception
were
assume
of
the
a relationship that
all
the
term
in
between 6_i's
the
J/
cancel
exceDt There
one,
are
unless
now
four
the
last
terms
in
of
one
the
itself
second
cancels type
too.
which
result
in
terms
t'(8_k8'_j--_j 8'=k).
For
purposes
The
4's are
of brevity,
developed
m I to designate table (8) :
I may
write
in powers
the
of
expression
them
the
which
in
the
following
' and fi
fi,
appears
in
of
_.
the
form
I have
second
employed
line
of
the
/120
m 2 may
be
deduced
from
m I by
interchanging
fk and
f'k,
603 may
be
deduced
from
601 by
interchanging
fj
f'j,
60_ may time.
be
deduced
from
601 by
making
same
cient
In order that
that
the
terms
in
these
and
two permutations
t 2 disappear,
it
at
is necessary
the
and
suffi-
X,- 4_- W+_,=o. ' (ll) If
will
this
condition
provide
us with
is
the
fulfilled,
following
Let
us now by
consider
pairs.
Let
terms the
of
four
terms
( 4_ - _ ) t '_k A'k[_
together
our
in
_'( "_J _'j)the
following
terms
t
_'_ _( Aj "_'j)]
fourth
type,
which
be
group
of
one
we
two
shall
group
terms
41tol + 42_, where is
41
the
602 is
and
42
may
expression that
which
be
developed
included is
on
deduced
in powers
the
by
fourth
of line
interchanging
C and of Ak
of
the and
the
table _
and
AkA_'s,where (i0),
and
changing
601 where
_k
to
-_k" In order
that
the
terms
in
t 2 disappear,
it
is necessary
that
+,--4, and
then
the
terms
in
t may
be
reduced
to
123
+_t[;(A_. A>.)g'=o --o"....
281.
Our
terms
in
t now
proceed
in powers
of
C,
and
of
the
AkA_'s
,
and according to the 6's and the 6' 's of s0, _k, C, AkA_. Wemust now make these terms vanish. I shall state that they are zero when we set C -o, without
assuming In
in
our
_C,
invariant,
(6fk6'f_ Let
that
let
6f_'f k)
D k be
that
d'C,
6'AkA'k,
B k be
becomes
which
A_A_ ----o,
that
when
the
and
D O that
which
set
the
zero. coefficient
of
the
term
C = AkA'k =0.
coefficient
the
are
which
we
f'_.(,;fk_'O becomes,
6'AkA _
of
the
term
containing
/121
-- 50 g'/,)
coefficient
of
the
term
containing
+
D_(_C
_'%--
_'C
(_,z, _'o -- _3o_'¢,). becomes.
We
must
also
+
of
For C and
purposes
of
have
identically
X D_-[_(Ah.A_)_'ao--_'(A1,
brevity,
A_.)_]
let
us
write
Yk
instead
of
o_o)
=
Ak_k,
o.
Y0
instead
O(u, v) instead
We
_u_'v--_v_u;
of
have EB_O(xz.,
7k)
+
ED,0(_.,
ao)+
Do0(7o:
ao)=
o
or
EE B,_ dek
J may
Under takeon When
the
setting
ttxo
sign E or the values the
d_o
EE, k may take on O, 1, 2, ..., n-1.
coefficient
of
3(_j,
the
values
Yk)
equal
d_z _ B_ -d_ _ Dk d_0 -B_ -d,[;. d',(_ d_,j +D_
124
d_0 ----o.
1,
to
2,
zero,
...,
n-1
we
obtain (12)
and
F
j
By
setting
the
coefficient d_
of
j
equations
indicate
yj)
G.y_y ° +Do
_
an exact We
set
constants.
¥j
= 0 in
but
The
a polynomial
_k'S
not
also
(12')
equations
(12)
and
is
an
(13)
order
expression the
as
this
is
may
exact
then
be
The
d_, s dy
are
there-
linear functions of the y's. may be developed in powers of /122 have just obtained is only valid if we stop the expansions In addition, the B's and
therefore
the
of D's
exactdifferentia]
degree. investigation
functions
functions
(13)
second
to carry
only
as
of
(12').
further,
let
us
express
the
of _o,
7,,
_o,
7t
-",
Y_-l,
of ....
,
_n-t,
In order to avoid any confusion, let us employ tives chosen with respect to the new variables, the derivatives chosen with respect to the old It
have
differential.
must
In
we
D;_0)d_A
if we neglect the squares of the y's, and the _'s at the terms of the first degree. are
zero,
that
fore constants. The _j's are therefore In actuality, as we have seen, the _'s the y's. However, the result which we
of
to
=o.
-- D0=0dy0-e x(Bk=_is
equal
d2o
dcx o
B_ _tTo These
3(¥0,
seen
differential,
3 to designate the derivaand the d's to designate variables.
that
which
entails
the
following
conditions (14)
dTk
_'li If these
one
knew
equations We
can
the would
express
relationships allow ZDjyj
us
as :xo,
while
between
to determine
a function "(t,
"_2,
• •.,
of
the the the
a's
and
the
coefficients
y's, B i.
variables
"ID*-t
writing
125
\
ED/7 i = Eo=o + XE_-/k. The
Ek'S
will
be
given
by
the
equations (14')
d=k
and
E 0 may
It certain
be
chosen
arbitrarily.
is necessary that conditions in the
equations case of n
(14) > 3
be
compatible,
which
requires
d=____ .O=i d,i = O=__L, dx__f7 d_k. &fj
0_[i
always
These conditions (15) will an integral invariant
f If there tically
for
values of the invariants.
are
several
the
periodic
coefficients
dyl,.
e_k
always
d"{l
be
(15)
d'_i
fulfilled,
since
there
is /123
X dxi dyi. integral
invariants
solution B i and
under E i must
which
do
not
consideration, correspond
to
vanish
a system each
of
idenof these
If equations (14) have q solutions which are linearly independent, we may calculate the corresponding values of the Ek'S by means of equations (14'). Since E 0 remains arbitrary, we shall have q + 1 systems of values, which
are
linearly
independent,
of
the
coefficients
B i and
E i.
We may therefore have q + 1 different integral invariants (if the periodic solution under consideration is not singular, with the meaning attributed to this word in No. 257), but we cannot have any more. 282. there may different invariant,
I stated above that conditions (15) were definitely fulfilled; still be some doubt on this point. If equations (14) have q solutions, we may have q + 1 invariants. If there is only one we could assume that q = 0. The presence of a single invariant
X_,dy, would tion.
not
This
enable
us
is
doubt
I would
126
the
first
to
like
state
that
which
I wish
to note
equations
that
to in
(14)
definitely
have
dispel. the
case
of
the
three-body
a solu-
J
problem,
there
In
Volume
this
are
not
I,
one,
but
Chapter
IV,
and
we
two
we
integral
studied
invariants.
the
variational
equations
of
problem. On pages
170
172
obtained
the
_, _dV m --_717
following
integrals (i)
_ = cona.
(2) X(2x_ In
the
same
+y$)--
way,
we
\_...._ m
could
obtain
_y,,, m __av _ X(_x_/+
Let then
us
multiply
y_,)_3t(_y<\--_ (2')
by
_,= const.
(i')
m -- EdV7_ _') = co,,st
(1),
(1')
by
(2),
(2')
and
let
us
subtract.
We
have
.
dx --E\ (y_m The
first
term
is
(16) d,') d__'
linear
X(2x_+y_)-----const.
with
respect
to
the
determinants
having
the
form
' We therefore may
deduce
and
it
In
the
is
283. not.
as
well
solution
have
from
may
Is Then as
it
case be
all
Therefore, under
'
bilinear
the
three-body
of
of
integral
a new
stated this all
an
that
the
variational
in
with
(15)
general which we
the
exception
the to
we are
the
attribute the
equations,
and
we
invariant.
problem,
conditions
Ek'S,
when we consideration
of
integral
still true the coefficients the
h-.%
'_
values and
xi's
therefore
have
at
least
q=l,
fulfilled. case? Let have called of
us Bi
assume that it must be zero,
E 0.
corresponding the yi's,
to i.e.,
the when
periodic we set
C = MA_ = o, the only
coefficients the
terms
of
the
terms
in
6fk6'f' k-
6f_6fk
must
vanish,
and
in
127
\ k_
f_ (_f,_'e -- _e _'/k) (_8'e
- _e_'_).
remain. Our
invariant
must
therefore
vanish
when
we
have
_e_- _'e=o. This
is
not
the
case
for
the
/125
invariant
Ed_idy_ to
which
the
Let
We
following
us
expression
corresponds
set
must
have
an
_0
:
Zai_xi
o'0
:
Zat'_'X
equation
+
of
E(_xi_i--_yi_'x_):
Vb
i +
i_yi,
Zbi_'yi.
the
form
E(aiSxi+
bi_yi)E(e,_'xi÷
--E(ai_'zi+ However, with
this
determinant We
284. several
must
is i,
therefore
Let
us
impossible,
and
now
the
conclude
try
since
second
to
is
that
ei_'yi)
bi_')'i)Z(¢i_£i+
the
first
a bilinear
whether
term
(15)
),
is
form
conditions
determine
ei_fi
a bilinear
with are
equations
always
(14)
solutions.
Let
be
these
two
solutions
Bz,
B2,
....
Ba.
w,.
w,.
....
B;,
and
let
us
Bk
and
then
the
two
assume
Bi
equations O"zt,.
O_i
= 8, B'k Oxk
128
02,
that
we
do
not
form
determinant
have
O.
fulfilled.
may
have
will
imply dTl
Then
the
d_
indices I_
will
be
divided
different
----o.
into
values
for
..._
51
a certain the
number of groups, as many groups Bi _F. Two indices will belong to
ratio
l
if
group,
In
they
order
correspond
that
i and
_k
to
the
depends
on
k belong
to
same
7i
the
value
(or
_i
the
on yk ),
ratio
_-_..
it
necessary
is
are
In order to formulate these ideas clearly, let us assume only two groups containing the indices, respectively,
will
dePend
only
same
of
indices
p ÷t, p+2,
the
there same
/126
RJ
the
Then
as
that
group.
....
n--i.
....
T_,,
that
there
on _o,
"i,
y-_,
and
will
depend
only
It then appears dent groups, in products
(or
AjA_
on
that such
the characteristic exponents ak form several indepena way that the ak'S of one group do not depend on the
corresponding
to
another
group.
The periodic solutions for which this condition will for which there would be one relationship between the
called
particular We
therefore
In
order
that
be produced _k'S) may be
solutions. arrive
at
the
following
there
be
another
conclusion:
algebraic
invariant_
in
addition
to
129
those
which
tions
be
with
the
not
occur
we
know,
it
particular meaning
I shall in
would
be
solutions, given
in
not
try
the
three-body
to
necessary
or
No.
thaL
that
they
all
all
be
the
periodic
singular
solu-
solutions,
257.
demonstrate
the
problem,
fact
but
that
this
this
would
condition
seem
to
be
could very
unlikely.
Quadratic 285. of
Let
view,
where
us
i.e.,
F
is
Let
the
now
study
a quadratic
us
the
integral
quadratic
invariants
form
/127
Invariants
with
invariants having
respect
to
the
the
from
the
same
point
dx i,
dy i.
form
differentials
set F = Elidx,.dxk,
where dxidx
the k
may
We (3)
H's
of
are
be
may
replaced
then
No.
functions
of
in
write
the
certain
the
x's
and
the
y's,
by
the
product
terms
following
equation
which
and
where dxidY
is
the k
similar
product
or
dYidY
to
k.
equation
278
v I1 3xt _,rk = const.
On
the
other
We
may
then
hand,
write
we
find
equation
in
No.
(1)
in
(i)
278
that
the
form
D -_- El q- Ft_ =: contr., where
D,
sines
and
quadratic
E,
F may
cosines with
be
developed
of
the
respect
in
multiples to
must
therefore
of
the
(t
+
gA'e-_t,
_C,
and E,
F
D
must
of
the
are
the Lkc_t,
We
at' A '_t_ Ae s , e and where D, h)_
powers 2_ of T"
gh.
have E -= F = o,
and,
130
in
addition,
D must
be
independent
of
t,
which
shows
that
be
/128
linear
with
respect
to
the
following
expressions
A_ _Ak _h, _C _h, or with A k and
respect A_,
The
to
or Aj
the
and
expressions
coefficients
286.
Let
the
the
expression
must
satisfy
by
their
us
same
will
of
i.
It must
in
to equations
ng
as
be
developed
reasoning
follow
expansions
be
that the periodic constant).
return
line
the
may
be
deduced
powers
of
by
interchanging
A_.
and of C (if one assumes zero value of the energy
pursue
which
(7)
given
as
given
conditions
when
functions
linear
solution
of
with
the
respect
"J
¢
fk, to
the
280.
xi's
f_,
products
Ak_ k
corresponds
in No.
in No.
the
the
and
_ and
to
280,
and
We
shall
Yi'S
the
let
us
find
are
that
replaced
O's;
following
quantities:
J
.r', f; af,. _:[,.
(8')
_,I, _0,
S'_ _/h.;o, where
the
the
coefficients
2.
It will
3.
If
time,
We included
not
these
Let find
that
given
in
us
the
in
on
the
in
0, but
are
Table
terms
on
fulfilled,
exponential
equations
following
powers
only
form
determine the condition either the exponential
consider
the
depend
developed
conditions
neither must now outside
are
(9) again will
of
the
fk_k'S
will
in
the
from
Section to
the
not
trigonometric
under which the or trigonometric
correspond
of
_.
_0.
expression nor
and
No.
include form.
time is not terms. 280.
different
We
shall
terms
!129
(8'):
131
\ %
\
+ Ak 3.), A7 A_ t _-_.
_=j,
A_.?_Ak _C ;- A_.A_t_
Let
us
first
The
entire
make
the
group
of
k _C,
terms
quadratic
form
The
coefficient
of
terms
could
four
terms
m2,
where In
must
is
_,
...,
be
zero.
6_k_ajt2
a quadratic
_n
must
with
respect
to
t.
therefore
introduce
form
the
be
product
zero.
However,
t26_k6aj;
these
there are
the
in
For ml,
which
t 2 vanish.
terms
_o,
are
in
these
This
(i0')
purposes m3,
41,
order
of
_4.
42,
43
that
brevity,
The
entire
and
44
the
let group
may
be
coefficient
us
designate
of
our
developed
of
these
four
in
t26ak6aj
four
terms
may
powers
vanish,
of we
expressions then
the
must
be
fkf'_s have
by
written
and
of
identically
+, + % + % + _, = o.
It
In the same way, arises from terms
the in
coefficient
_A.el;, For _1,
where
m_,
purposes m_,
4_, In
and
_2,
order
of the
4_ that
may
brevity, entire
be the
of
f-
group
developed coefficient
k
must
vanish.
/? V_.
us of
t262_
designate the
in of
these
three
powers t2_2a
terms
of k may
the
three
expressions
by
by
fk f''k s
vanish,
and we
of
must
_. have
(Zl)
132
/130
_.
For
the
periodic
solution,
A=A
we must
have
=A =A ...../. ,=/__,_o.
All the terms including as a factor one of the expressions appearing the 2nd, 3rd, or 4th lines of the Table (8') must then vanish,
on
because The periodic
each
of
these
expressions
includes
only terms of expression _ solution are therefore the 8fkSfk,
Equation the
term
The power,
(ii)
_6fkSf
_ must
that
also
Finally, all the terms the term in 6_ 2.
body
In the problem
general problem which we have
problem, and but no more
I may
8,1,',
vanish.
We
a factor.
vanish
for
write
invariant
the
80_.
_I contains
the
of H vanish
fkf_ then
the planar than one. energy
is nothing
for
of dynamics, designated as reduced
equation F
This
8,>_0,
not
as
as
have
a factor. only
the
Therefore, terms
it in
else
_
cons[
in
the
periodic
just as in the the restricted problem,
the
we
have
following
in
the
fact that this third term is the only one which zero. If it is zero, the second term will also that it is the only one which includes t.
except
reduced variant,
which do terms in
_k
first does not include t, the second includes and the third includes it in the second power.
Due to the it must be to the fact
t 2, due
shows
fk or
first
includes be zero,
solution,
case of problem,
the threethe general
a quadratic
in-
form
.
than
f ¢_l_, and
the
this
of
term
If there invariant
in
_2
which
does
not
vanish
is a quadratic invariant, must vanish for all points
In other words, the meaning given
this periodic in No. 257.
corresponds other than that of the periodic
solution
must
be
to
this
invariant.
which is known, solution.
singular
in
the
sense
133
/131
v
There
would
be
an
exception,
were not independent of each between them. In this case, respect
to
the
n exponents
other, but if there were one the coefficient of t 2, which
ratic
form
could these
vanish without all of its coefficients being n variables will no longer be independent.
To addition
with
if
the
relationship is a quad-
n variables
zero,
since
sum up, in order that there may be other Ruadratic invariants_ in to those which we are acquainted with, it is necessary that all
periodic
solutions
It
is very
be
singular
unlikely
that
or particular. this
will
be
the
case
for
the
three-body
problem.
Case
of
the
Restricted
Problem
287. We may conceive of another discussion method which we shall only apply to the case of the restricted problem. The discussion presented in No. 257 has presented the possibility of two quadratic invariants, of which one is known. Let us assume that these two quadratic invariants exist, and let _ be According
the quadratic form corresponding to to the preceding statements, _ may
A _f_, On the quantities
other
hand,
_
is
/_V?,
coefficients Following
are
_,t,_o,
o0-,
a quadratic _.c_,
whose
" "
are
the
the
variables
_2,
algebraic
_,,
one of include
form
;+,. with
respect
to
the
of x I , x2,
YI'
Y2"
_,
functions
x i and
these invariants. terms in
Yi which
we
shall
select.
/132 In
this
p_oblem, which I have called the restricted problem, two of the bodies describe concentric circumferences, and the third (whose mass is zero) moves in the plane of these circumferences. I shall refer this third body to moving axes turning uniformly around the center of gravity of the first two. One of these axes will constantly coincide with the line joining these
two
first
bodies.
ordinates
of
the
Yl
to
designate
134
and
Y2
third
I shall body the
with
use
x I and
respect
projections
to of
x2
to
these
the
designate moving
absolute
the
axes, velocity
coand on
the
V
moving
axes. Let
us
then
set ,b :-- F + (o G,
where in
F
the
velocity The
and
G
designate
the
energy
motion,
and
where
absolute of
the
equations
two
take
first the
bodies
The
integral
Volume
I,
Under
this
_ = No.
is
page
that
which
the
the
four
assumption,
our
entails
four
could
be
Let we
shall
All
the
x
Sxi,
and
and
us
one
consider
function rotational
center
of
gravity.
o
than
"the
Jacobi
integral"
y ..--
_
will
a quadratic
form
in
_y:,
algebraic
are
be
in
related
by
x i the
and
Yi"
If
we
assume
One
of
them
relationship
cons[.,
condition
_
no
will
point
longer
become
of
the
f_=f_=
o.
a
be
independent.
ternary
periodic
quadratic
form. For
solution.
this
point,
have
the
expressions
_,_,
If
Z)',,
be
$y i will
eliminated,
common
else
expression
will
following
variables
et'b dxl
.....
nothing
Lr,.,
'I_
which
their
dj.j = dt
our
coefficients variables
area angular
23).
Z.r,, for
the the
form
ct,P -- J)'i '
eonst,
9,
and
designates
around
canonical
d.ri dt
(see
function _
we
set
6_
=
0,
(i)
will
therefore
_o_,
_,p_o
and
they
will
_, _ Therefore,
for
The
group
entire
a point
of
all
and of
terms
vanish
with
the
exception
of
_,1,2.
vanish
with
the
exception
of
_O'.
the
periodic
for
t 2 will
solution,
therefore
let
be
us
set
reduced,
for
135
/133
this
same
(see,
point,
supra,
to
Table
10')
and,
since
fl
=
f'l
= O,
may
be
reduced
to
Ct:_%'. The not even
terms
in
t 2 must
vanish.
The
latter
is
vanish for the point under consideration; when the condition _ = 0 is not imposed,
provide
tion,
terms
in
However, we have
the
only
one
which
all the others because _6@ and
does
are zero, 6_ 2 do not
t 2.
_a 0 is not
also
zero.
d:% d_o d/, - _-_ -
For
one
point
of
the
periodic
would
assume
that
having
the
solu-
d_o _ =o,
da0 but is a which
we
cannot
be
continuous does not
sure
may designate Consequently, in Keplerian
from
terms
which
in
it
However,
of
it
I
The
having
infinity occur.
Nevertheless, which body, i.e.,
of
may
-
de periodic
be
noted
only
This
solutions da0 _--
that
by _ as a factor, it may be noted motion. t 2 can
0.
vanish
i.e., that
if
includes
the
the mass 6a 0 vanishes
we
same
there period,
small
quantity
of for
the second _ = O,
H
may
have
follows
this
latter
equation
would
indicate
that
be
reduced
to a binary quadratic form and, consequently, that its discriminant is zero. Thus, the discriminant A of H must vanish for every point of every periodic solution. 288.
However,
an
algebraic
relationship
such
/134
as
_==-O
cannot be valid, unless every periodic solution. If
136
the
relationship
it is
reduced
--O
to
an identity,
for
every
point
of
is supplemented by two other relationships F = _, (where
B and
functions arbitrary
y
are
two
arbitrary
which were designated algebraic relationship
G= y
(3)
constants, in
the
and
preceding
F and
G
are
section)
the
and
two
a fourth
II= o, the number of solutions whatever the constants
Yi
(4)
of these four algebraic B and y may be.
equations
will
be
Let us now consider a periodic solution, and the variables will be developed in powers of _ in the following form Xl
--:
the
s_me
way,
"_
x i and
-F'.
l_x_+ .... . '-:y_0+ t_.s'_
Iy, In
Xl
limited
F will
be
(5)
developed
in powers
of
_,
and
we
shall
have F
and
G and
H will
be
independent
_
Fo-t-
of
tsPi
state
....
_.
The quantity A remains. It may is algebraic in x i and Yi under the algebraically on _. If we
-,-
be stated that this function, terms of the hypothesis, also
that
is an integral invariant, we will be led to certain include the coefficinets of H , their derivatives, of the differential equations of motion.
We
assumed
which depends
that
H is
an
algebraic
function
relationships which and the coefficients
of
the
xi's
and
the
Yi'S.
We may assume that this algebraic function is included as a special case /135 in a definite type, not containing _ explicitly, but depending algebraically on a certain number of arbitrary parameters. The quantity f /_-H will not be an integral invariant no matter what these when these parameters take on certain special When
stating
that
f _/_-is
an
integral
parameters may be, but values dependinE on _.
invariant,
one
is
led
only
to
certain algebraic equations between _ and these parameters. These equations must be compatible, and it is apparent that the parameters will be obtained as algebraic functions of _.
137
k.J
The coefficients
of the form _ and A will
also be algebraic in _.
The equation A = 0 is therefore algebraic in _, and we may assume that it has undergone a transformation in such a way that the first term is a whole polynomial in _. Wemay therefore write Ao +
A =
would
In addition, A 0 will not vanish, A would contain The
by to
the the
function
expansions fact that
A must
l_i-_-
_
A2 _ ....
be identically zero, a factor _ which,could
vanish
when
the
xi's
and
unless A is. be made to Yi'S
are
If A 0 vanish.
replaced
(5). It may then be developed in powers of _ and, due the term which is independent of _ must vanish we shall
have _0(x_,y_) -_o
We
should
now
point
out
that
we
(2')
must
have
(3')
i IL(<°,y_) = _0,
where B0 and Y0 are constants. sufficient to recall that, for Keplerian
In
order
_ = 0,
that
the
this
motion
may
may
be
be
the
case,
reduced
it
is
to
motion.
Now,
for
example,
let
us
/136
take
1t =x_+x]--, and
let
us write
the
equation (4')
If we set _ = 0, we may then observe that the third body will describe a Keplerian ellipse. Let _ and n be the coordinates of this body, not with respect to the moving axes, but with respect to the axes of symmetry of this
ellipse. The
equations
of
the
Keplerian
ellipse
will
then
be written
(6)
$_== _,o+hCos',_-4$:cos27 r,, sin7 + r,_ sin2? .... + .... , The major
138
coefficients
axis
and
the
_k'
Bk will
eccentricity
of
depend
on
two
the
ellipse
constants and,
which
consequently,
are on
the B0
and ¥0- Weshall have where the meanmotion n I dependson B0 and where _ is a new integration constant. The intersection
will
of the ellipse
occur at two points which will = cosO,
We
will
then
_2 We
tions
is
a new
shall
(7)
be given by the equations
_ =-_sinO,
(7)
? = ± ?o-
have [ .r] =
where
(6) with the circle
and
_ cos(tot-+
_.)
integration
obtain
+
_2),
(8)
constant.
solutions
(8), which
-f- "t_sln(_ot
of the
equation
(4')
by
combining
equa-
yields
.r_ = cos 0 -_- _-]
[
,°
]
zo := cos -- _ ._ -(--?o -F-.,.1__,--_) + rJ2 It I (k is
an In
cient
arbitrary order
that
form
of
whole
that
the
B0
in
-_-mbe nl reduced
a fraction
that
such D may This
It may
(2'), matter
a way also can
solution
ratio
its denominator. solutions. Equations solutions, no
the
number).
that be
only
as
be
periodic,
commensurable. to
be
its most
seen
that
it Let
simple
is necessary us write
this
expression,
equation
(4')
has
and
and
suffi-/137
ratio
in
let
D be
the
2D different
(3'), and (4') must have only a limited number of what the constants B0 and Y0 may be. I may choose -_-_has nl large
occur
if
the as A0,
value
which
I desire,
and
consequently
A,
are
identically
H
is
I desire. and
consequently
if
zero.
and
Consequently, this form must
the discriminant having be reduced to a binary
the form form.
identically
zero,
139
it
It could be shown in the sameway that, in the sense of No. 257, is impossible that every periodic solution be a singular solution.
This has only been proven in a very special case, but it ble that this proof may be extended to the general case. 289.
is possi-
The form _ regarded as a binary form, must be reducible to
for one point of a periodic solution. The binary form will therefore be definite (i.e., equal to the sum of two squares) if the periodic solution is stable -- i.e., if the characteristic exponents are imaginary. It will be indefinite (i.e., equal to the difference of two squares) if the periodic solution is unstable -- i.e., if the characteristic exponents are real. Let us assumethat _ is very small, and let us reconsider equation
(4').
According to the given value of B0, we one is stable and one
be
the
corresponding
Let
us
principles outlined shall have at least is unstable. Let
values
of
the
O+
(?o--
'
in Chapter III (No. 42), for a _138 two periodic solutions, of which
constants
_I
and
_2-
set '=_',
nl
0 +--
and
equation
(4') will
give
(?o-_i)+_,=_',
us,
for
the
nl
and
for
the
first
periodic
/
second
nl
/
Without restricting the conditions of generality, _" > _' and that _' and _" are contained between form
1_0
_
will
solution,
be
we may assume that zero and 2___. Then D
the
definite indefinite definite indefinite
definite indefinite
which
shows
must
vanish
from
this
The since
for
it
every
these
that
form
the
2D
it
identically
H
must
may
for
x_:cos(_'+i_),
for
x_=c°s('_'+_D)'
for
_r_=:cos(,_'+JD),
for
x?=
for
._-__cos(#" + 2_.);
discriminant
least is
_z:T:_cos(#'+_),
times.
of Just
therefore
cos(_' i-2,_),
9,
considered
as
above,
as it
may
a binary be
form,
concluded
zero.
be
reduced
to
a
square
Therefore,
term.
equal
point
of
a periodic
solution,
it
must
vanish
for
all
/139
of
points. The
To is
that at
for
known,
same
sum at
line
up,
of
there
least
for
reasoning
is
no
the
would
other special
show
quadratic case
of
that
it
is
invariant the
restricted
identically
except
the
zero.
one
which
problem.
141
CHAPTER
POISSON
Different
ferent
XXVI
STABILITY
Definitions
of
Stability
290. The word stability has been understood meanings, and the difference between these
apparent
if we
recall
the
history
of
to have meanings
several difis clearly
/140
science.
Lagrange has shown that, if the squares of the masses are neglected, the major axes of the orbits are invariant. This means that, with this degree of approximation, the major axes may be developed in series whose terms have the following form
where
A, If
_ and these
B are series
constants. are
uniformly
convergent,
this
results
in
the
fact
that the major axes are contained between certain limits. The system of stars cannot therefore pass through every situation which is compatible with the integrals of energy and area, and furthermore it will repass an infinite number of times as close as desired to the initial situation.
This
is
complete
stability.
Carrying the approximation further, Poisson stability continues to exist when one takes into masses and when the cubes are neglected. However,
not
this
does
not
have
He meant that the major axes only terms having the form
the
same
meaning.
be
developed
may
demonstrated that accout the squares
in
series,
the of the
containing
h sin(_t -+ _), but
also
terms
having
the
form Atsln(=t+_).
The value of the major axis then undergoes continuous oscillations, but nothing indicates that the amplitude of these oscillations does not increase indefinitely with time. We
142
may
state
that
the
system
will
always
repass
an infinite
number
/ j
v
of times as close as desired to the initial situation. not state that it does not recede from it very much. The Lagrange
word stability as for Poisson.
does
not
therefore
have
It is advantageous to point out that the Poisson include one important exception: They the ratio of the mean motion is commensurable. The the
scientists
probability
It bility.
the
two
is
that
therefore
In order that three folowing 1.
None
concluded
they
are
from
that
commensurable
is
to provide
stability
three
bodies
can
recede
an exact
in
the
infinite
and if
because
small.
definition
of
sta-
three-body
problem,
indefinitely;
the
3. The system repasses initial situation.
the
If the third condition alone is fulfilled, first two conditions are fulfilled, I would
Poisson
an
may
for
exists,
2. Two of the bodies cannot collide with each other, and of these two bodies cannot desend below a certain limit;
tance
number
of
times
the
dis-
as desired
without knowing say that there
to
whether is only
stability.
A case has been known to exist for a long condition is fulfilled. We shall see that the filled
also.
I can
say
nothing
with
respect
This is the case given in the problem assumed that the three-bodies move in the the third is zero, and that the first two ences around the common center of gravity. shall
meaning
infinitely
there be complete stability conditions are necessary:
of the
same
we
theorems of Lagrange are no longer valid
it
advantageous
the
However,
call
this
the
restricted
291.
In the
order proof,
to provide I am now
to
the
second
the is
first ful-
condition.
of Section No. 9, where I same plane, that the mass of describe concentric circumferFor purposes of brevity, I
problem.
Motion
derlying
time for which third condition
of
a better going
to
a Liquid
/142
explanation present
of
a simple
the
principle
un-
example.
143
\
Let us consider a liquid which is enclosed in a vessel having an invariable form and which is completely filled. Let x, y, z be the coordinates of a liquid molecule, u, v, w the velocity components, in such a way that the equations of motion may be written _t,_ _ _
The components functions, of I shall depend
only Since
u, v, w are functions, x, y, z and t. assume
on the
x,
that y and
liquid
is
the
motion
which
is
other
is
an
words,
integral
the
(1)
I assume
steady,
in
to be
such
given
a way
that
u,
v, w
z. incompressible, du
In
_ _= __ dr.
we
dv
shall
have
d_v
volume
invariant.
Let us study the trajectory of an arbitrary molecule. I may state that this molecule will repass an infinite number of times as close as desired to its initial position. More precisely, let U be an arbitrary volume inside of the vessel, which is as small as desired. It may be stated that there
will
Let molecules U 1 at volume
the
be
molecules
crossing
this
volume
U 0 be an arbitrary volume inside of which fill this volume at the time
the time t, a certain U at the time nT. n
volume
The incompressibility of existence of the integral
U2,
...,
an
infinite
the vessel. 0 will fill at
the
the liquid invariant,
or, which indicates
Uo,
...,
time
is to
volumes
are
U_,
V be
the
total
volume
of
the
vessel,
V<(n-__OUo,
we
144
shall
of
times.
The liquid a certain volume 2t,
and
a certain
the same thing, us that all the 1143
UI,
U_
equal. Let
number
have
and
if
i-
V-( Uo÷ U, i- U:*-...÷
are
It all
is therefore exterior to
of
them,
U i and
It may will be
hold
be
M
since
the
U k,
for
stated
true
for
in
common
a point
point
impossible each other.
at
the
if
U 0 and
i_
have
U i and
Uk_ i
is,
at
M belongs
Uk. the
On
the
since
M belongs
it
to
is
Uk,
U k have
The
common.
a part
for
0,
in
example,
molecule
time
In the same way, the molecule is, at the time (k-i)T, at the hand,
in
at
common, k
which
a point
the
> i).
is at
same
Let
M
the
M 0 belonging
to U 0,
to U i.
k_
other
a part
(assuming,
to U i and
time
point
that all the volumes U 0, UI, ..., U n It is necessary that at least two
example,
that
U,,
at
the
and we
which point
time
must
is at the M0, since
0 at
a point
conclude
point M at the motion
the time is steady.
M I belonging
from
this
that
to
U 0,
M 0 belongs
to
Uk_ i • Therefore,
Uk_ i
Therefore,
it
U 0 and
U s have
Let as we way
that
the
in
part
U I, U 2,
I_0 and
U 0 have
is possible
a part
U'0 be
formed
and
points
to
in
choose
common.
the
q.e.d.
number
e in such
a way
that
common. in
common,
..., with
and
U O. We
b_
have
a part
this
part
in
in
let may
us
form
obtain
I_I, U_2, ..., with
a number
B in
such
I_0, a
common.
!
Let
_0
be
common. !
We may in
obtain
a number
Y in
such
a way
that
and
Lf0
l_#
have
a part
common. This As
procedure
a result, U0(p+l)
general, definitely,
the
According perhaps to U0,
U_,
then
be
is
part
of U 0, U"
of
be
part
of
When
U_ will volume
continued.
U (p) u
U_ p) must
to a well-known
several, to
may
therefore theorem,
or
perhaps
an
to U_
, ...,
and
infinity, to
U_P),
U'0, and the
number
become
there which however
U_' of
U_,
be
belong large
In
p increases
smaller
will
. ""
and
smaller.
at
least
at
the
p may
in-
one
same
point, time
/144
be.
145
X
\ This the limit itely.
group of points, toward which the
which volume
v
It may be composed different. For example, having a finite volume.
of isolated points; it may happen that
A molecule which will be of Us, at the time zero, will
of
U'
inside of be inside
E, will be in a measure when P increases indefin-
however, it may be somewhat E is a region in space
U_, and, of U 0 at
A molecule which will be inside of U_ and, at the time zero, will be inside of _0 at
consequently
inside
A molecule
of
U 0 at
which
will
the
be
side
of
U_
at
the
time
-yT,
of
U 0 at
the
time
-(a
+
U_' , U_ , U_
are
part
Since U 0 at
four
different
times
inside
the
same
way,
and
more
at
the
time
zero
will
be
will Since
equal E
the
is part
cule which, at the number of different There number
of
are
of
of
+
this of
inside and,
the
the
time
time
zero
-(B
+
will y)T,
be
in-
and
in-
molecule
will
a molecule
which
of U 0 at p different
multiples however
be
inside
of
T).
generally, inside
u_P),
consequently, the time -BT,
inside
B)T.
U_'at
U 0' at
of
consequently, the time -aT.
y)T. of U0,
negative
of
is
inside
previous
of times
T).
large
p may
be,
as
a result
a mole-
time zero, is part of E will cross U 0 an infinite times, which all equal a negative multiple of _.
therefore
times,
fl+
-(_
(multiples
In
(which
time
inside
side
uIP)
I shall call U_P) tends,
however
molecules small
which
this
cross
volume
may
the
volume
U 0 an infinite
be. q.e.d.
The
equations d= _ dy -- dz = dt tl
P
iv
become d,_:
when
t is As
146
changed
into
a consequence,
-t.
dy
They
according
¢1_ --=<(t, ----
therefore to
the
retain
same
the
reasoning
same which
form. we
have
just
employed to show that there are molecules which cross U0 an infinite number of times before the time zero, we should be able tO show that there the
are time
molecules
which
cross
U 0 an
infinite
number
of
The
preceding
U s, at
the
Due of
after
zero. line
of
reasoning
provided
us with
the
U0 is crossed by a molecule which, at the time zero, to the fact that it is inside of E and, consequently, of
tines
to
U'_ and
time
the
of
zero,
fact
I_B at
it will
that
the
it
time
is
be
inside
inside
zero,
of
of
U 0 at
E and,
it will
be
times
is part inside the
of of
which
E. U_
Due and
time
consequently,
inside
at
of U_
inside
and
U s at
the
time
and
inside
of U 0 at
the
time
-(= _ _)_ It will Since
therefore it
be
is part
of
inside E and
of U 0 at of
U_'at
two
the
times
time
-B_
zero,
and
-(_
it will
+
B)_.
be
part
of U" 0 at the time -y_, of U_ at the time -(B + y)_, and part of U 0 at the time -(_ + B + y)T, so that it will cross U 0 at three times
At
the
time
of U_.
At
it will
therefore
To
the
sum
up,
-y_
it
is
part
be this
part
of
molecule
- ==,
_,
the
B, Y,
will
coefficient
of -T
and,
consequently,
part
of
U_
and
U 0. must
cross
U 0 at
different
--7=,
times
;.,
.......... , ........... .. ,
is an
arbitrary
combination
of
the
numbers
....
Among all of these not only be inside It
U_
-- _=,
_-(=_-_÷y)_,
where
of
time
may
be
readily
times, there are now times when of U 0, but also inside of U_. seen
that
it
is
sufficient
to
the
select
molecule
combinations
147
which do not include the number _. The times at which the molecule will pond in the same way to the combinations the
number 292.
e or Let
the us
number
again
be inside of U_will which
do not
include
correseither
B.
consider
the
volumes
U0, U,, Us, ..., U#. For purposes of brevity, I would like is the conseNuent of that preceding it in antecedent of that following it. Thus,
I may consequents
U2,
U 3
will
continue of U
be
the
the
second
series
and
(1) to state that each the series (I) and
the
(i) beyond
third
Un,
consequent
compiling
the
of them the
of
U O"
successive
n U_+i,
I may antecedents
also extend of U 0
it
....
Uz_+2,
to the
left,
and
may
compile
the
in such a way that the molecules which are in U 0 at the be in U_ 1 at the time -T, and in U_ 2 at the time -2T. Under this assumption, I shall always volume of the vessel and k to designate an we have
use V to arbitrary
successive
time
zero
designate the whole number.
will
total If
kV<(n+OUo, there will be points at the same time.
If no time,
may
will I may
148
are
part
of k +
1 volumes
of
the
series
(i)
The sum of the volumes of the series (i) is equal to (n + l)U 0. point could be part of more than k of these volumes at the same this sum must be smaller than kV. We
which
which
therefore
have
obtain
a part
conclude
in
from
k +
1 volumes
in
the
series
common. this
that
the
k +
i volumes
(i)
f
/
Uo, have
a part For
in
U:q-_.,
U_:-_o,
we
example,
may
let
us
set
k = 2
obtain
three
will
have
a
part
+OUo,
volumes Us,
which
Uai-_.
common.
aV <(n and
....
in
U_,
common.
Ur
The
indices
_,
8,
Y will
satisfy
the
three
volumes
conditions o_n;
It may
be
deduced
o5_5n;
from
this
o_ySn;
that
Uo, have
or
a part
the
in
three
common,
and
that
the
U_-_, the
_ < _ <_.
three
volumes
Uy-=
same
holds
true
for
the
volumes
293. We saw above that there are molecules which cross U 0 an infinite number of times before the time zero., and others which cross infinite number of times after the time zero. I propose to establish the as
fact after
the
Let we may second
that
there time
U 0 be
an
are
zero
as many
which
an
infinite
arbitrary
volume.
always obtain positive, and
a part
in
U 0 before of
the
According
common.
Let
U 0' be
Uo,
time
zero
times. to
the
two numbers, a and _, the first such that the three volumes Ua,
have
cross
number
an
preceding negative
section, and
the
U_x
this
part
in
common.
T
at
Every molecule the three times
which
will
be
in U 0 at
the
time
zero
will
be
in U 0
the
last
is
positive.
/14____8 Of
these
three
times,
the
first
is negative
and
149
\ %
time
Our molecule will therefore cross U 0 at least zero, and at least once after this time.
Following the same procedure with two numbers b and B, the first negative that the three volumes
have
a part
in
common.
U "0 be
Let
once
before
the
U_ as with U0, we shall obtain and the second positive, so
this
part
in
common.
I!
Every molecule three times
the
which
will
--
and,
consequently,
in
U 0 at
these
times,
the
_'.j
in U 0 at
O)
the
--(_+_)=: Of
be
the
time
zero
will
be
in
the
last
two
U_
at
--- a=)
five
times
--B:, o, --b=, --T(,_ b)=.
first
two
are
negative,
and
are
posi-
tive.
least time.
Every molecule which will be in U_ at the time 0 will cross U 0 at two times before the time zero, and at least two times after this
This One every p
procedure could
form
moleculewhich
times
before
the
have
However, a group
part
of
also
Every molecule be inside of
all
will U_'
then with
will time
E
continued. U IV 0 with
U_,
be
in
zero,
u_P)
and
the
volumes
U_ p)
which,
is part
time
Therefore, O, and an
times
There both
of
all
it will infinite
at
the
U'o,
U ....
these
U_', and
the
least
time p
it
could
0 will
times
after
be
cross this
seen UO
that
at
least
time.
of U6, and so on. We shall therefore at least one point) which will be
the
at
at
at
U6 is part of U0, U_ of points E(containing
lJo,
since
be
same
time
l
,
time,
zero,
UIv'
wherever
will
be
p may
inside
be.
of E will
ad il_f.
volumes.
cross U 0 an infinite number of times number of times after this time.
are therefore molecules which cross before and after the time zero.
U 0 an
infinite
before
number
the
of
q.e.d. 150
294. The group groupEconsidered in single point (be that cules crossing U0 an
E, which was defined in No. 291 (just like the the preceding section) may be composedof a as it may, there is always an infinity of moleinfinity of times).
It may be composedof a finite number of discrete points.
number of points,
/149
or of an infinite
It could be assumedthat this group E has a finite see what the consequencesof this hypothesis would be. the group E defined in No. 291.
volume. Let us Let us discuss
I shall consider the series of whole numbers
which were defined in this section,
and it may be stated that we have _
The quantity U is the first commonpart with U0. U' is the first commonwith U'. 0
of the consequents of U0 which has a
of the consequents of U' which B has a part in 0
!
However has
' U 0 is part
a common
a part
in
In
same
the
part
common
way
The numbers _, never decrease. On
the
with with
of
U_, U 0.
we would
B, y,
other
6,
hand,
U 0 ' and
U'B is part
U B is one This
of
entails
We
clearly
... are
and,
since
we
the
inequality
of
if
U 0 which
U'B has
according
to No.
Ot <
Uo,
always
291,
increasing
we
_
<
at
least,
have
V I +
or,
V
U_o.,
O
0
have
if E has be,
consequents
therefore
Uo> u_>
may
the
Therefore,
obtain
V l-t-
of U B.
a finite
have
E is part
volume
which
u_>
I may
.... also
call
E,
no matter
what
p
E < U_ 3 of
U(p). 0
151
The numbers _, 8, y, ...
are therefore all V E
smaller than
/150
Therefore, they cannot increase beyond any limit, and we may conclude that, starting with a certain order, all the terms are equal in the series of numbers _, 8, .... Let pe order.
us
Then and
u_P +I) so on.
and
the
UI(P+I) will
the
the
group
same
all
the
u(P X ) will
E l be
E is at
that
u_P)and
and
Let
assume
time.
terms
have
a P art
have
a part
i e consequent of
points
E l will
of
which
be
are
the
equal
in
to %,
common
in common
starting
which which
will will
with
be be
the
U 0(p+I) ' u_P +2) ,
E. are
group
part
of U 0, U_,
of points
which
U_,
...,
ad
are
part
of
inf.
TT
UX,
U_,
group
UX,
...,
of points
at
the
which
same
are
time.
part
It may
also
be
stated
that
E is
the
of (i)
U_"_", U_P*_'....
at the same time, since each of the regions U0, U_ is only a portion of the preceding region. In the same way, E l is the group of points which are part of
uip_,u_p_,, ..... at
the
same
time.
However, Each
term
Therefore
U_ p+I)
in E
series is
However, finite volume. xth
consequent
sam__.__e volume. coincide.
(2)
is (2)
a part
we
a part is
of
a part
El,
or
assumed that E Due to the fact
E l must E cannot
also
of
be
therefore
ux(P) , and of
the
u(P +2)
is
a part
corresponding
coincides
with
term
of in
- (p+l) uI series
E X.
is a certain region in space having a that the fluid is incompressible, its a certain he
region
a part
in space
of E X.
having
Therefore,
the E and
EX
If we assume that E is a certain region in space having a finite volume, it must be stated that E coincides with one of its consequents.
152
. (i).
_j
295. I shall
Following
confine
are
myself
some
theorems
to discussing U_,.
be
those
numbers
consequents _
are
U_,,
of U 0 which
arranged
which these
have
in order
of
are
all
but
theorems.
....
U_:,,
a part
and
... in
increasing
obvious,
Let
common
with
magnitude.
O 0. We
The
shall
have
V
Then
let Uy,,
be and
the
_
with
small
consequents U 0.
of
I have
as possible.
U0,
chosen We
shall
Uy,,
...,
which
have
these
numbers
Uy_
a part y
Let U
us employ the to designate
and
shall
U8__
in such
with
a way
each
that
y_
other is
as
V O.
notation given in No. 291 once the first consequent which has
with U0, U_ to designate sequent of U_ which has _, we
common
have , _ %jI+y_L
employ
in
this common part, U_ a part in common with
again, a part
and let us in common
to designate the U_. If 8 is not
first equal
conto
have
will
have
a part
in common
with
U 0.
Probabilities 296. We saw in No. 291 that infinite number of times. On the U 0 only these
a finite
latter
precisely,
number
molecules the
of must
probability
there other
times. be that
are molecules which cross U 0 an hand, there are others which cross
I plan
regarded one
to demonstrate
as
molecule
unusual crosses
or,
the to
fact
state
U 0 only
that
this
a finite
more number
of times is infinitely small, if it is assumed that this molecule is inside of U 0 at the initial instant of time. However, I must clarify the meaning which I am here attributing to the word probability. Let _(x, y, z) be a positive, arbitrary function of the three coordinates x, y, z. I may state that the probability that a molecule may be located within a certain volume at the time t = 0 is proportional to the integral J=:f_(x,y,z)dxdydz
153
extended over this volume. Consequently, it equals the integral divided by the sameintegral extended over the entire vessel V.
J
Wemay arbitrarily choose the function _, and the probability is thus absolutely definite. Since the trajectory of a molecule depends only on its initial position, the probability that a molecule behaves in a certain way is a completely definite quantity, as soon as the function ¢ has been chosen. Under this assumption, I shall simply set _ = i, and I shall try to find the probability p that a molecule does not cross the region U0 more than k times between the time -nT and the time zero. Therefore, let o0 be a region which is part of U0 and which is defined by the following property. Every molecule which will be within o0 at the initial instant of time will not cross U0 more than k times between the times -n'T and 0. If we assumethat our molecule is within desired probability will be
U0 at the time zero, the (1)
qO
P== Uo Let
be the region
n first consequents common to more than
of o 0. It is not Dossible k of the n + 1 regions O'O)
0"i)
,'_,) )
...)
therefore
a
O'n)
since, without this stipulation, every molecule this common region at the time zero would cross more than k times between the times -nT and 0. We
to have
which was located in o 0, and consequently U0,
have (_t-I-,)¢o < kV,
and,
consequently, P < (n.t-
kV ,)Uo"
No matter how small U 0 may be, or how large k may be, we may always take n large enough, so that the second term of this inequality is as /153 small as desired. Therefore, when n tends toward infinity, p tends toward zero.
154
Therefore the probability is infinitely small that a molecule which is located in the region U0 at the initial instant of time does not cross this region more than k times between the times --_ and 0. In the sameway, the probability is infinitely small that this molecule does not cross this region more than k times between the times 0 and +=. Let us now set n = k3 + x. The probability that our molecules does not cross U0 more than k times between the times -(k 3 + x)_ and 0, will be smaller than • _V (k3+x+I)U0 It tends toward zero when k increases indefinitely. The probability P that our molecule does not cross U0 an infinite number of times between the times --_ and 0 is therefore infinitely small. In reality, this probability P is the sum of the probabilities that the molecule crosses U0 only once, that it crosses U0 twice and only twice, that it crosses U0 three times and only three times, etc. However, the probability that the molecule crosses U0 k times and k times only, between the times --_ and O, is obviously smaller than the probability that _t will cross U0 k times or less than k times between the times -(k 3 + x)_ and 0 -- it is consequently smaller than kV (k_+z+ The
total
probability
P
V P < (x_-.2)Uo
terms
is
t)U0 therefore
_V + (x + 9)Uo +'"+
smaller kV (,_3___x +,)U,
The series of the second term is uniformly tends to zero when x tends to infinity.
series In
tends
to
zero,
and
P is
the
same
way,
the
probability
cule does and + _. The function
not
same _,
Equation
cross
results instead
infinitely
U 0 an infinite
are of
(i) must
obtained
setting then
be
_ =
is
+ ....
convergent. Therefore the
Each of the sum of the /154
small. infinitely
number
when
than
any
of
times
other
small between
choice
that the
is made
our
mole-
times
0
for
the
i.
replaced
by
the
following
155
k_2
J(_o) P -- J((u0)' where
J(o
o 0 and
U O,
and
J(U O)
designate
and
the
integral
J
extended
over
the
regions
respectively.
I shall assume that not become infinite, have
does then
we
O)
the function and I may
_ is assign
continuous; an upper
limit
consequently, _ to it.
it We
since
may
deduce
the
following p<
No matter how small J(U 0) is, or how large k is, we may always take n large enough that the second term of this inequality is also as small as desired. We again obtain the same results which are therefore independent of the choice of the function 4. To sum up, the molecules which cross U 0 only a finite number of times are unusual, in the same way as the commensurable numbers which are only an exception in the series of numbers, while the incommensurable numbers are the rule. Therefore,
if Polsson
could
provide
an
affirmative
answer
to
the
sta-
bility question which was posed, although he had excluded the cases in which the ratio of the mean motion is commensurable, we have the right to state that the stability which we have defined has been proven, although we are forced to exclude the unusual molecules which we have Just discussed.
vides
I would llke to add that the existence of asymptotic solutions prosufficient proof for the fact that these unusual molecules exist
in reality.
Extension
of
the
Preceding
Results
297. Up to the present time, we have limited ourselves to a very special case -- that in which an incompressible liquid is enclosed in a vessel, i.e.i -- to employ analytical language -- the case of the
156
/155
following
equations ,Ix @ dz X- = -y- = -_£,
where X, Y, relationship
Z are
three
functions
dX
and such have
that
on
every
point
which
dY
of
are
interrelated
by
the
following
dK
a
closed
surface
(that
of
the
vessel)
we
lX + mY + nZ = o, where i, m, surface.
However,
n are
all
the
of
direction
the
preceding
more extended cases without reasoning leading to these
tions
Let
the
n variables
cosines
x2,
...,
Xn
are n dMX,
...,
Xn,
such
a way
that
equations
arbitrary,
fM
Finally,
I shall
assume
valid
even
the
the
closed
in
line
the
of
differential
equa-
(i)
_
functions
satisfying
include dxt dmt..,
=o, the
integral
invariant
dw,,.
(2) We
may
then
state
that
/156
(I) are such that, if the point a certain region V at the initial
the same role that enclosed), it will
that
this
dMXn
I shall assume that equations x2, ..., x n) is located within
instant of time (which plays recently where the liquid is this region.
still
satisfy
In addition, I shall assume that M is positive. equations (i) have a positive integral invariant.
(xl,
to
including
uniform
dMXa
(i)
normal
dw,, _-X'7'
_----7"+ "-g_ ÷...+ in
are
a thing,
,!x, dx2 _;t --- _X-,-= X7 ...... where XI, X2, the condition
the
results
changing results.
Xl,
of
the
the vessel played remain indefinitely
just within
integral
157
k.j
i,'.
extended
over
Under we
may
this
these
select
region
conditions,
the
initial
infinite
number
of ways,
infinite
number
of
at
random
not
within
cross In
respect
if we
so
times.
U 0, the
other
words,
if the
Nothing obtain
needs the
of
be
as
close
in
the
inequality
and
V
and
U 0,
U 0 designate
the
this
(x I, x 2,
of
times
will
as
are
word
desired
preceding
not
--
its
proof.
Xn)
will
infinitely
the
small.
-- with point
initial
For
an
is made
...,
unusual
above to
be
in
U 0 an
position
point
this
in V,
x n)
region
initial
conditions
position
example,
we
again.
V<(n-t where
the
...,
the
number
to
U 0 contained
(x I, x 2,
crosses
of
that
initial
changed
a region point
point
choice
I attributed
following
the
this
the
infinite
(xl, x_, ..., x ) will return an inflnite number of times.
may
consider
probability
U 0 an
meaning
that If
region
the
finite.
position
the
to
is
,)U0,
integral
(2) extended
over
the
regions
V
respectively.
The
same
results
may
be
deduced
from
this.
Due
to the
fact
that
the
integral (2) is basically positive by hypothesis, it will have the same properties as the volume, namely, when extended over the entire region it will be larger than when extended over only a part of this region. 298.
How
may
we
now
determine
whether
the point (x I, x 2, ..., x n) always at the initial instant of time? Let
Let
us
us
assume
consider
the
that
equations
region
remains
(i) have
V defined h_
where
h
and
h'
are
two
arbitrary
by
the
there within
an
is
a region
this
V
region
if
such
that
it occurs
/157
integral
inequalities
F < h',
constants
which
may
be
as
close
as
de-
sired. It is apparent that if these inequalities are satisfied initial instant of time, they will be always satisfied. The therefore closely satisfies the proposed conditions.
158
at the region
V
Application 299.
We
shall
apply
to
the
these
Restricted
principles
Problem
to
given in No. 9. -- a zero mass, the circular and zero inclination. If we refer the zero
the
restricted
problem
motion of two other masses, mass, whose motion we are
studying, to two moving axes turning around the common center of gravity of the other two masses, with a constant angular velocity n equalling that of the two other masses, if we employ _, n to designate the coordinates of the zero mass with respect to the two moving axes, and if we employ of motion
V to designate as follows
the
force
potential,
d_
d,,
=_"
we
may
write
the
equations
d-7 =_"
d_' dt --- 2n_'+ --
dV -_,, d;
n_+
(i)
dt It may
be
immediately
seen
that
they fd_
On
the
other
hand,
they
have
have
h
is
Since
integral
invariant
d,_d_'d_'.
the --V+
where
a positive
Jacobi
(2) integral
-- (_' -.,-,.,') + Z,,
(3)
a constant. _,2
+
n,2
is
necessarily
positive,
nl
V -t- --(_-I-
we
must
J158
have
_,_)> -- h.
(4)
2
We are
therefore
led
to
compile
the
foIlowing
curves
V -t- -- ($_-+"q') = const.
The first we have
term
in
relationship
(4) is necessarily
r;
where are
m I and their
finite
for
m 2 are
distances rI =
0,
the masses to the for
zero
r 2 = 0,
of
the
mass. as well
positive,
because
F_
two The
principal first
as at
bodies,
term
infinity.
of
and
r I and
(4) becomes It must
r2
in-
therefore
159
have vanish
at
least without
a minimum, and two points where its there being a maximum or a minimum.
two
first
derivatives
More generally, if there are n relative minima or maxima, will be n + 1 points where the two derivatives vanish without a maximum or a minimum.
there there being
However, it is apparent that these points, where the two vanish, correspond to the special solutions of the three-body which Laplace studied in Chapter VI of Book X of his Mecanique (Celestial Mechanics).
derivatives problem Celeste
Two of these points may be obtained by triangle on mlm2, either above or below the for the axis of the _'s. The third apex of of the solutions in question. All the other points satisfying of the _'s. It may be readily seen minima, and only three minima, when
the condition are located on the axis that the first term of (4) has three _ varies between --= and +_. The
first minimum is located between infinity located between the two masses m I and m2, infinity and the mass m 2.
The these
derivative
intervals, The
dV _+
since
n25 it
is
only the
constructing an equilateral line mlm 2 which we shall use this triangle represents one
vanishes sum
of
and and
the the
(for
three
mass ml, third is
n - 0)
terms
the second is located between
once
which
in
all
equations
each
of
increase. /159
dV
dV
d_-+_'_" = _ +_'_
=o
indicating that the first derivatives of the first term of (4) are zero, have only five solutions, namely, the points B 1 and B 2 which are the apexes of the equilateral triangles, and the points AI, A 2 and A 3 located on the axis of the _'s. We shall assume that these points occur in the following order
We must now determine which of these points and we know in advance that there are two.
correspond
to
a minimum,
We should point out that if we vary the two masses m I and m 2 continuously, any of the five points A and B will always correspond to a minimum, or will never correspond to one. One may only proceed from one case to another if the Hessian of the first term of (4) vanishes, i.e., if two of
160
the points A and B coincide, which will
never occur.
It is sufficient to examine a special case -- for example, that in which mI = m2. In this case, the symmetry is sufficient for indicating to us that the two solutions A1 and A3 must have the samenature, just like the two solutions BI and B2. It is therefore A1 and A3 alone, or B1 and B2 alone, which correspond to a minimum. Therefore, A2 does not correspond to a minimum. It can be seen that A1 does not correspond to a minimum. The two minima correspond therefore to B1 and B2. Let us now assumethat mI is a great deal smaller than m2, which is the case in nature. For sufficiently
large values of -h, the curve 2
will be composed of three closed branches C 1 encircling ml, C 2 encircling m2, and C 3 encircling C 1 and C 2. For smaller values, it will be composed of two closed branches, C I encircling m I and m2, and C 2 encircling C I.
For branch
values
leaving Finally,
symmetrical
shall
which ml for
are
and
m 2 on
even
curves,
still
each
The statements therefore put
still the
smaller,
we
outside,
and
smaller
of which
values,
encircles
shall
have
only
encircling we
below will only apply to the the last two cases aside.
B 1 and
shall
B 1 and
have
B2,
two
one
two
closed B 2. closed
/160
respectively.
first
cases;
we
In the first case, the group of points satisfying the inequality (4) may be divided into three partial groups: The group of points which are inside of C I, the group of points which are inside of C2, and the group of points which are outside of C 3. In divided of CI,
the
The group group
second
case,
into two partial and the group of
of of On
statements points points the
are are in
group
of points
satisfying
(4) may
groups: The group of points which points which are outside of C 2.
below
which which
contrary,
the
do
not
apply
outside outside the
first
of of
either
C3, nor C 2.
case
this
in in
the the
applies
are
be inside
first
case
to
the
second
case
to
the
to
the
group
of
161
k
points which are inside of C1 or to side of C 2 and, in the second case, inside of C I. In the
order
first As
to
case
the
formulate
and
the
region
V we
these
group shall
.+h + _>
We
shall
have employed definition of (_, n)
is
It region
assume
of
that
ideas
_/'
E is
more
points
take
_':+
the group of to the group
V_
clearly,
which
the
points which are inof points which are
are
region
let
inside
defined
by
located
the
inequalities
n__(_ + _,)> + h _ E.
(5)
small
and
that
h has
a value
within
the
curve
which
clear that, if initial instant
the of
f
the
region
V is
we
the the point
C 2. point time,
(_, n, _', n') is located in the it will always remain there.
In order to illustrate the fact that the results preceding paragraphs may be applied to the case which we must now show that the integral
over
consider C2.
in the first case. Finally, in order to conclude the region V, we shall impose the condition that
is then V at the
extended
us of
presented in the we are discussing,
_ dn d_'dr/
(2)
finite.
How may this integral become infinite? Due to the fact that the curve C2 is closed, $ and n are limited. The integral can therefore only become infinite if _ and n' are infinite. However, because of the inequalities (5), _ and n' may only become infinite if
becomes
infinite,
However, point
m]
is
--
V becomes outside
of
since
_ and
infinite C2,
we
_ are
for
need
limited
r I = 0 and
only
examine
--
if V becomes
for
r 2 = 0.
the
case
Since
infinite. the
of
therefore evaluate the portion of the integral which is in of the point m 2. If r2 is very small, _2 + N2 is equal to ml (0 m2) 2 , and the term-is also constant, so that if we set rl
the
162
Let us vicinity
or
/161
_
n 2
h+
--(_,_
_,,)+
-mA= H,
2
H will
be
regarded
as
If we then
a
/'!
constant.
set (_--Om2)=rtcos_,
inequalities
(5) will
_=r,
sin_;
....
>H--_
,l
the
integral
(2)
will
add
the
(5')
rt
become (2')
pr_ dp dr_ dto d?.
f We shall
inequality
to the inequalities (5'), where a is very small, of the integral which is close to m 2 which must the other part is definitely finite. If we
integrate
7l'= p sin?,
become II+_>
and
(_'= p cos?,
with
respect
to _
and
_,
since it is be evaluated,
the
integral
the and
part since
(2') will
be-
come
4_' for,
Let
us
integrate
first
with
a_ dr,
respect
to
(2")
p.
We
must
calculate
the
/162
integral
fpclp=
which
is
chosen
between
the
limits
_. II-_--I-72_-
which
provides The
us with
integral
?_,
)
and
P-_
V(
2 li÷_+
E.
(2I') may
therefore
be
reduced
to
163
kJ
_7;:=- 1
It
is
therefore
The which we as close
r2dri:_
2_I£x
i ,
finite.
theorems which were proven above may be applied to the case are discussing. The zero mass will repass its initial position as may be desired an infinite number of times, if one does not
impose certain unusual, is infinitely small.
initial
conditions
for which
the probability
In the restricted problem, if we assume that the initial conditions are such that the point 6, n must remain within a closed curve C 1 or C2, the first of the stability conditions, which were defined in No. 290, is fulfilled. However, the stability exists. 300. attraction If
The may
the
third
result be.
motion
condition
will
of
is
clearly
a material
also
be
fulfilled;
the
point
therefore,
same
whatever
the
_, _ is
governed
by
Poisson
law
the
of
equa-
tions
d t_
or,
in
the
case
of
relative
dV
d' _
motion,
d,_
by
an
dV
the
equations
dV
dt_ -- 2 ri -27 = -d_ ' d t _ dl
in such
a way
that
the
t
energy
'
d_ +
:in
arV
_
=
integral
--d-_ p
may
be
/163
written
' ,
:_'Lt-di ,I -1-t m l J- v _--_ h and
if
the
function
and
of
_ remain
However,
V and
limited, this
is
not
the we
constant
shall
all.
have
The
h
are
such
Poisson
same
holds
that
the
values
stability. true
in
the
more
case. I%
Let
164
x I, x2,
..., x n be
the
coordinates
of
of _material
points.
extended
Let V be the force potential
depending on these n variables.
Let mI, m2, ..., mn be the corresponding masses, in such a way that we may employ mI, m2 or m3 at random to designate the mass of the material point whose coordinates are Xl, x2 and x3. The equations may be written d'.T_
and
the
energy
In
virtue
of
are such that stability. What
integral
must
the
be
is finite when the V, which is defined
may be written
this
equation,
coordinates
if
is
dx',dx'2...dx'_dz,
us
call
A
the
over
the
fact
dx,...dz,
V and there
that
the
(x_=
integration is extended by the inequalities
--_-\dt/
over
the
will
constant be
integral
h
Poisson
Invariant
dr,'_ dt / the
region
I have
called
integral
fd_,i extended
function
limited,
the
V+h--_< Let
the
x i are
demonstrated
f
dV
region
dx; ... dx;,,
defined
by
the
inequality
.r_l t s <,. -/-x,
The
will
same
integral
obviously
extended
over
the
region
/164
be AR
n.
165
k_j
When be
extended
over
the
region
A
or,
since
e is very
defined
[< V-i-h
by
the
inequalities
(i),
it will
o] ,
1-_)5--(V-_-h--_-)
i
small, n
nAE(V+ Our
integral
invariant
therefore
n A _/(
and
the
integration
must
h)i-'. equals
V -_-h)_-'d.rj
be
extended
dr:..,
over
(2)
ar:r.,
every
point,
such
that
V + h
is positive. According
or
It may infinite.
then
It will zero.
then
Let
to my be
readily
always
us now
hypothesis,
be
assume
the
verified
finite
that
if n =
n is
region
V + h
whether
the
2, because
> 2,
and
that
the
quantity
under
the
sign
f in
limited.
integral
the
large of the order p when the distance between and x4, x5, x 6 becomes infinitely small of the Then
> 0 is
(2)
exponent
of
V + h becomes the two points first order.
the
integral
is
finite
V + h
is
infinitely Xl,
x2,
(2) is of
x 3
the
order
subset
has
n
which
1
The
3 dimensions.
the
integral
is
The
integral
finite
may
is of
the
therefore
be
n--(n--3)>p(n from
which
it
follows
n;
the
condition
under
_t),
/165
that P<
166
order written
6 n--'_
This is the condition under which there is Poisson stability.
Application 301.
The
following
preceding
to
the
Three-Body
considerations
Problem
apply
to
the
case
in which
the
equation
(l) results
in
the
fact
Unfortunately,
that
the
xi's
this
is not
can
the
only
case
vary
in
shall employ the notation presented in No. to designate the coordinates of the second x4, x5, x 6 to designate the coordinates of the center of gravity of the first two, a, tances of the three bodies, and MI, M2, M 3 Finally,
to
I shall
designate We
the
shall
the
between
finite
three-body
limits.
problem.
I
ii. I shall use Xl, x2, x 3 body with respect to the first, the third body with respect to b, c to designate the disto designate their masses.
employ
quantities
then
which
I have
called
B and
B'
in No.
ii.
have V = M'M'_ _ ,'-_ )13.MI -_'--5I' 51,
Equation
(i) entails
the
inequality V i-h>o.
The
function
V is
h is positive, the question is whether that the inequality coordinates
x i.
essentially
(2)
those
which
This
amounts
are
imposed a+b>c,
can and
if
the
constant
inequality will always be satisfied. However, the we may assign small enough negative values to h so can only be satisfied for limited values of the to inquiring _J_r,
with
Therefore,
positive.
at
the b+c>ct,
whether
_t,C _t,
>
three
sides
ct+c>b
only be satisfied for finite values of assume that it is very large; we shall
the
/166
inequality (3)
of
a triangle (4)
a, b, c. Let us set a = c, assume that b is very small.
167
The inequalities
(4) will
be satisfied
With respect to inequality
(3) which becomes
5!_M..j-_-3ttM_,
_I:M, = ---!-h>%
+
_L
no matter of a. No assume
These
what
h may
be,
by them.
b
it may
be
"
satisfied
matter how small h may be, that b is small enough that
by
arbitrarily
or how large a may the first _erm may
The existence of area integrals integrals may be written
does
not
modify
large
values
be, we may always be positive. this
conclusion.
p(_,_', - _, _'_ )+ _'(,_o_: - _,_; _-- .,, I _(z,_"_-- _',) + p'(T_z_--x,_,;) = _,, p(z,x;--x,._', )+ p'(_, _:,-- _',) = _. In
virtue
of
these
equations,
we
(5)
have
(6)
- (_? + x'_'+ =_:)+ ¥ (x? + x'_'+ x'£-)> "T .,_ _;+., where I is the moment of inertia two material points whose masses
of a system which is formed of are $ and B' and the coordinates
with respect to three fixed axes are Xl, x2, x3; x4, xs, x6. I repeat, that I is the moment of inertia which this system would have with respect to the line serving as the instantaneous axis of rotation for a solid, which would coincide momentarily with this system and would rotate in such a way that the area constants are the same as for the system. Inequality
(2) must
then
be
replaced
by
the
following /167
V + h > _ + a_+_|.
However,
this
equality,
just
like
inequality
(2') (2)
itself,
may
be
satisfied
by arbitrarily large values of the xi's , because -- for very large values of the xi's -- the moment of inertia I is very large, and, due to the fact that the second term is very close to zero, we return to inequality (2). We must therefore preceding section are
late
168
conclude that not applicable.
In order to provide a better the integral invariant
the
considerations
determination
of
this,
given
let
in
us
the
calcu-
f
extending
it
over
a
dx', dz'_
region
@
@
defined h --¢
The
s's
of
equalities
of
(6).
Let
are
very
(5),
us
first
T
by
the
following
a,---z,<
K,
z, <
K, < a, + %,
a_ -- ":a<
K_ < as -4- ¢,.
quantities.
is
integrate
inequalities
+%
a,--
small
and
dz' s dx, dx_ . . . dx_,
@
(7)
The
the
reduced
over
the
Ki's
energy
' ' x i s,
are
the
'--
i.e.,
we
obtain
and
first the
terms
first
term
s
3 ES, E_S3
(_'F where
Ii,
12
for
this
are
chosen
and
13
/(
V+h--
represent
passing,
I would
parallel
definition
the
of
I,
like
to we
the
It
shall
may of
be
seen
values
j
v'l,l,i,
principal
moments
of
inertia
is
infinite,
of
the
gration
Xl,
field
T, + I-7 = the
doubly
the
x 2,
is
that, axes
if of
the
axes
inertia,
of
the
coordinates
according
to
over
every
the
l
integral,
'
which
is
extended
that
although
points
note
have
that
such
to
V-4-h
x3
then
a]-4-a_
4-a]
aI
denominator or
triply
xq,
>o
_i1213
x 5,
x6
infinite,
becomes
infinite
recedes
indefinitely.
and
denominator
the
when The
only
one
inte-
/168
becomes
infinite. 302.
no
Even
longer
conclusions Let and
that
the
third
will
three
principal
_;
are
dx_...dxs
______
)
=,
system.
In
system
- .... ._ a_*-a_-+-a_\_-d:c __
if
applicable, from
us
the
considerations we
may
existence that
the
third
body
recedes
body
will
essentially
no
longer
presented nevertheless
of
assume
the
become
the
the
distance
in draw
integral b
of
two
the
preceding
certain
sections
interesting
invariant. of
indefinitely. disturb
the
the
Due motion
of
bodies
to
its
the
becomes great
first
small,
distance, two,
which
elliptic.
169
center
This of
third body will essentially gravity of the first two.
describe
a hyperbola
around
the
In order to elucidate this point, I shall present a simple example. I shall assume that we have a body describing a hyperbola around a fixed point. The hyperbola is composed of two branches. One of these branches jectory
is the is only
analytical extension of the other, although the composed of one single branch for the engineer.
We may then cal extension in defined.
xl,
inquire whether the case of the
The coordinates of the x2, x3; the coordinates
of gravity of the first motion of two imaginary fixed axes, are Xl, X2,
the trajectory has three-body problem,
second of the
an and
tra-
analytihow it may
body with respect to the first third body with respect to the
be
are center
two are x4, x5, x6, so that we must envisage the points whose coordinates, with respect to three X 3 for the first and x4, x5, x 6 for the second.
The first of these points will essentially describe an ellipse, the second essentially a hyperbola, and it will continue receding indefinitely on one of the brances of this hyperbola. In order to obtain the desired analytical extension, let us construct the second branch of this hyperbola, point.
and
let
us
relate
it to
the
ellipse
described
by
the
first
Let us then consider two special trajectories of our system. For the first, the initial conditions of motion will be such that, if t is positive and very large, the point x4, Xs, x 6 will be very close to the first branch of the hyperbola ellipse, in such
and the point Xl, x2, x 3 will be very a way that the distances of these two
to the hyperbola indefinitely.
or
and for
to
the
ellipse
--
tend
to
zero
close points
when
p
to the -- either
increases
Let us take the asymptote of the hyperbola as the a_is let V be the velocity of the point which describes this a value of t which is positive and very large. Then
of the x4's , hyperbola,
x_--.Vt will
tend
toward
a finite
and
determinate
way, and
n be the mean difference
limit
X when
t increases
in-
definitely.
the
In the same mean anomaly,
let the
l -- nt
170
motion
on
the
/169
ellipse
and
£ be
will
tend toward a finite
and determinate limit
_0.
If we specify the ellipse and the hyperbola and, consequently,V and n, and in addition if we specify X and _0, the initial conditions of motion corresponding to the first trajectory will be completely determined. Let us now consider the second trajectory, and let us assumethat the initial conditions of motion are such that, for t which is negative and very large, the point x4, x5, x6 is very close to the second branch of the hyperbola, and the point Xl, x2, x 3 is very close to the ellipse, and that these two points cometogether indefinitely from these two curves when t tends toward --_. The differences :r,.-- V t, tend
toward
toward
the
finite
and
l--lit
determinate
limits
X'
to
the
and
£b when
t tends
infinity. The
initial
completely
conditions
defined
when
corresponding
we
specify
the
ellipse,
X = X',
/o:: l_,
second
trajectory
the hyperbola,
are
and
X' and
_'0If we
the two other. Let
have
traiectories
us
now
may
be
consider
regarded
a system
of
d:j_, = x dt where the functions the relationship
X i, which
as
the
analytical
differential
(i :: ,, _, ...,
depend
solely
extension
of
each
equations
n),
(i)
on Xl,
x2,
...,
Xn,
satisfy
dS.,: = o .
dXi
These
equations
will
have
the
f Let
us
assume
must remain within which was considered
that a
we certain in the
integral
d.v,
know
invariant
d#,,. arbitrarily
region preceding
(2) that
the
point
xl,
x 2,
...,
Xn
V, which is similar to the region V sections, but extending indefinitely
171
so that the integral (2) extended over this region is infinite. conclusions of Nos. 297 and 298 will no longer be applicable.
The
However, let us replace equations (I) by the following {9: = x_ = x;, dt'
where M is xl, x2, ..., the The
a
given arbitrary x n, whose motion
same trajectories differential
function of is defined
as that whose equations of these
x2, ..., equations
x n. (1'),
The point will describe
by equations in both cases
(i'), we may see but obey different
that these laws.
If I employ t to designate trajectory, and t' to designate these two times will be related
We
two
points
describe
the time when P passes the time when _ passes in the following way (lt dt'
(1).
dx,, .-_,_"
However, if I employ P to designate the point whose motion by equations (i) and P' to designate that whose motion is
by equations trajectory,
by by
is dedefined
the
same
a point of its this same point,
t 3I
have d( ,_IX',)
-_l_e-which
Xl, by
motion is defined trajectories are
dw, dx, Xt _ X, .....
fined
(1')
M
indicates
that
the
= o,
equations
d._, _ x_
(i')
tlt'
have
the
integral
invariant
f
tends
172
rapidly
enough
that
(2')
dxt...dx_.
Let us assume that the function toward zero when the point Xl,
and recedes V is finite.
(i'). define
M dx,
the
M is always positive, and that it x2, ..., x n recedes indefinitely, integral
(2' 7 extended
over
the
region
The conclusions presented in Nos. 297 on may be applied to equations These equations (i') therefore have Poisson stability. Since they the same trajectories as equations (i), it may be stated in a
v
certain sense that the trajectories stability. I shall clarify
of the point P also have Poisson
this point.
Wehave = /"d,'. t
Jo
(3)
_I
Since M is essentially positive, t increases may vanish, it may happen that the integral is infinite.
be
For example, infinite for
let
us
assume
that
t' : T or
into time t
T
Let two t'=
us consider the trajectory parts, the first which P' T; the second C which P'
with t'. However, of the second term
M vanishes
for
for
t' = T ; then
t will
t'> T.
of the point P'. traverses from the traverses from the
We may divide time t'= 0 to time t'= T to
it the
_
The point P will describe the same trajectory describe the part C, because it can only reach the finite time t. For
the
engineer,
the
trajectory
of P would
For the analyst, it would be composed which is the analytical continuation. Let point P1 occupies equality
The motion of the point P1 will describe points P and P1 may be other. Let us now assume initial instant of
unusual,
not
only
as P', but it will only part C' after an in-
only of
be
C, but
composed also
of
of
C.
C',
/172
us imagine a point P1 whose position is defined as follows: The will occupy at the time t I the same position that the point P' at the time t'. With respect to tl, it will be defined by the
t,=/
the
since M of (3)
in
the
sense
d,,
t, dr' 7_i-
(where
t'0>T).
point P1 will conform to equations (i), C', in such a way that the trajectories regarded as the analytical continuation
and this of the of each
that the point P is within a certain region U 0 at time. If the initial conditions of motion are not attributed
to
this
word
in No.
296,
the
trajectory
173
of the point P and
its
successive
analytical
continuations
will
cut
across the region U 0 an infinite number of times, no matter how small it may be. However, it may happen that the point P never re-enters this region, because this region is not traversed by the trajectory, strictly speaking, of the point P, but by its analytical continuations. 303. We
which
This
saw
we
may
above
have
be
that
reduced
applied
to
we
consider
to
must
the
the
sixfold
three-body the
problem.
integral
integral 3_
y(V
is
However, infinite, Let
we and
have this
us write
+
h--
a_
the
Xi's
the
Then
let
and us
_ dx:dr,...dxs
seen that this integral, extended over the region V, has prevented us from arriving at Poisson stability. equations
of motion
d.ri Xi, 5t-t- = where
W,r,t, "
4-al-+_a])
the
Yi's
are
in
the
form
of
the
xi's
d.e_ Yi, -dr- -
functions
the
new
d.vi
Xi
(l:r;
7lt _ = 7q" new
equations
x' i 's "
will
+,)'
all f?,l
Yi
(-l-t _ = Tii"
have
the
dx,
d:rGdx',
m
The
the
equations
following •
write
W
us
B
let
and
set
_.x, + _l + x_, +...+_: and
_/173
as
the
integral
invariant
dx'_
or
Jk However,
point
174
this
integral
is
finite.
Therefore, if the initial situation P in space has 12 dimensions whose
of the system is such coordinates are
that
the
and if this point P is within a certain region U0 at the initial instant of time, the trajectory of this point and its analytical continuations -- such as we have defined at the end of No. 302 -- will cut across this region U0 an infinite numberof times unless the initial situation of the system is not unusual, in the meaning attributed to this word in No. 296. 304. It may first appear that this result is only of interest for the analyst, and has no physical significance. However, this point of view is not entirely justified. It may be concluded that, if the system does not repass arbitrarily close to its initial position an infinite number of times, the integral/174 ft
t= "
dt
-0
will
be
finite.
This proposition is tories whose probability word in No. 296.
valid, if we overlook certain unusual is zero, in the meaning attributed to
trajecthis
If this integral is finite, it may be concluded that the time during which the perimeter of the triangle formed by the three bodies remains less than a given quantity is always finite.
175
CHAPTER XXVII THEORY OF CONSEQUENTS 305. Wemay obtain other conclusions from the theory of integral invariants which will be of use to us below, although they will be presented in a somewhatdifferent form.
J175
Let us commenceby investigating a simple example. Let us assume a point whose coordinates in space are x, y and z and whose motion is defined by the equations dy _ _ :y, dt
_I_C=X, dr
d._ -..... dt
z.
(1)
where X, Y and Z are the given, uniform functions assume that X and Y vanish all along the z axis,
X
is
a solution Let
us
of
equations
then
-=
_
::
equations
(i).
set
(i)
will
y=
we
R, fl and Z are the period 2_
It is advantageous may do this with I
shall
example,
have
Let the
now
always us assume solution
the with
no
for us difficulty
assume
in
remains
176
letters
Let
usset
d_ _ =a,
functions respect to
b
assign since
addition
_
x
that Then
that equations in the following
a and
dz _-/=
of 0, to _.
positive.
p =_(,.,, The
psin_,
become dp _ .-_ --R,
where having
Let us that
O
x=pcos_,
and
of x, y, z. in such a way
,,
represent
_
and
(2) z which
only = y
positive = 0 is
fl can
never
will
(2) have form b),
Z,
been
_ -_(,,,,
integration
are
a
values solution.
vanish
always
periodic
to
and,
increase
integrated,
0,
for with
and
and
that
t. we /176
_, b). constants.
v
po =_ft(o, ?, ::ft(_-, Let
M 0 be
the
point X
and
M 1 be
the
point
a, b), a, b),
whose
coordinates
::
y.-_
whose
_Oo,
These two the side of The
point
If we
points both the positive M 1 will
consider
be
the
O,
_
coordinates .z" =-
on
z0=f_/o, :: = f,('_,
_l,
the
to
are
.:
.70,
are .3 =:
)." : : o,
belong x's.
of
gl.
the half-plane
consequent
bundle
a,b), _, b).
of
the
xz's
located
the
differential
of M O.
curves
which
equations (i), if we pass a curve through the it until it encounters the half-plane (y = O, definition is justified by the fact that this
satisfy
point MO, and x > O) again, new encounter
if we extend the preceding will occur at
M 1.
quents
If
an arbitrary figure F 0 is of the different points of
called
the It
is
consequent evident
of
that
drawn in this half-plane, the conseF 0 will form a figure F 1 which will be
F O. Pl
and
z I are
continuous
functions
of
PO
and
z0 • Therefore,
the
consequent
curve, the consequent consequent of an area is connected n times. Let
us
now
of
a continuous
of a closed curve which is connected
assume
that
the
three
curve
will
will be a closed n times will be
functions
X,
Y
and
be
a continuous
curve, and the an area which
Z are
related
as
follows dMX a_
where
M
is
a positive,
Equations
(i)
then
uniform have
dMY dMZ *--dT--y+_=
function the
integral
o,
of x,
y,
z.
invariant
M.dxdyd_
177
and equations (2) have the following
invariant
/177
_I?d?dt,_dz.
Let
us
now
consider
the
dp
where
_
is
They
regarded obviously
as
equations
R
r[_
the
have
Z
d_o
independent the
(3)
variable.
integral
invariant (4)
(see
it
No.
is
3I t._o d? dt,J d:
253).
Since it was assumed a positive integral Let
F 0 be
an
above that invariant.
arbitrary
area (y
and
let
F1 be
Let
J0 be
its the
_o,
M,
_ and
located
in
p are
the
essentially
positive,
half-plane
x>o),
consequent. integral
(5) extended over the planar area F0, tended over the planar area F I .
and
let
Jl be
Then let _0 be the volume produced by the tated around the z axis by an infinitely small tegral (4) extended over _ will be J c. 0 0
it
is
tended
_I,
In the same way, let _I be the volume turned around the z axis by an angle over
_i will
be
The integral invariant and we must have
178
Thus, the consequent.
inteBral
same
integral
area F 0 when angle g, and
ex-
it is rothe in-
produced by the area F 1 when c, and the integral (4) ex-
Jl e. (4) must
Jo
its
the
=:
(5) has
have
the
same
value
for _0 as
for
Jl,
the
same
value
for
an
arbitrary
area
and
v
This is a new form of the basic property of integral
invariants.
306. Let us then assumea closed curve CQ located in the half-plane (y = 0, x > 0) and encompassingan area F0. Let CI be the consequent of CO• This will also be a closed curve which will encompassan area FI, and this area F1 will be the consequent of F0. If the integral (5), extended over F0 and over FI, has the value J0 and J1, we shall have from which it follows that F0 cannot be a part of FI, and Fl cannot be a part of F0. Four hypotheses may be formulated regarding the relative of the two closed curves CO and CI. i.
C1 is within
CO;
2.
COis within
CI;
3.
The two curves are outside of each other;
4.
The two curves intersect.
The equation J0 = Jl excludes the two first will
position
hypotheses.
If the third is also excluded, for whatever reason, the two curves definitely intersect.
For example, let us assumethat X, Y, Z depend on an arbitrary parameter _ and that for _ = 0, CO is its own consequent. For very small values of _, COwill differ very little from CI. Therefore, it could not happen that the two curves CO and C1 are outside of each other, and they must intersect. Invariant 307. invariant
Any curve curve.
Invariant point
of
the
which
curves half-plane,
may
will
be and
be
readily let
Curves its
own
consequent
formed.
Let
M 1 be
its
will
M 0 be
consequent.
an Let
be
called
an
arbitrary us
connect
M 0 to M 1 by an arc of an arbitrary curve C O . Let C1 be the consequent of C O , C 2 be the consequent of C I, and so on. The entire group of arcs of the curve CO, CI, C 2, ... will obviously constitute an invariant curve.
179
But we may also consider invariant more natural.
curves whose formation will
Let us assumethat equations (i) have a periodic solution. x = ?,(t), be
the
tions
equations _i are
I shall
curve
of
this
periodic assume
periodic
in that
y = ?2(0,
z = ?_(t)
solution,
in
t, having when
the
period
t increases
Equations (6) represent a curve. intersects the half-plane; this
be
Let
1179
(6)
such
a way
that
the
by
27.
func-
T.
by
T,
m
increases
Let M 0 be the point where point M 0 will obviously be
this its own
consequent.
close
Let us now assume that there are asymptotic to the periodic solution (6). Let x=
be
the
equations
of
these
The functions _i may efficients are themselves is a characteristic In equations
4,(t),
y = 42(t),
which
are
z = %(0
very
(7)
solutions. be developed in powers of Ae at, and the coperiodic functions of t. In this expression,
exponent,
(7),
solutions
the
and
three
A is
an
integration
coordinates
x,
y,
constant.
z are
therefore
ex-
pressed as a function of two parameters, A and t. These equations therefore represent a surface which may be called the asymptotic surface. This asymptotic surface will pass through the curve (6), since equations (7) may be reduced to equations (6) when we set A = O. The asymptotic certain curve which an invariant curve. 308. Y,
ceases
_.
Let
Z depend I shall to be
on
us the
assume closed
Let A 0 be For _ = O,
surface passes
consider
will intersect the half-plane through the point M 0 and which
along a is obviously
an
assume
parameter
invariant _,
as well
that for _ = 0, the for small values of
a point of K. the curve K is
curve as
K.
the
curve _.
I shall
curve K
that
X,
K.
is closed,
but
that
The position of this point will depend closed, so that, after having traversed
it
on
this curve starting with A0, one returns to the point A 0. If _ is very small, this will no longer be the case, but one will pass very close to A 0. Therefore, on the curve K there will be a curve arc which is
180
different from that where A0 is located, but which will pass very close to A0. Let B0 be the point of this curve arc which is closest to A0. /180 I shall
join
will
Let A 1 and be located
line
A0B 0 . We
must
AoMB 0 of A0B 0. In the
four
B 1 be on K.
the consequents of A 0 and B 0. These Let AIB 1 be the consequent curve of
consider
curve
What
A0B 0.
K,
will
order
to
points
the
closed
included its
consequent
define AI,
curve
between
our
A0,
C O which
A 0 and
B 0,
is and
two the
composed of
the
of
small
points small
the
arc
line
be?
ideas
more
B I, B 0 follow
precisely, each
other
let
us
assume
on K
in
the
that
order
AIAoBIB0.
curve
The consequent K and of the
C 1 of C O will be composed of small arc AIB I, the consequent
Several
hypotheses
i.
small
The
may
curvilinear
then
be
the of
arc the
AIMB 1 of the small line A0B 0.
formulated:
quadrilateral
AoBoAI_
is
convex,
that
is,
none of these curvilinear sides have a double point, and the only points which the two sides have in common are the apexes. In this hypothesis, the form of the curve would be that indicated in one of the following figures
A c A,
....
B,
Figure
I
This hypothesis must be rejected, because it integral J is larger in the case of Figure 1 for smaller in the case of Figure 2.
Figure
2
is apparent C 1 than for
that the CO, and
181
2. The arc AoAI or B0B1 has a double point. If this were the case for the invariant curve K, there would have to be a double point on the arc joining an arbitrary point on the curve to its first consequent;
/181
we shall assume that this is not the case. Actually, this condition would not occur in any of the applications which I have in mind. It does not apply, in particular, in the case of the invariant curve produced by an asymptotic surface, as I explained at the end of the preceding section. It may be readily stated that the asymptotic surface does not have a double line if we limit ourselves to the portion of this surface corresponding to small values of the quantities which I have designated as Ae _t above.
On the other hand, the line A0B 0 does not have a double point, and the same must be true for its consequent AIB I. To sum up, we shall assume that the four sides of our quadrilateral do not have a double point. 3.
The
case
includes
will
then
arc
A0A 1 intersects
that
have
the
in which form
the
shown
the curve
in
4. have
The the
arc
A0B 0 intersects
form
shown
in
Figure
B0B I.
K would
Figure
Figure
then
arc
(As be
a special
closed.)
Our
case, curves
3.
3
its
consequent
AIB I.
Our
curves
will
4.
There are cases in which this hypothesis example, let us assume that X, Y, Z depend on for _ _ 0 the curve K is closed and that each consequent, so that for _ = 0 the four apexes coincide.
must be rejected. For one parameter _, and that of its points is its own of the quadrilateral
Then the four distances AoB0, AIBI, AIA0, BIB 0 will be infinitely small quantities if _ is the main infinitely small quantity. Let us assume that AIA 0 is an infinitely small quantity of the order p, A0B 0 an infinitely small quantity of the order q, and that q is
182
this
/IS2
V
A
A_
Figure
larger
than
4
p.
Since AIB I is the consequent of AoB0, the length of the arc AIB 1 must be of the order q. Then let C be one of the intersection points of A0B 0. In the mixtilinear triangle whose two sides are the lines AIA 0 and A0C , and part of AIBI, the others. must be The 5. AIA 0 and
It of
whose third side AIC is
should therefore the order q.
hypothesis
must
Two adjacent AIB I. It is
side is the larger than
be
of
the
therefore
be
order
sides of the quadrilateral then necessary that AoB0,
Figure
is
apparent
p,
that
A_
and
of the curve AIC which difference between the and
we
have
seen
that
is two it
rejected.
of AIBI, intersect K itself. If A_ is and A_ is the intersection of Al_with sequent of A_, and we shall obtain the
It
arc the
intersect_ for example, which is the antecedent
the intersection of A0B 0 with K, the arc AoAI, E l will be the confollowing figure.
5
A_ may
play
the
same
role
as
A 0 and
AI,
183
k_J
and
that
we
This
therefore
new
return
hypothesis
To sum up, the two hypotheses 2 and 4 must
to
must
the
therefore
case.
be
1183
rejected.
arcs A0A 1 and B0B I will intersect every be rejected, for one reason or another.
We must now examine the case follow one another in a different BoBIAoAI, AoAIBoB just studied.
first
I do not
differ
in which the order on K. essentially
points A1, The orders
from
that
time
that
A0, BI, B0 BIBoAIA 0,
which
we
have
Orders such as AIBIBoA 0, AIBoBIA0, AIBoAoBI, ... will not appear in the applications which follow. We shall always assume that, if _ is very small, the distances A0A I and B0B I are very small with respect to the
length The
no
longer
of order
the
arcs
AIAoBoBI,
discuss
AoMB 0 there 309. solution
will For
AoMB 0 or
them. be
or
the
It
is
a point
example,
AIMB I. equivalent apparent
which
let
us
assume
x = ?,(t),
and
asymptotic
_,
and
us
that
assume X,
be
if
its
that
y = ?2(t),
remain,
they
own
and
appear,
we
on
shall
the
arc
consequent.
equations
(i) have
a periodic
z = ._(t)
(6)
z = %(t).
(7)
solutions x = %(t),
Let
will
orders,
that
Y,
that Z may
equations be
y-=- _(t), (i) depend
developed
on
in powers
a very of
small
this
parameter
parameter.
For _ = 0, let us assume that the asymptotic solutions (7) may be reduced to periodic solutions. This may be done as follows. We have stated that the _i's may be developed in powers of Ae _t, with the coefficients themselves being periodic functions of p. However, the exponent
_ depends
on _;
let
Then for _ = 0 the functions and the solutions (7) may be
us
assume
_i will reduced
that
it vanishes
184
The 307.
_ = 0.
become periodic functions to periodic solutions.
The asymptotic surface intersects the half-plane curve C O which passes through the point M 0, which is the half-plane with the left curve (6).
No.
for
of
t,
along a certain the intersection
curve C O is obviously invariant, as I stated at Fo_ _ = 0, each of the points of C O is its own
the end of consequent.
of /184
In addition, = 0.
I shall assumethat the curve CO is closed for
Let us refer back to Chapter VII, Volume I. Wesaw from Nos. 107 on that, in the case of dynamics, the characteristic exponents may be developed in powers of _, and are equal pairwise and have the opposite sign. Weshall assumethat this is the case. In reality, we then have two asymptotic surfaces corresponding to the two equal exponents having opposite sign _ and -_. Wetherefore have two curves CO which will intersect at the point M0. Wemay distinguish
between four branches of the curve C_.
all
four
the
exponent
of
which a,
end C_
at
and
the C_
to
C" "_U_
C' all
C|
point
Mo;
the
exponent
C_
and
C'_ will
correspond
to
-_.
Bo
These
different
branches
Figure
6
of
curve
the
are
shown C "O is
branch C_ is the branch MoPoPIAoA I, the branch the branch C_ is the branch MoQIQ 0 and the branch MoRIRoBIB 0. These
four
branches
of
the
curve
are
in Figure the branch
C'_ is
obviously
the
6. The MoEoE I,
branch
invariant. I!
Now, for _ = 0, C_ is identical to C_, C_ is identical to C1, and (if we assume that the curve C O is closed for g " O, which we shall 0 call CO, ) these four branches of the curve will coincide on the closed 0 curve C0. It may
be
deduced
from
this
that,
for
very
small
_,
these
/185
branches
185
of the curve will differ very little from each other, that C'0 will deviate very little from C'I, C_ will deviate very little from CI," and that,if C_ is sufficiently extended, it will pass very close to C", if it is sufficiently extended. 1 I have indicated on the figure different points of these branches of the curve and their consequents. Thus, AI, BI, El, PI, QI, R1 are, respectively, the consequents of A0, B0, E0, P0, Q0, R0. Wewould first like to note that the points AI, A0, BI, B0 do follow each other (as we assumedat the beginning of No. 308) in the I
order AIAoBIB 0 when and Cy is traversed This the
invariant
closed In
As we
the invariant curve from A 1 to B 0.
curve
this
have
curve
is not
formed
closed,
but
of
it
the
two
differs
branches
very
CO
little
from
C_.
connection,
seen,
the
let
first
us
must
examine be
the
five
rejected.
hypotheses
The
second
of
will
No. no
308.
longer
occur.
It
could
only
occur
if
the
asymptotic
surface
(7) had
a double
line. We have stated that Therefore let us set
the
_i's
may
'I_g= ,l_t ° -;- Aeat,l,_
If satisfy
our surface had equations (i).
a double Actually,
an infinite number of lines that, if two layers of this could
we
only
be
one
of
these
where
A
three
identities
186
and
B are
two would
of Ae _t.
_-:k-'e_-_tq,_-_-....
lines.
a double
+_(t,
in powers
line, this double line would have to the asymptotic surface is produced by
time
t and
,I,,.= q,Jt, were
developed
satisfying these equations in such a way surface happen to intersect, the intersection
Since _i depends on the may show this by writing
If there identities
be
line,
A)=
have
A at
the
same
time,
have
to have
the
three
(i = ,, a, 3),
and where
to exist
parameter
A).
we would
4,_(t', B)
constants
the
t'
no matter
is
a function what
t may
of be.
t.
These
/is6
Performing
differentiation, d'Pt
we shall dq, i dr'
d--F=-yFWi However,
in
view
of
have
•
equations
d.t,, = x[+_(,, dt
(i),
we
shall
have
A), +.(t, A), +,(t, A)]
and in the same way d4:,t -- X[+,(C, dr'
from
which
it
follows
B),
+_(t',
B),
d'l_i
dr'
-_F = -dF' which
we
B)],
that
d¢i from
+_,(t',
-37 = "
have t'=t+h,
where
h We
is
a
would
constant. thus
,1,,°(t)
obtain
the following
+ A e _t,t,_ (t) + A' e TM q,', (t) = ,;,f(t
+
h)+
C e _t q,l (t + h) + ....
where C=
The
identity
must
be
valid
for
t =
Be _h.
--®,
from
which
it
follows
that
Ae =t= _e =t= op and
we
have
• ?(t)= ¢_(t + h),
from which we have h = 0 and
+_ (t)+ AeatOl (t)+-.., = _,_(t)+Ce=t,_l(t)+.. or
AOl (t)+X'*"+,'(O+...= C *f(t)_-Oe='¢_(t)+..
or,
setting
t =
--_,
we
have A = C = B.
187
J
Due
to
the
double
fact
that
the
two
values
A and
B
are
equal,
there
is no 1187
line. The
third
hypothesis
may
be
adopted.
Let us pass on to the fourth hypothesis. In order to determine whether it must be rejected, we must try to determine the order of magnitude of the different
as we
the distances applications
Finally, the have seen.
fifth
AIA 0 and A0B 0. which follow.
hypothesis
Extension 310. tions
We
(i), but
formulated all
of
of
the
very
them
is
always
Preceding
special
are
not
This
is what
reduced
to
the
shall
first
do
in
one,
Results
hypotheses
equally
we
above
concerning
equa-
necessary.
Let us consider a region D which is simply connected and which is part of the half-plane (y = O, x > 0). Let us assume that we know arbitrarily that, if the point (x, y, z) is located at a point M 0 in this region at the initial instant of time, m will constantly increase from 0 to 2_ when t increases from 0 to to, in such a way that the curve satisfying equations (i) and passing through the point M 0 -is extended from this point M 0 up to its new intersection plane
--
is
Just
never
tangent
as in No.
M0, and it applicable
305,
to we
a plane may
is apparent that all to the figures which
then the are
passing define
through the
assuming that it with the half-
the
z axis.
consequent
of
preceding statements will located within the region
the
point
still D.
be
It will not be necessary that the curves satisfying equations (i) and intersecting the half-plane outside of D be subjected to the condition of never being tangent to a plane passing through the z axis. It will
no
longer
be
necessary
that
x = y = 0 be
a solution
of
equations
(i).
quent,
Then, the
The invariant
if C O is a closed curve inside of O and if CI is two curves will be outside of each other or will
results curves
given which
in No. do not
curve leaves the region will still be applicable this
D when it is sufficiently to the portion of this
applicable to the If even one invariant
extended, curve which
the results is within
region. 311.
188
308 will be equally leave the region D.
its conseintersect.
Let
us now
consider
a curved
surface
S which
is
simply
/188
v
connected, instead of a plane region D. Let us pass a curve Y satisfying equations (i) through a point M 0 of this curved surface, and let us extend this curve until it again intersects S. The new point of intersection M I may still be called the consequent of M 0.
other There
If we consider two points M 0 and M_ which are very close to each their consequents will be, in general, very close to each other. would be an exception if the point M i were located at the boundary
of S, or if the curve y touched the surface point M 0. Except for these exceptions, the tic functions of the coordinates of M 0.
at the point M I or at coordinates of M I are
the analy-
In order to avoid these exceptions, I shall consider a region D which is part of S and such that the curve Y, proceeding from a point M 0 inside of D, intersects S at a point M I which is never located at the boundary of S -- so that the curve y does not touch S either at M 0 or at M I. Finally, I shall assume that this region D is simply connected. Let E, n and
us
adopt
_, for
a special
example,
system
and
of
coordinates
for which
I shall
which only
I shall
assume
the
i. When [_[ and IT][ are smaller than i, the rectangular x, y and z will be analytic and uniform functions of $, n and are periodic with the period 2_ with respect to _. 2. a point
No more than one system (x, y, z) in space, such
of values that
of
$,
n,
_ can
call following:
coordinates _, which
correspond
I_I<,, l_l<,, o<_<2= 3.
When
we
set
_ = 0,
-i and +i, the point x, this surface containing
or
_ =
2_
(x)
and when
y, z describes the region D.
the
we
Equations form
(i) may
be
d_
I shall
assume
that
Z*
=
transformed
_'
l_l<,,
=
positive
I_[<,,
S,
_ and or
n between
a portion
that the functional z is never infinite
by writing
dt--It,
remains
vary
surface
4. It results from conditions (i) and (2) determinant A of $, _, _ with respect to x, y, zero when the inequalities (X) are satisfied. 5. following
to
Z o.
them
in
of
nor
/189
the
(i')
for
_--o.
189
The
equations
(i')
will
have
the
integral
invariant
f _t d_ dn d_,
and
the
equations (3') d_ ,4" =
will
have
the
integral
-m _'
dn dE
Let
F 0 be
from
0
to
volume
¢,
¢0,
an
arbitrary
d_d_
d r..
figure
which
is
part
of
D
the different points n remain constant and
with
small.
and
¢ being
the
very
figure
F I will
f_1z.
have
:
Let us assume that a way that _ and
the
same
value
for
00
The
figure
produce
d_an a_ =
d-Xwill
de d_
invariant
MZ*
consequent. move in such
H z='
a
let
F 0 will
volume
F 1 be
its
01 •
produce
a
The
integral
the
double
d_d_,
f MZ* .I_-
and
and
of F 0 and of F 1 that _ increases
for
Therefore,
01 .
inte-
gral MZ*
-_-- d_an, which for
is
similar
F 0 and It
to
No.
tion
308 of
It
Even
is
integral
this
closed
curves applied
be
if
curves an
which
the
312. Let
us
Let
formulate
F be the
within
invariant
a
of
leaves will
region.
to
Equations
function
of
D,
the
_'t
in
and K,
have
No.
that or
region
the
306
same
may
the
at
value
be
results
least
D when
be
applicable
of
Dynamics
four
variables
equations dF
will
to
given the
por-
D. the
still
this
canonical
given
curves
inside
curve
d_t
190
located
results
Application
305,
positive.
results
is
within
No.
the
to
which
of
essentially
that CO
invariant
is
(5)
therefore
may
extended, curve
the
from
these
ciently this
F I.
follows
applied in
to
dF
it to
is
the
suffiportion
of
/190
xl,
x2,
Yl,
Y2.
I shall
assume,
i.
this
as
That
I usually
F is
2. That parameter
do:
a periodic
function
of Yl
and
Y2;
F depends on a parameter in the following form
_ and
may
develop
F =: Fo-4-
3.
That
Under
F 0 is
this
a function
assumption,
_Ft
of
we
in
powers
of
:-- l.t'- F_ +...;
only
shall
x I and
have
of x 2.
the
integral
F=C, where
C is Under
once and ordinates
a constant. this
assumption,
for all are
to C,
and
let let
us
attribute
M be
['+'i'(x_)cosy,]cosy2,
The function comprehensive
(2)
a moving
a value point
[,+?(xt)cosy,]siay_,
_(xl) is a function of xl, determination below.
which
whose
is
determined
rectangular
co-
?(z,)siay,.
of which
I shall
make
a more
Let us first assume that F, which will depend arbitrarily on x2, may be developed in increasing powers of x I cos Yl and x I sin Yl. For x I = 0, it will result that the function F will no longer depend on Yl and, in addition, that the function F will not change when x I is changed into -x I and Yl is changed into Yl + w. We shall then assume that _(Xl) is an odd function of x I which increases from 0 to i when x I increases from 0 to + = We may set, for example
?(x,) =
If this hypothesis of radius i, which
is is
adopted, tangent
the point M will to the z _xis.
An infinite number of systems correspond to each point M within will
not
differ
the other by into -x I and
(2).
essentially
increasing Yl into Yl
from
Yl or + _"
'_'
Y2
always
of values of Xl, Yl this torus. However, each by
other,
a multiple
since of
one 2_,
be within
a torus
and Y2 will these systems passes or
by
from changing
/191
one
to
x1
If Xl, Yl and Y2 are given, x 2 may be deduced by means of equation Let us assume that the variables x and y vary in accordance with
191
equations (i), and the corresponding point M will curve which I shall call the trajectory. One
and
only
one
The
form
For
_ = O,
trajectory
of
these
passes
through
describe a certain each
point
inside
the
torus.
the
trajectories
differential
for
equations
dx_ d--F= o,
tories
The xi's are are located
_ = 0 may may
be
be
readily
reduced
determined.
to
dYl dFo d--f = -- dx---_"
therefore constants, which indicates on the tori, and the Yi'S are linear
that our functions
trajecof time,
because dF, dx i = nl
depends
only
If curves.
the ratio nl:n2 is commensurable, Conversely, they are not closed
Let
ml,
on
the
xi's
m 2, Pl,
and
P2 be
is
four nltp2
Let
us
a constant.
whole --
nttFt
the trajectories are closed if this ratio is incommensurable.
numbers, =
such
that
l;
set
y] = y_ = x] = X_
The
=
--
m,yl ply,
+ m,y,, +PRY,,
p_xl
--ptx,,
nll_I
+
nl|
_2,
identity x',y'_ + x, t y, e = x_y, -- x: y2
indicates that when one passes xi, Yi, the canonical form of We certain shall
from the variables x i, Yi to the equations is not changed.
variables
shall
assume
than
a
limit
a.
we
/192
have,
for
that n2 does not vanish when Xl remains less dy2 Then d-_will always retain the same sign, and
example
d_,,
d--7> o.
192
the
This inequality, values of _.
which is valid for _ = 0, will
still
be valid for small
The relationships 9"2= o,
will then a circle.
define
a certain
Ix, ] < a -
plane
region
D which
will
have
the
form
of
The trajectories starting from a point in this region will never tangent to a plane passing through the z axis, at least before having cut the
across the half-plane Y2 = 0 again. role of region D in No. 310. The
equations
(i) have
the
Our
integral
region
may
therefore
be
play
invariant
./d_,dx2dy, dy,,
from which const.
we
may
deduce
the
following
j = __f
However, J
is
the
then
dF =7u_2
equals
a positive
The results curves drawn
dy 2 --dt
m I and
, and
The
which
of
the
integral
F =
is
consequently
The
negative.
invariant
invariant.
given in Nos. in the region
m 2 are
means
d.v, dy,d.r,
306 D.
and
Under this assumption, let b be a - _, and such that the corresponding the following relationship
where
by
two
prime
numbers
308 may
therefore
be
applied
to
a value of x I which is smaller than values of n I and of n 2 satisfy
with
respect
to each
other.
curve
is a circumference
If we always assume different points on this
will
_Z'|_
b.
be
an
that _ = 0, circumference
invariant
curve
the trajectories will have the
for
_ = 0.
emanating from general equation
/193
193
,}'l
from
which
we
=
'rtlt'_-const._
In
order
sufficient
to have to
set
=
n_
,Y2 +
successive the
y2==o,
to
n_l
]-COrlSt._
have
Y_
be
_'1:--
In order to pass increase Yl by
consequents
following 2"2= 2_,
from
const.
y2=_l= ....
a point
of
a given
point,
it will
successively
to
,
its
y2:=2h=.
consequent,
it
is
sufficient
7 _2
from which it follows that b will coincide with their
n_l
all points on the ml th consequent.
invariant
circumference
This point and its m I - 1 first consequents are distributed on cumference in a circular order, which may be readily determined two whole numbers m I and m2 are known. I shall call the order Let us no longer assume Chapter III, will still have from the solutions Xl=b,
They
will
have
at
least
that _ = 0. The equations (i), periodic solutions which differ
._i _-:/_i [ -r- COflS[.,
two,
of which
xI =
this cirwhen the _.
according to very little
yl----n_tq-const.
one
is
unstable
and
the
other
is
stable. A closed trajectory will correspond to each of these periodic solutions. I shall consider one of these trajectories which I shall call T and which will correspond to an unstable solution, so that two asymptotic surfaces pass through T.
Y2
Let M 0 be the point where this = 0, and let MI, M2, ... be its M 0 will
coincide
with
trajectory successive
The
point
The
I shall join the point M k to the center radius which is thus drawn will intersect
penetrates consequents
its mlt--h consequent
M m.
of the circumference the circumference
point M_ which is very close to M k. The different points each other on the circumference in the circular order ft.
194
the half-plane (Figure 7).
_k will
x I = b. at a follow
V
/194
In order to formulate these ideas more precisely, I have drawn the figure on the assumption that ml = 5, m 2 = 2. The closed trajectory T intersects the half-plane at five points M0, MI, M2, M 3, M_. Two asymptotic surfaces which intersect pass through this trajectory. The intersection of these asymptotic surfaces with the half-plane will be composed of different curves. We shall have two curves intersecting at M0, two at MI, two at M2, two at M3, and two at M 4. All these curves are shown in the figure.
Figure
We
In particular, may distinguish
let us between
MoB2, MoP0, M0Q 0. last two are shown the second and the other. In the same of the points M. two are shown by of the other. Let A 0 A 0 going to this radius line M3B 0. very little very
7
consider the four branches
two of
curves which pass at M 0. the curve, i.e., MoA0,
The first two are shown by a solid line, and the by a dashed line. The first and the third, just fourth, are each located in the extension of the
like
way, four branches of the curve will end at each Two of these branches are shown by solid lines and a dashed line, and each pair is located in the extension
be a point of the branch M0B 0. Let us draw a radius through the center of the circumference x I = b, and let us extend up to B 0 where it intersects the curve shown by the solid Since _ is very small and since all of our curves differ /195 from the circumference x I = b, the segment A0B 0 will be
small.
195
Wemay then see that MIAI, M2A2, MBA3,M4A4, M0A5 are the successive consequents of MoA0, that MqBI, MoB2,MIB3, M2B4, MqB5 are the successive consequents of M3B0, and finally that AIBI, A2B2, ..., AsB5 are the successive consequents of A0B0. The arcs AIBI, A2B2, ..., A5Bs are no longer rectilinear general, but are very small arcs of a curve.
in
Figures 1 or 2 shown in No. 308 reproduce the part of the figure shownby the solid line. The entire group of our curves shownby the solid lines represents an invariant curve K. I have drawn the figure based on the first hypothesis, which -- as we have seen-- must be rejected along with the fifth hypothesis. According to the statements I madein No. 309, this also holds true for the second hypothesis. Wemust examine the fourth hypothesis in greater detail. In order to do this, let us try to determine the equation of our asymptotic surfaces. Based on the statements presented in No. 207, this equation may be obtained in the following way. A function S is formulated which may be developed in powers of in
such
a way
/_,
that p
S = S04-(yS,÷...+
Regarding respect to y_, We
shall
Sp, it is a periodic and 4_ with respect
funtion to y_.
dS
If
(4) is
the
equation
the series S were the condition that
significance
the formal point of terms of the series
196
period
2_ with
dS
convergent, our curves
does view, S, so
as
of the
points A 0 and B 0 must coincide. 225, and the following). What
the
*; = dT,
_s
entail
of
have
_" = _,y';'
Equation
s_Sp + ....
i.e., that
asympototic the must
However,
equation if
(4)
surface.
periodicity of the be closed and that this
is not
(4) have? 2p is
the
sum
the
It may of
the
S_'s would t_e two
case
only
be
(see
valid
p + 1 first
No.
from /196
P
xp = So + ,,/_ S, +...+
the equation
dZp
a_- Sp, dXp
_' = " 7tfi_+ m, dy_
(4') 2
will
be
valid
up
However, bitrarily
to quantities
equation
(4')
of
the
order
represents
a closed
surface,
and
p
is
ar-
large.
We must therefore conclude that the distance A0B 0 is an infinitely small quantity on the order of infinity (see Nos. 225 on). In addition, the distance A0A 5 (or BoBs) is on the order of /_, and is consequently infinitely small of the order of ½.
AoAI,
The distance A0B 0 is therefore infinitely small with respect which indicates that the fourth hypothesis must be rejected. The
only
possible
Therefore
the
two
hypothesis arcs
Application
is
A0A 5 and
to
therefore
the
to
third.
B0B 5 intersect.
the Restricted
Problem
313. I am going to apply the preceding principles to the problem presented in No. 9, and I shall employ the notation given in that section. Consequently, we shall have the canonical equations dx_
dF'
dg_
a_ = aye,' based
on which
we
may
dF'
-d? = - a_--i,'
set
(5) I y_ x'I=L, l, and,
in
x_=G, ' yz=g--t
addition, F'.=
R t G-----F, + _Ft+..., I
Fo -----_x,,--7 + x;.
Let
us now
set
197
xl=
L--
G,
,2.y_:= l--
and
the
equations
will
xz=L+G,
# + t,
retain
the
dx: dF' dt -- _ ' We
will
2 y_ ----l + g -- t
canonical
dyt dt --
2
which
it
follows
nlIf we
assume
differ very ties x I and
(x,+
that + 4 i (xt+ x_)3 +-, 2
that
like
indicate
that x2
are
G
is
X!
+ ----a
the
to note
always
essentially
(X
eccentricity
in
4- i
----
ni---
i
_- X,
)3
,
_
-.
2
is very
value.
small,
Therefore,
addition
L : '/a,
and
that
the
one
L
and
G will
of
the
two
G = V_a(--T _ e' )
smaller
than
L
in absolute
value.
Therefore,
positive.
v/_ cosy, It will
be
periodic
with
us now
equation
of
assume energy
that
our
be a of of x 2
_I_ sin.y,.
period
2_ both
in Yl
and
If, on the other hand, it is x 2 which is very small, be a function of x I and of Yl, which may be developed
Let
198
and
the
_/_ cosy,
the
quanti-
equations
Let us assume that x I is very small. The function F' will function of a and of £ + g - t which may be developed in powers e cos g and of e sin g. Therefore, this will also be a function and of Y2 which may be developed in powers of
will
/197
become
dF' dxj " .T l --
x,)'
little in absolute x 2 is very small.
I would
xl
and will
have Fo--
from
form
and
in Y2. the function F' in powers of
V'_ sin.y,.
four
variables
F= C
.
x
and y
are
related
by
This equation may be approximately reduced to F0=
nates
Let of
us construct the curve a point in a plane.
The
equation
may
C.
F 0 = C,
taking
x I and
x 2 as
the
coordi-
be written (x,.i-r:)2C2C+ xl--x2_ = 4-
This
curve
and
it
which
has
two
is symmetrical
/198
asymptotes
with
respect
the
first
However, it should be noted that the only is of use to us is that which is located xj>o,
in
to
Based the two
on the values of C, following figures
of
these
portion in the
two
asymptotes.
of the curve first quadrant
x_>o.
the
Figure
curve
may
have
one
of
the
forms
shown
8
The axes of the coordinates are represented by the dot-dash line, the asymptotes and the utilizable portions of the curve are shown by the solid line, and the portions of the curve which are of no use are shown by the dotted line.
199
D
Figure
We
shall
assume
the form shown AB and CD. We
that
a value
in Figure 9 and shall no longer
9
is
assigned
so that consider
to
C, so
We creases OB
that
it contains two the arc AB.
should point out that when one traverses constantly from OA to zero, x 2 increases x2 and--increases constantly from zero to + Xl
the
curve
utilizable
this arc constantly
AB, x I defrom zero
If we now construct the curve F = C, assuming that Yl and Y2 are constants and x I and x 2 are the coordinates of a point in a plane, the curve will differ very little from F 0 = C and can still be represented by
Figure
this
arc
We
9. the
thus
It will have x2 ratio--will Xl
a utilizable
arrive
at the
following
system are
will
The location of the tangular coordinates
These _sin
200
three
functions
may
Yl,
if x I is
very
increase
be
be
AB,
constantly
method
of
represented
developed
small,
arc
and
in may
when
one
from
zero
to +_.
geometric by
powers be
and
the
of_xl
developed
traverses
representation.
point
cos
has
arcs
whose
rec-
Yl
and
in powers
of
to
_199
V_
cos
and _2sin Xl ratio-. x2
the
Y2
Thus, of
one
values
of
The to Yl,
and
Yl
only
and
functional
Y2,
and
Y2,
if x 2 is very
one
point
of Y2
and
determinant
with
respect
in
space
to each of
to xllx]
corresponds
point
the
They
small.
on
three
, always
the
only
depend
to each
utilizable
coordinates retains
the
system arc
with same
on
AB.
respect sign.
_x2
tion
We may within
therefore apply the results all of the region D where
However, But,
n 2 vanishes
if we
have
for
xI + x2
obtained n 2 does
in the preceding not vanish.
sec-
/200
x I + x 2 = 2. =
2, x I
>
0, x 2
0, we
>
shall
obviously
have X:--Xl
2
2
=
(xt+
_-4-
x_)= -I-
X 1
2
3
= 4
However, the first term of this equation is F 0 and, when compiling the curve F 0 = C, we assumed that we were dealing with the case presented in Figure 9. However, the case shown in Figure 9 assumes that
cannot
Since F 0 differs very little have at the same time
from
3
c>_, (unless
C is
Under have
very
the
close
conditions
to
F,
and
consequently
from
C, we
3
Fo<_
its
limit
3 _,
which
with
which
we
are
we
now
have
not
dealing,
assumed).
we
shall
not
n 2 = 0.
Thus, the results presented in the preceding section are applicable, and if we construct the asymptotic surfaces and if we consider the intersection of these surfaces with the half-plane Y2 = 0, the two arcs which are similar to those which we designated as AoA 5 and BoB 5 above will intersect.
201
I would like
to add one word to this.
The coordinates of the third body, with respect to the major axis and the minor axis of the ellipse which it describes, are -- according to the well-known formula
It may thus be seen that, when G changes sign, the second of these coordinates changes sign. As a result, the perturbed planet turns in the samedirection as the perturbing planet if G is positive, and it turns in the opposite direction if G is negative.
202
CHAPTER
PERIODIC
314.
Let
us
SOLUTIONS
consider
the
Xi's having
OF
a system dx/ dt
where periodic
XXVIII
are functions the period
THE
of
Xi
SECOND
TYPE
/201
equations
(f:,,2,
...
of Xl, x2, T with respect
..., to
n), Xn, t.
(1) and
of
t,
which
are
Let x_ ---:?i(t) (2) be
a periodic
solution
of period
We shall try to determine solutions which are very close T.
of
These solutions, the second type.
to
Let (2).
be
the
us consider Let
value
of
if
they
a solution
x i for
t = ?,(o)--
be
the
value
of x i for
0, _i+
t = kT
T of whether to (2)
equations
(i).
equations and whose
(i) have other periodic period is a multiple of
exist,
will
of
equations
and
be
called
periodic
(i) which
is
solutions
very
close
let
+, = ?i(kT) (k is
+ _i+
a whole
_i number).
The Bi's and the ¢i's, whose definition is the same as that given Chapter III, will be very small. Just as in Chapter III, it will be found that the _'s are functions of the B's which may be developed in increasing powers of the B's.
In order that the solution may is necessary and sufficient that +1 = '_ .....
be
periodic
q= = o.
having
the
period
kT,
in
it
(3)
203
L202
vanish
Due to with We
the the
shall
fact 8's.
that
the
_i(t)'s
are
assume
that
the
functions
periodic
X i which
functions,
appear
in
the
_'s
equations
(i) depend on a certain parameter _. Then the functions _i(t) will depend not only on t, but also on _. As regards t, they will be periodic of period T, with T being a constant which is independent of
Under these conditions, the functions the same, will depend not only on the _'s, that
4, whose but also
are
n +
coordinates
(3) will kT, will
this
of
a point
represent correspond
Since curve
in
space
having
a curve in this space. to each point on this
the _'s all will consist
vanish when the of the straight
definition remains on _. If we assume
1 dimensions,
A periodic curve.
B's all line
vanish
equations
solution,
at the
of
same
period
time,
(4) The
solution
straight line. tion of period kT.
(2) will
correspond
to
different
Due to the fact that this solution T, it is for that reason a periodic
points
on
this
is a periodic solusolution of period
But we must try to determine whether there are other periodic solutions which are very similar to the first or -- in other words -- if curve (3) includes, in addition to the straight line (4), other branches of the curve which are very close to the straight line(4).
which
In other branches
words, are there points on the straight of the curve (3) pass, other than this
line line?
(4)
through
Let
be
a point In
P
order
of
the
that
line several
point P, it is necessary or the Jacobian, of the
204
(4). branches
of
the
curve
may
pass
through
the
that at this point P the functional determinant, _'s_ with respect to the B's_ vanishes.
v
This condition is not sufficient, point, P.
for
several
real
Let us formulate let us add -S to
B's,
roots
of
_ is one
this
of
In order sufficient
and
equation
(see
the
means
No.
characteristic
is
a multiple
$'s,
We is
us
is not
_
try
shall zero.
_,
to develop
of
equa-
equation
(i).
be zero, it is necessary must therefore have
2i_. branches of the curve of the characteristic
sufficient,
the
B's
in whole
assume that the Jacobian This Jacobian vanishes
and
equal
In the same this Jacobian
tions
the
and
a more
pass through exponents be
extensive
dis-
set
to
We shall therefore that its derivative,
but
of
of
obtain
I,
or
fractional
In this (3) we
a multiple
this may be exponents 2i_ of _--.
assume that the Jacobian with respect to %, does
way, we shall assume that the do not all vanish at the same
case, based on may derive n -
powers
of
%.
of the _'s, with respect to the for % = 0, but will not be identi-
cally zero, in general. In order that necessary that one of the characteristic dent
point"
are
may We
315. This condition cussion is necessary.
let
80)
/203
the
respect to the us set the de-
thus
that the functional determinant that one of the roots he zero.
Therefore_ in order that several point P, it is necessary that one 2i_ a multiple of k-T-"
and
shall
of
k_T
through
_'s with and let
exponents
that
us
to pass
of the terms,
the
Let
curve
I_
C_ _T
which
the
equal to zero. We the equation for S.
e _T-
where
as we shall see at a later
of
the determinant all the diagonal
terminant thus obtained tion which is known as The
branches
the theorem in No. 1 of the quantities
the case, it is be constant, indepen-
vanishes for not vanish.
minors time.
of
the
% = O,
first
order
30, from n - 1 of equaB in the form of series
205
k.2
developed in whole powers of % and of the tn-_hquantity B, for example of 8n. Let us substitute
thus
derived
equation in
in
will
the
the
be
the values of
tn!b-hequation
developed
following
the
other
hand,
contain
a term
does
Finally, since not vanish, we
factor.
the
of
degree
the
total
group
A and
B are
constant
be
seen
that
If p imaginary,
that
this
tn_-h
us write
of
divisible
0
with
by
8n,
the
it
respect
because
to
now same and _
assume
terms
coefficients may
derive
8n
respect
_p+l _n ' _8_,
containing which from
to the powers of series is real.
that
not
holds true for p - tl_horder. order
are
the
8n must
are this
not in
),_', , and
or
not
only
as
the
Jacobian
zero
at
the
terms the
vanishes
time.
a
zero. of
a series
problem
is
A and B have opposite signs, of the second type exist.
all of its minors of I shall nevertheless all
to
%2B n
is odd, and if A and B have opposite signs, the series and there is no periodic solution of the second type.
second, etc., minors of the
206
Let
vanishes. For _ = O, 0 does not Let us assume that it no longer ..., p - tl__ degree, but that it
is even or if, p being odd, real, and periodic solutions
I shall but
of
8n.
p.
of
we
whlch progresses according to determine whether this If p is
term
the derivative of the Jacobian with shall have a term containing %B n. write
It may
series
derivative
therefore
C is
of
,%,) -_ o.
since the Jacobian the first degree. the second degree,
does
where
first
% and
point out that @ must be part of the curve (3).
vanish for _ - O, contain a term of contains terms of
I may
The of
form.
I may first (4) must be On
(3).
in powers
o(L
line
/204
for
the
is
% = 0,
the first, the assume that the
1205
According to the statements presented in No. 57, under these conditions, there will be not one, but p, characteristic exponents which will
be multiples
of k--_--.
Fromn - p of equations (3), we may then derive n - p of the quantities B in the form of series developed in powers of % and of p
last
quantities
the
_.
For purposes of brevity, I shall employ the _''s to designate the n - p first quantities B, and the _"'s to designate the p last quantities B. We shall therefore have the B"s developed in powers of k and of the B" 's.
the
p
Let us substitute these expansions last equations (3), and we shall
in the place of the obtain p equations
B''s
in
o,= a_ ...... _p=o, whose
first
orders degree
terms
will
be
developed
(5)
in powers
of
k and
of
the
B" 's .
Due to the fact that the Jacobian and its minors of the first p - i are zero, these first terms will not include terms of the first in B" which are independent of %. We must now determine whether
the first terms with respect to respect to k. Let
8i be
of equations (5) will contain terms of the first degree the B"'s, and at the same time of the first degree with
the
total
degree with respect to in powers of k. Let
group
the
of
B"'s.
terms
of
It
apparent
is
@i which
are
that
of
the
8i may
first
be
developed
Ol= o, °_ kO_÷ _,0_'J+...
be
this
first
expansion.
degree
with
According
to
The respect
the
8_k)'s the
will the
preceding
be
homogeneous
polynomials
of
the
_"'s. 80 i will
statements,
be
identically
zero,
Y
but
we
must
The
now
Jacobian
determine of
the
whether _'s
with IlO--
The product indicated to the n characteristic
by
the sign exponents
the
same
respect
holds to
the
true
for
_'s
equals
ei.
e_T), H extends _.
over
n
factors
corresponding
207
Let _I,
_2,
_n
"'',
be
these
n
exponents,
and
J206
let
?(=) ---_, -- e_=T) . The
Jacobian
will
equal
the
product ?(=0_(=z).
?(=n).
In order that % = 0, the Jacobian vanishes as well of p - I first orders. As a result, p of the exponents 2i_ of kS"
Therefore,
consequently, therefore be We
p of
divisible divisible
shall
assume
the
factors
by %. by %P.
that
what we already assumed the _(_)'s are divisible by xp+l. Thus,
the
The
for
_(_)
vanish
product,
% = 0 none
i.e.,
is
divisible
by
_ = 0 and
the
d_ the _
of
previously. Under by %2. Therefore,
Jacobian
for
as its minors are multiples are,
Jacobian,
vanishes,
will
which
is
these conditions, none of the product is not divisible
%P,
but
not
by
%p+l.
l!
As and
a result,
the
consequently
The second
none
simplest
degree
determinant of
case
do not
the
is
degree
us
of
assume
0 i for
It will
be
that
ni
8 i s is different
in which,
in
the
these terms of the second degree all the B" 's vanish at the same Let
the
e il' s vanishes
that
vanish
of
for
% = 0,
expressions
cannot time.
is
the
total
to
consider
from
zero,
identically.
for
vanish
group
at
of
the @i,
terms and
the
same
terms
of
of
the
in which time,
the
unless
second
% = 0. sufficient
the
algebraic
equations
T,i + ),0_ := o, whose
first
respect
to
If
of
I shall
208
are
homogeneous
the
B"'s.
these
solutions
plete
terms %
this
and
equations the
second
not
extend
discussion
have
real
polynomials
solutions,
of
we
the
second
degree
shall
have
periodic
with
type. the when
discussion treating
the
to
the
other
equations
of
cases,
but
dynamics.
shall
com-
Case 316. (i) do
Let
not
As
In
us
have is
Time
assume
depend
we
exponents
in Which
on
addition,
that
time
seen
the
Not
Enter
functions
Explicitly
X i which
]207
appear
in
equations
t.
in No.
always
Does
61,
in
this
case
one
the
same
also
of
the
characteristic
zero. if z_=?dt)
is
a periodic
solution
of
period x_:
whatever
the
In might
be
constant
the --
h may
?i(t+
holds
for
h)
be.
preceding
section,
there
a periodic
was
T,
we
assumed
that
-- no
Xi's
were
matter
what
solution
_i=?i(t)
and of
the t,
period
could
only
be
of
period
T.
The
period
was
therefore
The
same
is not
true
no matter
what
_ might
T,
We However, t = k
shall we
in this
be,
(T + Then,
z) the
of
case.
equations
We
designate
designate
(and #i's
not
for
will
be
periodic
functions
_. shall
(i) have
always
assume
a periodic
that,
solution
?dr).
period will depend on T O the value of T for the definition of the
always
shall
the
independent
_i= However, the T the period, and shall then modify extent.
since
the
U, in general. _ = _0, i.e., quantities 8
value
the value
of x i by
of x i by
I shall call for X = 0. We and @ to a certain
_i(0)
_i(0)
+
+
8 i for
8i +
@i
t = 0.
for
t = kT). functions _,, _ .....
_,
of
the
n +
2 variables
_, _.
209
If we of a point
continue
to
in
having
space
assume
that
n +
,'._ will
no
longer
may
vary
the
represent two
a curve,
parameters
the
$'s and
1 dimensions,
are
the
coordinates
equations
/208
o
(3)
but
• and
%'s
the
will
represent
% independently
However, we should point out that curves whose different points correspond to periodic regarded as being essentially different.
a surface,
and
since
we
continuously.
are drawn solutions
on this surface which may not be
If
is
a periodic
solution,
the
same
will
.ri=_(t no
matter
from
what
the
the
first
in
The
following
constant
h may
the
for
be,
and
this
new
solution
will
not
differ
reality, point
following
true
÷-h)
corresponds
to
_i=:_(o) and
hold
point
the
first
-_i(o),
corresponds
to
the
second
_--_(h)-?i(o). When
h
different
is varied
points
continuously,
do not
In particular,
let
correspond us
consider xi=
The
which
following
belongs The
point
to the
following
line
the to
will
second
point
solutions
the
which
describes are
a curve
actually
solution
_(t)
correspond
to this
solution
(4).
point _l= ?l(h)- ?do),
which respond
belongs to
to
the
a certain
surface
(4')
(4) making
solution z_=:?i(t_-h),
which
210
is not
whose
different.
actually
different
from
the
first.
up
the
surface
(3), will
cor-
Wemust now determine whether the surface (3) includes layers other than (4') approaching very close to (4'), i.e., whether there are points on in
the surface addition to
(4') through the surface
which other (4') itself.
layers
of
the
surface
Without limiting the conditions of generality, we B1 = 0 (or we may impose another arbitrary relationship In
actuality,
the
We a way
and
it
may
the
constan_
choose for
is
sufficient
h
which
we
to
take
p bJtrarily,
one
and
of
we
them
may
into
take
consid-
it
in
such
example, /,(h)
from
may assume that between the B's).
xi::fi(t-_-h)
different,
that,
/209
pass
solutions xi_f,(t),
are not eration.
(3)
= _,(o),
have
_1
--_-
O.
q.e.d.
be
If we reduced
impose this condition B1 = 0, the two surfaces to curves, and the surface (4') may be reduced
in
particular.
We would like (3) passes through
to again determine whether a point of the line (4).
another
(3) and (4') may to the line (4),
branch
of
the
curve
For this purpose, let us combine equation _i = 0 with equations (3). These equations will represent the curve (3), or a curve of which (3) is only a part. In the region under consideration, in order that this curve may not be reduced to the line (4), it is necessary that the Jacobian _I,
_I, _2,
_2,
"'',
..-,
_n,
_n with
_I with respect
respect to
_2,
BI,
_2,
"'',
-..,
Bn,
_, be
to
_3,
8n,
_,
zero
and for
that % =
of
0.
Since nothing distinguishes _I from other $'s, the Jacobians of the _'s with respect to T and with respect to n - 1 arbitrary B's must all vanish. That is, all the determinants included in the matrix of Nos. 38 and 63 must vanish at the same time. By pursuing a line of reasoning similar to that presented in No. 63, we may see that the equation for S must have two zero roots. As
a result,
2i_ k-_--" This exponent
is
must
two
already be
of
the
true
a multiple
characteristic
for
the 2i_ of-_--.
one
of
exponents
them
which
must
is
be multiples
zero.
of
A second
211
If this condition is fulfilled,
we shall formulate a system of /210 equations (3) and el = 0. We shall derive of a series developed in whole and fractional
n + 1 equations including and the _'s in the form powers of %. If second
the series are real, we shall have periodic type; if the series are imaginary, this will
I shall 317.
not
Let
continue
us
now
this
assume
discussion
that
the
any
solutions of the not be the case.
further.
equations
(i)
dt --Xi, where
time
enters
explicitly
have F--
in such
a way
that
we
a uniform
integral
C_
have dF x,--= o. _/._
We saw in No. 64 that spect to the B's vanishes, is zero. The
in this case and that one
the gacobian of the of the characteristic
_'s with reexponents
equations
+,--- +, -...-%
-- o (3)
are
not
then
different
since
F[7i(o) They
this
do not
However, case we
we
have
+ _i-+- +i]--
represent
F[?i(o)-_-
a curve,
according to have a double
identically
but
_i] = a. rather
a surface.
the principles presented in Chapter III, infinity of periodic solutions of period
in T
x,. = ?_(t), since
there
= _0 +
no
is one
_ and
to
which each
corresponds
value
of
the
to each constant
value
constant solutions
zi = _i(/),
212
each
of
them
corresponding
to a value
the
parameter
C.
We shall assign a fixed value C O to the longer have a simple infinity of periodic
with
of
of
X.
C, and we shall of period T
Due to the fact that equations (3) are not different, replaced
by
n - i of
them
--
for
example,
they may be
/211
by
+,_=+,..... +n, - o Let
us
then
consider
the
._,=_,
part
Equations of which
system
.....
___,=o,
F[_(o)+
(3') no longer represent is formed by the line
_] = C0.
a surface,
but
(3') rather
a curve,
_, = o.
the
In order line (4),
that another branch of the curve it is necessary that the Jacobian
+_, %, with
respect This Let
to
the
assume
may pass of
through
a point
on
. , +n ,, F,
B's
vanish.
may
be
written
that
we
have
condition us
(4)
in
still
solved
another
form.
equation
F(x_)= Co with
respect
to Xn,
and
that
this
solution
xn :: O(x,, x: .... Let us substitute this substitution. Equations
8 in place
(i)
are
thus
of
These
equations
(i')
, x,,-t).
x n in Xi,
replaced
,_- -= x_
by
(i=,,_
will
have
yields
the
..... the
and
let
X_
be
the
result
of
following
n-,).
(1')
following
periodic
solution
xi = ?_(t).
which Gl,
The is _2,
number assumed ...,
of characteristic exponents of this periodic solution, to belong to equations (i'), will be n - i. Let
C_n-I be
these
n -
i exponents.
These
will
be
the
same
as
those for this periodic solution x i = _i(t), which are assumed to belong to equations (i), suppressing the n exponents which equal zero. In order that equations (i) have periodic solutions of the type in the vicinity of a point on the llne (4), it is necessary sufficient that equations (i') have them, i.e., that one of the
second and n - i
characteristic
at
exponents
_i,
_2,
"'',
_n-I
is
a multiple
of-_-
_212
a
213
J
point
on
the
Thus,
line
the
(4).
condition,
of 41, 42, ent manner.
..., In
two
of
the
exponents
one
of
them
Let we
tional whether
time
which
us
shall
_n-l, order
assume
derive
powers these
F is that be
is
zero;
in
presented
above,
zero may be expressed it may be fulfilled,
must
be
condition
series
true is
which
I shall not are real.
is
for
Jacobian
true
a second
ordered
this
the
always
fulfilled.
are
extend
that
in a completely it is necessary
2in k--T-" This
of
this
this
B's
of %. series
was
multiples
that
the
which
for
the
exponent.
From
equations
in whole
and
discussion,
318. Let us now assume that the Xi's do not and that equations (I) have an integral
differthat
to
depend
(3')
frac-
determine
explicitly
on
F=C.
In this case, according to No. 66, two of the characteristic exponents are zero. If the equations have a periodic solution for a system of values of _ and of C, they will still have it for the adjacent values, so that we shall have a double infinity of periodic solutions _t = ?,-(t) which depend on the two parameters constant; it will be a function of Let
be
the
tween
us
then
values
We shall the B's
assign
a fixed
of x i for
_ and _ and
value
t = 0 and
for
Without
limiting
as was
given
shall
thus
We
The C.
C O to
C,
t = k
(T +
add equation F = C O , and then an -- for example, BI = 0 -- to the _, : +, .....
reason
C. of
the in No.
obtain
the +i= o,
we
and
T will
not
be
let
_).
arbitrary equations
relationship
_ ::o
conditions 316,
period
of may
(3)
generality, assume
be-
and
that
for
the
same
/213
$I = 0.
system F ::Co,
_,: o.
(3")
These equations represent a curve. The number of equations equals n + 2, but the n equations (3) are not different, and may be replaced by n - 1 of them. This is Justified by the same line of reasoning that was presented
214
in the
preceding
section.
System
(3")
may
thus
be
reduced
to
n + i equations.
The number of variables is n + 2 -- i.e.,
This curve (3") includes the line (4)
_i_O.
Let 8 i = O, _ = _0 be a point on this line. In order that another branch of the curve may pass through this point, it is necessary that the Jacobian of the first terms of equations (3") be zero or -- which amounts to the same thing -- that the Jacobian of n - 1 of the _'s and of F with respect to 82, 83: ..., nothing distinguishes B1 from the bians of F and of n - 1 arbitrary bitrary B's all be zero. This
from
condition
may
be
8n and T be zero. Finally, since other B's, it is necessary that the Jaco_'s with respect to r and to n - 1 ar-
expressed
Just as in the preceding the equation F = CO
in
section,
another we
shall
way. derive
the
following
::,, :_0(z,,x_..... v,,_,), and
we
shall
obtain
the
equations tl_i
According necessary that 2i_ kT
[if
(i')].
to No. one of
it
is
assumed
In
other
sary that of the periodic solution be
8's
a multiple Let us and the
or fractional point.
of
,t
dt
=Xi
316, them
of be
that
the
--
which
words
(i')
(i:_l,_,...,n--O. the n - i characteristic zero and that the other periodic amounts
solution to
the
exponents, it is be a multiple of
belongs same
thing
to equations --
it
n characteristic exponents [if it is assumed belongs to equations (i)], two be zero, and 2i_ k--T-"
assume that this condition is fulfilled. T from (3") in series which are ordered powers
of
Application
_.
I shall
to
the
still
Equations
forego
We shall according
a discussion
is neces-
that the a third
/214
derive the to whole of
this
of Dynamics
319. I would like to discuss the equations of dynamics in greater detail. However, in order to do this I must first present an important property of these equations.
215
Let _i and _i be the values of xi and Yi for t = 0. be the values of xi and Yi for t = T. Weknow that
Let Xi and Yi
f f ,z,+, is
an
integral
invariant.
We
f f with
the
double
This
may
x(
integral be
written
other
words,
dYi
the
--
an
exact As
--
over
an
Y, dXi--
_idr,
Let
area
exact
along
= o,
the
contour
we
find
Yi,tXi--
-
area
A,
_, d-r,i-_-
rdd';i
)
that
vary T, it is the derivative
dXi
_t)d(Xi
b-_,.)1
apparent that S will be a function of T. of S with respect to T by means of the _215
dF
d¥l
a'F = _iYi'
_
dF
-
ax,
have
ds 7t'f=: f
E
[dx d,l_d(Y
_)--dl_d(X+_) dY +(X--_)dyf
dV --(V
-- -q)d
d'r]
,
or
,IS JIT
:Jf-._[dV _._ d? d(y.,__).+_dr 7rE d(X dF
--(x
216
the
expression
equations
or,
of
differential.
320. If we us calculate
We
A.
differential.
a result,
an
arbitrary
i q- _id_i)
dS = _[(Xi--_,),/(Y_-_-_,,.).-(V,.-is also
have
== o,
d_f
is extended contour.
E(XidYi-
is
i
therefore
as follows
the simple integral an arbitrary closed In
dY
extending
E(x,
where i.e.,
shall
integrating
by
parts,
we
have
--_)d
g£ -
__) -(Y -- _)d
dF] _
d_j '
dS
v
[(X--_)_-/_+(Y--5)_y dF
+:t
dX+
dY
)
or finally _dS
and
-_F
-Z [ (X--_)_X dF +(y___)dF]
We shall set we shall have
For
T = 0, we
the
arbitrary
,lS
_(F--C)--x
have
dS
3Y
+ arbitrary
function
of T equal
function
of
T.
to a constant
-2C,
S will
identi-
dF (X--_)3_+(Y--_)_?
= 0 and
consequently
S ==const. We cally
Let
shall
for
take
T = 0.
this The
constant
function
321. Let us determine us first consider T as
a maximum a uniform
to be S is
zero
thus
the maxima a constant.
so
that
completely
and In
vanish
determined.
the minima order that
of the function S. the function S has
or a minimum, assuming that this function S may be regarded function of the variables X i + _i and Yi + ni in the region
consideration, variables are
it is necessary that zero -- i.e._ that we
The corresponding solution period T, and this period T is at hand.
has
We shall a maximum
and
in
no or
is one
Xt == _i, dS dT
if
with
X =
$, Y =
n, we
respect
to
these /216
therefore a periodic solution of of the known quantities of the problem
longer regard T as a known quantity. a minimum, it would be necessary that
addition
However,
its derivatives have
as under
In we
order first
that have
S
Yi = rJt,
--0.
still
have
dS dT=2(F--C)' from
which
it
follows
that
217
The corresponding solution will period T.
still
be a periodic solution
However, the period T will no longer be a given quantity. energy constant C, which did not enter the preceding case, will given quantity.
of
The be a
The two methods for determining the maximaof S are related to the two methods of interpreting the principle of least action, that of Hamilton and that of Maupertuis. This will be clear to the reader after the following
chapter
322.
The
following
has
been
definition
read. of
the
function
S may
also
be
modified
in
the
way.
In a large number period 2_ with respect assumed to be periodic
of applications, F is a periodic function of to the Yi'S. In this case, a solution may be when X i = $i' and when Yi - ni is a multiple of
2_. It
is
then
apparent
that
dS _= X[(Xt-where ml, m2, will still be We
We
shall
..., m n an exact thus
dS dT
_F
shall
set
set
_i) d(Y_--_- r,i) --(Yi
are arbitrary differential.
- _'--
whole
am_)d(Xi+
_i)],
numbers,
the
expression
dS
]217
obtain v (X
_) _t_ dF +(Y
dS _-1_---- 2(F--C)--z For
if we
T = 0, we
_tF] --_,--:;nr.)dy]+
(x--_)_Lf_(YdF
arbitrary
_,--2m_)d?
dF
function
of
r.
]
have dS =: Z _mix d_i.
We
which
be
218
shall
concludes
set
the
S = 4_.Xmi_r,
determination
of
the
function
S.
Assuming that T is a given quantity, the maxima and minima of obtained by setting its derivatives equal to zero, which yields
S will
X_ = _1, The Yi-
_i
corresponding
is
solution
a multiple
If
T is not
in
addition,
of
27.
given,
it
is
The is
Xi
and,
Yt = _l + _ m, _.
_-
still
period
first
_i_
a periodic T
necessary
Yi
_-=
_i-'k-
solution,
since
is given. that
2mi_
dS
from
which
we
have F
='=
C.
323. It is now necessary that we learn to distinguish between the real maxima and the real minima of S. Up to this point, we have only determined the condition for which the first derivatives of S are zero, but it is known that this condition is not sufficient for providing a maximum. It is still necessary that the second derivatives satisfy certain inequalities. Let and
let
us us
first regard
assume T
that
the
conditions
presented
in No.
319
hold,
as given.
Let x_ = ?dt),
be
a periodic
solution
)'_ -. ?_(t)
of period
T,
?i(o)= ?t(T), A maximum
or a minimum
of
the
so
that
/218
?_(o)= ?_(T). function
S may
correspond
to
this
solu-
tion. Let x_=
?i(t)+xi,
xi=?i(t)+
be
two
solutions
I shall neglect the variational
which
xi,
very
little
Y'i'
x"i'
Y"i
squares and may assume equations (see Chapter
that IV).
these
assume
differ
r
that
x'i,
from
this
are
periodic
small
enough
quantities
solution. that
satisfy
we
may
the
219
Let _'i and _ be the values of xi' and Yf' for t = 0; _i and Y'i -the values of x_ and y_ for t = T. In order to determine whether S has a maximumor a minimum, it is sufficient to study the total group of second degree terms in the expansion of S in powers of the _'_'s and the n_'s. It may be readily
seen that this group of terms may be reduced to
Let us study the expression ...... - (x,-yi--yi,rl). According What If
tions
56,
is the
form
there
solutions
The
to No.
are
having
ak'S
are
the
We
shall
of
expression the
n degrees the
of period
this
must
general of
to
of
the
We
shall
freedom,
finally
Therefore,
we
of
shall
to
the
a constant.
variational
have
n
characteristic
exponents,
n - i other
the
exponents
n - i exponents have
the
- 1 particular
and
the
O's
are
solutions
the
the
2 tn_-h
220
func-
the
form
y_ .= e-axtO_._(t)
-a k which
particular
general
having
are
equal
and
have
solution
A, B,
C, D's
r_= d-_ solution
will
be
d_i
d,_;.
x_ = t _[ ÷ +,.,
y_ ::: t _: + +;..
solution
are
the
opposite
/219
may
be written
XBke-a_tOO&i(t)+C
_-+ d_t
D (d?t) t _.[ + +i ,
),,i = E AA.e'_lt O'&i(t) -4- E Bi, e-a_ t O_i ( t ) + C -_1_ d?_ *- D' ( t _[ d?_ + _', ) , the
periodic
ak.
obvious
x_= XA_e_ttOk.,.(t)+
where
equations?
form
x,= -d[' and
reduced
solution
x_ = e-attO_.i(t),
sign
be
T. have
corresponding
(i)
integration
constants.
In the sameway, we shall have x_-="A'_-e='tO_i(t)+XB',.e-='tO_i(t)+C . .
'd?:--_ +D'(t--dt
_d?t ++,)
1!
with
a formula The
Since
is
similar
A' , B' , C' , D' s are
Let us become
will
which
this
substitute a bilinear
form
this form will in the matrix
must
be
for
new
Yi"
constants.
these values in expression (i). This expression form with respect to the two series of constants
vanish
A,
B,
C,
D,
A',
B',
C',
D'.
identically
A,---- A_.,
B/, = B_.,
a linear
form
C -- C,r
with
At B,, ,% R, A2 A', B2 B; -.. ....
The coefficients of (i) must be reduced
In general, none of these exponents
the
It follows determinants
from
this
as
that
The
CA),,
coefficient
a factor, only
we
that
I may
must
cannot
and
cannot
AkB)--
be
have
B_,%.,
determinants
contained
c'C D'D .
constants,
since
a term
expression
be
zero,
containing
one
and
of
B_.B)-- BiB_ ,
A_D'--DA_.,
BkC'-- CB),,
of
this
must
be
reduced
determinants
which
term
to may
A_B_.--BkA_, so
",Bn-t B.,_,
the
/220 A_C'--
the
A,,-t _'._,
to
of the characteristic exponents will will not be equal to each other.
A_.A':-- AjA'k,
because
D --- D'
respect
this linear form to a constant.
two
for
BkD'--
contain
DB_,,
one
of
the
exponentials
a constant. enter
in
our
form
are
therefore
CD'--DC',
write n
i
-(xiyi--y,x_)
t
= X M_.(Ak [_ -- Bk.A_-)+- N(CD'--
DC'),
(2)
221
where the Mk and N's are constants. I may state that Mk cannot be zero, otherwise expression (i) would not depend on the constants Ak, A_, Bk, B_. If we then assumethat all of the constants _ and B', C' and D' are zero, with the exception of the two constants A_ and B_ to which we may assign given values which are different from zero, we would have a relationship
which would be linear with respect to the unknownsx_ and Yi' ' and where " " would be given functions of time which are the coefficients xi and Yi different from zero. Such a relationship cannot exist, since the 2n x 'i and
variables If we variational changing
Yi' are
independent.
M k cannot
change t into t + T, we shall obtain new equations, and these new solutions will
the
be
zero.
solutions of the be obtained by
constants
A,t-,
into
A/.c_
In
Therefore,
order
13s,.,
Bke-:_
C,
T,
C-_-
D DT,
D.
to have :: (x_-,,_-- v', _',.),
it will
be
sufficient
to
set
A_ = A_e_ r, from
which
we
324.
B_
=
following
also 2.
(i)
D'= D, f221
In
order
to
,t t-#
Mk(e-:qr-
discuss
equation
+_k
real.
e'xtr)A,t. Bk -- NTDI. (3), we
must
(3) distinguish
between
The
exponents
are
01.i,
The
functions
_,
real. The
exponents
+e k are
purely
imaginary,
and
the
square
e_
k.i
are
imaginary
is
negative. Then
conjugate. 222
C'= C -+ DT,
Bke-_i T,
O_..l, %i,
and
expression
cases :
i.
are
in
have ,f_ ,t ,l ~(Xir, i -
several
the
the
functions
0k. i and
8_.i,
8'k.i and
and
real
3.
The
ponents, and
exponents--+_k
we
shall
conjugate
of
then the
are
have
complex.
the
Among
exponents
exponents--+_k,
the
characteristic
_+_j which
will
be
ex-
imaginary
and
o+.,., o;.,, o;,., +'7., will
be
imaginary
and
conjugate
of
Ok.i,
%,.i,
01./,
O_.i.
are real. In order to x'i's and the Yi' 's B, C, D, we shall have 2n equations which we by setting the following in the equation for
Let us now assume that the calculate the constants A, shall obtain, for example,
xl t --- o,
t=T,
t = a'r,
These 2n equations are linear with D. The second terms are real, and and conjugate pairwise. When
and
we
change Bk
do not
2.
A k and
Bk
interchange
when
3.
Ak
Bk
change
Aj
and of
change
into
and
Bk
are
real
2.
Ak
and
Bk
are
imaginary
3.
Ak
and
Aj,
Let Bk,
second
i.
C and
us
_k
is
is
real;
purely
and
Bj when
assign
C, D, of
o k is
real;
and
and
Bj
conjugate
imaginary;
_k
is
complex
and
imaginary
values as
(2) must
are
conjugate
imaginary of
when
_k
is
purely
and
conjugate
when
and
Y'i to be
real.
imaginary; _k
is
/222
aj.
real.
are
as well
when
and
D are
conditions
term
following
Bk
imaginary
Finally, These
_k
:
Ak
and
when
_j.
i.
complex,
A, B, C, or imaginary
_ __Ff-_..
A k and
conjugate
t =(_n--1)T.
respect to the 2n unknowns the coefficients are real
i.
Therefore
Ak,
_-_into
...,
sufficient
for
satisfying
these
to be
the
constants
real,
and
in
_i
conditions A_,
order
B_, that
to
C' , D'. it may
the
constants
Then be
the
real
the
is necessary: That
M k is
real
if
_k
is
real;
223
2.
That Mk is purely imaginary if
_k is purely imaginary;
3. That Mk and Mj are imaginary and conjugate if complex, and imaginary and conjugate.
_k and oj are
Form (3) contains a term and does not contain another term depending on Ak or Bk. If the exponent _k is real, the presence of a term containing AkBk is sufficient for providing that the quadratic form (3) can be defined. Therefore, if only one of the exponents ok is real, cannot have either a maximumor a minimum.
the function S
Let us now assumethat two exponents ok and aj are complex, and imaginary and conjugate. Let us cancel all
the constants except for Ak,
and the
form
(3) may
be
two Let
terms us
are
assume
Aj)
reduced
_I_ (_-=_'r These
Bk,
to
_ c=_T) Ak Bk _i- _lj (e-_, _ - _=/r) Aj B7.
imaginary that
B j,
Ak
and
does
conjugate,
not
change,
so
that
form
and
that
Bk
(3) is
changes
Aj,
which
is
imaginary
and
conjugate
of
Ak,
will
change
no
Bj,
which
is
imaginary
and
conjugate
of
Bk,
will
change
into
Therefore,
form
(3) will
change
sign;
Therefore, if only one of the exponents S cannot have either a maximum or a minimum. Let imaginary In that
all
us
now
and order of
assume
that
conjugate, that
the
S have
and
ok
is purely
the
a maximum,
it
quantities 51k slnjkT,
224
product
--NT
therefore, _k is
it
real. sign.
longer, -Bj.
cannot
complex,
and
the
Then
AkB k is
sum
of
and
sufficient
is necessary
two
defined.
function
imaginary. the
A k and
be
Bk
are J223
squares.
be negative. In order that S have a minimum, it cient that all these quantities be positive. Mk
It should be pointed out that all and _k are real.
is necessary and suffi-
these quantities
are real, because
325. Howmay these results be modified if it is assumedthat the energy constant is one of the given quantities of this problem? We then have identically \ d_ ._'+ -j-fy: where
we
assume
that
periodic
functions
The solution
constant
dF in _x
_i(t)
dF and_y , x i and
and
value
of
for
the
infinitely
the
function
relationship
is
have
close
a linear
F must
yi=
been
replaced
by
the
be
the
same
for
the
periodic
_(t)
solution
xt==?i(t)÷ x_, This
Yi
O[(t).
x_= ?,.(t),
and
=o,
y_
?_(t)+y_.
equation
between
the
constants
Ak, Bk, C, D and
the
coefficients
must
be
independent
It follows from this that A k and lationship, since these constants are since
this In
exponential
addition,
cannot
C is no
t.
Bk must not be included in the always multiplied by e±_k t and
re-
vanish.
longer
x_ = ?dr)+
of
included,
C di"
since
7_- : ?i(t)-t
the
solution
C dr" .'
where C is a very small constant, may be deduced from the periodic solu- /224 tion by increasing the time by a small amount C. Consequently, this solution corresponds to the same value of the energy constant as does the periodic solution.
fore
Our relationship, be reduced to
which
cannot
be
reduced
to
an
identity,
may
there-
D_o.
225
However,
if D
In that
is
order
the
zero,
that
the
S may
term
have
-NTD 2 vanishes
a maximum
or
in
the
form
a minimum,
it
C3). is
sufficient
quantities _M_ sin ==cT
¢_ all
have If
these
the
same
sign.
there
are
only
degrees
of
freedom,
only
two
degrees
there
is
only
one
of
quantities. Therefore,
purely
if
there
imaginary, 326.
so
two
¢,
Let
us
are
the
function
now
assume
S always that
of
has
the
freedom
a maximum
conditions
and or
given
in
if
ak
is
a minimum. No.
322
hold,
that dS
and
let
maximum
=
Z[(Xl--_i)d(Yi--_i)--(Y_--_i--_mi_)d(Xi+_i)]
us
assume
or
a minimum,
that
T it
is
a constant.
is necessary
In that
order we
that
have
S may
a periodic
have
a
solution
where 9i(t + T)--=_(.t);
Let
us
then
consider
?_(t-i-T)::
a close
and are
this the In
that that
have
all all
the
discussion same. order the the
same
If it the problem sufficient
is
that
will
proceed
there
exponents quantities
be
_k
Yi='_'i(t)
in
the
a maximum
are
purely
Mk
. _kT
_-Yi; '
same
or
way
as
a minimum,
imaginary.
The
above,
It
it
is
is
then
results
necessary necessary
sign. assumed
that
the
energy
at hand, D is zero, that the quantities Mk
226
solution
s.
Zi=?i(t)_-xi,
?_(t)-r- _mi_.
the .
constant term
_kT
is
a given
-NTD 2 vanishes,
quantity and
it is
of
/225
_J
all have the samesign. 327. What will now take place if the equations have other uniform integrals in addition to the energy integral and if, consequently, some of the characteristic exponents are zero? A discussion similar
to that presented above could still
be employed.
For example, let us assumethat our equations have p other uniform integrals, in addition to the energy integral: Ft, F:, in
such
a way
that
the
at a time are zero. then know that 2p + assume that all the We
shall
then
brackets
(3) will
have
n
- p -
ENkTD _ is If we
now
F k]
of
these
then
a sum assume
of
1 pairs
of
of
have
a maximum
taken
two
terms
that
or
which
constants
are Ck
similar
and
D k which
become
the
c_LT)A_B k_v _Nh-TDk, similar values
to the of
our
quantities of the question ounder consideration, all be zero, the terms NkTD _ will vanish, and S may
integrals
constants
and p + 1 pairs C and D.
E3l,(e__fr. where
[Fi,
Fp,
Based on the statements presented in No. 69, we 2 characteristic exponents are zero. We shall other exponents are different from zero.
to the constants A k and Bk, are similar to the constants Form
...,
a minimum
will
still
term p +
NTD 2.
1 integrals
are
given
the constants D k will the condition under which stipulate
that
all
the
/226
quantities
have
the
same
sign.
I shall not insist upon this point, because -- in the case of the three-body problem -- either we shall be dealing with the restricted problem presented in No. 9, or we shall be able to decrease the number of degrees of freedom by applying the procedures given in Nos. 15 and 16. In the case of the reduced problems of Nos. 9, 15 and 16, there is no more than one single uniform integral, that of energy, and there are only two zero exponents, as we saw in No. 78.
227
Solutions
The
of
the
Second
Type
for
Equations
328. Let us change T successively function S defined above depends on
of
Dynamics
into 2T, 3T, T, and let
...,
mT,
...,
S,,,==S(/,_T). Let T is
us
try
to determine
the
maxima
and
minima
of
Sm,
assuming
that
a constant.
If we consider a periodic solution S are zero.
a periodic of period
solution of period T, this will also mT. Therefore, the first derivatives
be of
m
In that
order
all If
the all
that
there
exponents the
may
be
a k are
a maximum
purely
or a minimum,
it
is necessary
imaginary.
quantities ___ , m_.T MI: s,n--
are negative, there will be a minimum. This
is
the
will
first
be
a maximum;
point to which
If we assign all the possible the n - i quantities (i) will have tions of signs. Let
us
set,
for
(1)
purposes
if
they
I wish
are
to draw
all
positive,
there
attention.
whole values to the whole number m, in general all the possible combina-
of brevity, akT _
and
[ok,
/227
let _k
Let
us
assign
...,
"- 2"qk_:"
possible
Zn_ 1 are
whole
assume
that
n - 1 dimensions, we shall obtain that there will be an infinity of having
z2,
the
//20)4
having stated
of space
Zl,
all
_:
n - i dimensions,
the
no
values
coordinates
to m of
and
a point
228
quantities
given
in
the
matter
following
in
an infinity of points. these points in every how
small
it may
In order to demonstrate this, I need only refer to reasoning employed to establish the fact that a uniform real variables cannot have n + i different periods. The
to m k.
table:
If we space
It may section
be.
the line function
of of n
Oj
will
play
the
There
role
would
of
be
i.e., if one of the generally, if there one
single
period
--
periods
an
will
the
b's
Let us equal
are
. . ._,
in
this
exception,
if
_
llne
of reasoning.
these
periods
were
not
quantities _ were commensurable with were a linear combination of the z's i.e., _|c,)l+
where
01
whole
disregard
if
b2(o
there
!-_...4-
were
b,_--l_O.--t-_
a relationship
different
--
27, or, more which had only having
the
2nb._O_
form (2)
numbers.
the
case
of
this
exception.
The
quantities
(i)
We may choose the whole number m in such a way that these quantities represent a combination having a given sign -- i.e., that there are numbers zk which satisfy inequalities having the form a,
made
the
ak's
equal
This results above.
I *_, a,
or
.... , a,_-t<=,,-t
(3)
n.
directly
from
the
statements
which
we
have
just
Let us move on to the case in which we have a relationship of form (2). We may always assume that the whole numbers b are primes themselves. In this case, the expression
bt
has
only
the
period
zt
q-
b:z2
q-.
• • "Jr-
bn-!
z..-I
/228
the among
(4)
27.
In order that there may be no numbers zk satisfying the inequalities (3), it is necessary and sufficient that the difference between the largest value and the smallest value which expression (4) takes -- when all values which are compatible with the inequalities (3) are assigned to be Zk'S -- is smaller than 2n, i.e., smaller than a period of this expression (4). This
difference
is
obviously
as
follows
229
kJ
_(lb, I-4-lb:l +... 4-lb,,-, !), and
we
must
therefore
have
lb, l + Ib,l +.-.+ The one
of
must
inequality
them
which
In
this
be
zero,
We
have
The two
of
can must
case
since
Then
the
can
of
multiple
of
2_¢-_t T
of
the
This
2_.
is
determined
the
case
in which
be
equal the
we
valid
note
that
of
b's
one
if
2_.
up
all
are
zero,
except
This
means
that
to a multiple
of
the
the b's
between
the
express
characteristic only
the
for
_k
of _¢_T
=k'S
is
zero.
are
zero,
the
_k'S
except
for
+i.
difference
may
of
a multiple
only
If we
the
all
(5)
+i.
equal
only
stun of
if
is
must
multiple
Two
_k
excluded
which
hold
equal
_k must
equation
them
only
Ib,,-,1G_.
_k'S
this
which
are
of
only
result
exponents
exception
two
will
still
determined
in
be
will
another
be
up
a
to a
way.
equal.
exists,
and
it may
be
readily
excluded. 329. eration
Let
depend
three-body When
us now on
an
problem, we
vary
assume
as we _
that
arbitrary
the
in
also Chapter
vary
always retain interested.
230
of
just
dynamics
as is
the
under case
consid-
for
the
continuously,
the
continuously,
periodic
solution
yi = ?i-(t)
as we
may
determine
from
the
discussion
III.
The quantities plained in No. 323
of
_,
know.
xt = ?_(t), will
equations
parameter
the
M k will also vary continuously, but -- they can never vanish. Therefore_ same
The energy constant the problem at hand,
_,
and
will be but this
_t
is
their
sign
regarded as one of given quantity may
alone
-- as was exthey will in which
we
are
the given quantities depend on _, and we
shall choose tion remains
it in such constant.
The exponents tinuously. Let us
a way
_kwill clarify
that
the
period
T of
the
periodic
also vary continuously when we to a certain extent the manner
solu-
vary p conin which this
variation should be handled in the case of the three-body problem. For = O, all the exponents are zero. However, as soon as _ ceases to be zero, the exponents cease to be zero also. One of these exponents can only to
vanish,
another
or become
equal
characteristic
to
a multiple
exponent
for
of
2=q_-7 T
certain
or become
'
special
values
equal
of p.
330. Let us consider a periodic solution of period T, such that all the exponents ak are purely imaginary. This is what we designated above by a stable solution. In Chapters III and IV, we proved the existence of these solutions. Let
us
consider
continuously, 27 -_ an infinity
one
_--_, of
of
the
-- which
times.
Let
exponents,
is
real
us
assign
a!
al,
-- will
for
example.
become
a value
P0
When
p varies
commensurable to p,
such
with
that
2k_
V--7= "_' where
k
ition,
in
and this
p are value
At a later the numerator In
number
the prime does
not
whole
correspond
point_ in No. t and not k_.
334,
any interval, no matter of similar values.
If m
is
an arbitrary
numbers
whole
zero.
minimum
In of
P0 -- E to P0
being
addition, _, __,
since
this
P0
shall
number,
see
it
for
or
why
may
this
In
a minimum
I have
be,
there
value
P0
addof
placed
is
the
/230
_"
2k_
an infinite
expression
T
does
expression
themselves.
a maximum
small
sin_ is
to
we
how
among
not
will
correspond change
sign
to
a maximum
when
or
p passes
a
from
+ c.
For example, positive.
let
us
assume
that
it
changes
from
being
negative
to
231
kj
that
Pursuing the line of reasoning we may choose the whole number
presented m in such
M_ ,iapm_T
72_ have
all
possible
Under
this
a maximum, B =
B0 +
of Sm. p,
assumption,
since e,
our
since
all
our
periodic one
of
of for
In
order
signs,
_ =_0
expressions solution
these
and
that
-- e, our will
will
be
no
they
negative.
longer
the Maxima
subject
all
have
However,
become
further,
negative.
Sm. p will
correspond
Considering this
are
function
will
to pursue
will find expressions
n --l)
expressions
Theorems 331.
(k=_,3,
_--_-_ .....
combinations
in No. 328, we a way that the
it
have
for
to a maximum
positive.
is
necessary
illustrate one property of the maxima. Let V be a function variables Xl, x2 and z, which may be developed in increasing these three variables. I shall assume the following:
to
of the three powers of dV
i.
For
no matter 2. z
Xl
what For
= x2 z may
=
0, V vanishes
as well
as
its
derivatives
dV
dx I, dx 2,
be;
x I = x 2 = 0,
V has
a maximum
for
z > 0
and
a minimum
for
/231
<0. It may
be
stated
that
the
equations dV
dV
_l_. = _t_ = o have
other
Let
real
us
solutions
develop
V in
in
addition
powers
of
to
the
z, and
solution
let
V = Vo+ z\rl + z_V_ + .... The functions V 0, VI, V2, ... may themselves be developed in powers of x 1 and of x2. However, these expansions will contain neither terms of degree 0 nor terms of degree i, because -- no matter what z may be -- we must have dV dV V = 71_ - d_ -- o for x I = x2 In
232
= 0.
addition,
V 0 does
not
contain
terms
of
the
second
degree
either.
Without of
the
the
z
second
maximum
degree
to
the
terms,
case
of
it
the
is
impossible
minimum,
to
when
pass
going
from
from
the
z
> 0
least
we
case to
< 0.
Conversely, assume
this
V I will
is
the
contain
case.
Let
first us
then
degree
terms,
consider
the
dVo
dV,
dV_
dVo
dVt
-_- z_ dV,
at
shall
equations
(1) o = _7_ + : 7lT, which
must
he
solved.
Let U 0 and the statements
to and
U0
is
of
U 1 be the lowest which we have
the
degree
p
(p--a)l_=t; W
may
be
--
developed
in
We obviously
=
-,
Ult-P of
depending
will
also
yl t
and
divide in
H 0 will
of
be
will
The
=
the
will
t may
not
x2=y2t, of
t.
±Utt-p+
than
2.
V=XVtP;
Let
us
be
first
Uot-p=
are
two
other
I have
in
of
set
Let
us
set
a=_lP-'
set
± U_+
+__tP-2.
dV
dUt
dV
dU,
dxt
dxt
dxt
and
of
a H
tl :
Ito+
be
the
I
shall
The
t when
certain
let
polynomials
p.
dzt
powers
U_,
homogeneous degree
z =
include
factor,
the
larger
According degree,
_Vo.__t_Vl._.t_V2-1-....
U0t-P
how
It
this
being
of V I. second
/232
developed
y2 t. by
powers
which
U_
2 and on
be
p
V 0 and of the
have
and
degree
with
powers
Wo=
one
degree terms of discussed, U I is
xt=ytt,
_r=
U_
_T, +""
x I
power
in
employ
Yl
the
and sign
Y2 +
-or
expression
and of
quotient.
x2
are
t as
a
This
replaced
by
factor.
Let
quotient
developed
us
written
of
the
tilt
+ t 2H,+_..;
expressions
dW,
dU_'
dWk
dU_
ay,
,ly,
a.r,
art
vanish.
equations
dV
dV
2_ =_,
=o
233
kJ
may
be
replaced
by
the
following
equations
dW ll----o _l_ _o. I shall
prove
that
we
may
derive
the y's
from
these
equations
form of series which are ordered in fractional and whole which vanish with t and which have real coefficients.
powers
In order to do this, according to statements presented and 33, it is sufficient to establish the fact that for t = equations have a real solution of odd order. For
t =
0,
these
equations
may
be
reduced
in
the of
in Nos. 0, these
t,
32
to
dW0 _ _o,
II0_o, or
d\V_. dU_ dW,. dU_ dy---,," 'O'---', -- -2yq'dy----1 ---o
(2)
and
dU', + dU; +-- dr--; ,ty, = o. (3)
assume
Equation (2) indicates that W k has a maximum or that Yl and Y2 are related by the relationship For
the
present,
if we
assume
that
Yl
and
Y2
a minimum, if we U_ = const.
are
the
coordinates
of
a point in a plane, the relationship U_ = const will represent an ellipse, because the quadratic form U I (and, consequently, the form U_) must be defined in order that V may have a maximum or a minimum. Due to the fact that an ellipse is a closed curve, the function W 2 must have at least a maximum and a minimum when the point Yl, Y2 describes this closed curve. Therefore,
whatever
the
constant
UI, equation (2) will have at least because we have seen in No. 34 that ponds to a root one independent evident. Under First identically
234
case.
value
may
be
which
is
assigned
to
two roots, and two roots of odd order, a maximum or a minimum always corres-
of odd order. At this point, where we have no more than variable, the theorem presented in No. 34 is almost selfthis assumption, we may distinguish between two cases: U' 0 is
not
a power
of
UII.
dV¢o dU',
d\Vo dU_
-
dr,
_0,
&,
In
this
case,
we
do not
have
We
shall
therefore
have
Wk
"°Equation
H 0 =
constant
value
same
0
is is
then
for
the
We
derive--
and
dU'o dU't
dU o dU',
dy, <¢y,
_zy, a r, - o.
homogeneous
in
Yl
is
to
U_,
which
values
= W0,
ratio
assigned yl --. Y2
and it
Yl.
No
will
provide
matter
what us
the
with
the
Yl may
from
equation
(2)
and,
according
to
the
preceding
Y2 statements,
we
Let =
_2u
Y_j_l_ Y2
and
let
shall
obtain
at
_i _2
one
these
us
be
of
substitute
least
in
two
solutions.
equation
equation
(3)
may
be
reduced
of
Let
(3).
U'o = A u., and
solutions
us
We
shall
us
a
odd
set
order.
Yl
/234
_I u,
Y2
have
U'l := B u'
to AuP-I-_-B--=-o.
If
p
-
2
is
odd,
this
If
p
-
2
is
even,
we
If
A
and
B
have
the
equation
may
will
give
distinguish
same
sign,
between
we
shall
take
real
value
two
cases.
the
lower
for
u.
sign
A up- i -- B = o. If
A
and
B
have
opposite
signs, A
and
we
In
shall
have
every
Thus,
case,
two
these
equations
Second
case.
real
(2) We
ztp
values
real
and
we
-i-!-
B
for
shall
::
will
the
upper
sign
Op
u.
solutions
(3)
take
are
always
simple.
have
solutions
of
odd
order.
have P
u; = h(u; We follows
shall
begin
by
solving
equation
p --1
BA(U_),
)r. (3),which
may
be
written
as
-±,=o.
235
This real
equation
and
simple,
definite
shall
order
that
the
value
The
We
is not that
of
U_ having
to
U_,
maxima
sufficient
and and
thus
been
in order minima
of
have
therefore
established
odd
always have real solutions beginning of this section let
V be
we
assume
i.
V may
2.
For
have
the
be
the
of has
because
solve
_I
is
value
is
a negative
suitable, it is as a consequence,
we
may
equation
As we
the
have
assign
(3) we seen,
fact
that
we
equations
odd order. The theorem thus been proven.
a function
this
need
only
shall
]235
derive
of n +
(2)
presented
and
(3)
at
the
i variables
..., x,,
and
=
and
of
following:
developed
following,
This
order.
v,, _, I shall
U_.
determined,
to
of W k.
solutions
Now
of
solution may be ' be negative; for U 1
two
332.
value
+.
value the
the
the
found
sign
value
determine least
the
us with
this
In
choose
constant
at
but
form.
necessary we
provides
in powers
no matter
of x
what
dV dxl
dV ax,
v ==-=_-
z may
be
dV
.....
z;
=o.
3. Let us consider the group of terms of V which are second degree terms with respect to the x's. They represent a quadratic form which may be equated to the sum of n squares having positive or negative coefficients. When z changes from positive to negative, I shall assume of these n coefficients change from positive to negative, and n - 2 other coefficients do not vanish. Under
these
conditions,
it may dV
dV
dx-_ = _ have
real
solutions
which
differ X|
236
be
stated dV
.....
X 2 ----- • • . :_-- ,._tt
--2
the
two the
equations (i)
d_
from _
that
that that
O,
Let us develop V in powers of z and let us set
Let U0 and U1 be the group of second degree terms of V0 and VI. The group UI is a quadratic form which may be decomposedinto a sum of n - 2 squares, because we know that, for z = 0, two of the coefficients which were in question above vanish. 1236 tional
Therefore, if we consider the discriminant determinant of duo
dUo
of U0, i.e.,
the func-
dUo
with respect to
this determinant order. However, third coefficient
vanishes, as well as all of its minors of the all of the second-order minors do not vanish, is zero, which we have not assumed.
We may also assume that a linear change in performed, so that U 0 is restored to the form
Consequently,
the
functional
determinant
dUo _,:x3
with
respect
is not
_
us
then
consider
the
equations dV
are
n
- 2 of equations
dV
(i). *T31
For
¢[,c.
to
dV
the
been
duo " " " _
Y
d_ = a,_ ....
in
has
zero.
Let
which
variables
a
of
dUo '
the
first unless
form
this
of
series
purpose,
which
in view
of
are
the
-- _,,
We may
l'& 7
. • • _
ordered
(2)
-'0
derive
_L_tt
according
statements
to powers
presented
in No.
of
30,
it is
237
-,,.j
sufficient spect to
that the functional
determinant of equations (2) with re-
does not vanish when we set Whenwe set z = 0 and when we limit ourselves to first-degree with respect to the x's, equations (2) maybe reduced to duo dUo d-_; "- dx_ ..... and not
we have zero.
Let equations in
the
just
seen
us replace (2). We
preceding
that
the
We
2.
The
3.
Equations
have
x3, x_, ..., shall then be
no
more
function
x4,
..., In
X_s
than
V may
three
be
(I) may
represent as
effect,
we
in view
replaced
4.
In terms
238
For
a maximum order
with
is
values derived from conditions stipulated
> O,
in powers
of
z, x I and
these
x 2.
variables;
by
O_
derivatives
taken
with
respect
of x 2 defined
by
to
the
equations
x2,
(2).
have dV
dV
d.r3
(2),
V --
dV
it
--
these
two
to
illustrate to x I and
_
dV
follows
from
d_
this
that
of
x 1 and
dV a, dx L dV d,vt
regarded
when
respect
variables
(3) _
of x I and
of equations
z
determinant
OV Od'l
--
OV d.vI OV dx_
has
independent
developed
be
the
functions
OV
and,
functional
x n in V by their dealing with the
OV
3's
==0
corresponding
O'rl
the
dUo -dx_
/237
section:
i.
where
_
terms
as
variables this, x 2 in
we V.
a
function are
must Let
of
x2,
--
zero. try
to
find
the
second-degree
W0
be
these
terms.
In
order
to
_- :,_Vt +
obtain _V
which terms
are
the
only
ones
which
0 -I-
._ _V
I shall I may
neglect derive
the the
other
following
in the form of series ordered I shall only retain the terms
me,
I shall
take
the
two
/238
zUt,
terms
of V which
from
X,1,
and x 2, and of lected, because
I
interest
U0+
and
z°- \V_ +...
X._
cannot
equations
. . ._
W0 +
zW I.
(2)
3_
in powers which are
degree 0 with respect they do not influence
influence
of x I and of degree
to
z.
The
x 2. In these i with respect
other
terms
may
series, to x 1 be
neg-
We + = W,. Equations
(2)
may
then
be
reduced
to
dUt
2 A 3x3 -+ z _/-_'s = o, dUt
dUt
2 A,, z,, + z _,
Xn,
If we in UO,
substitute we shall
U I,
it may
be
the values thus obtained find that U 0 is divisible
reduced
U 0 is
none
x S, x4,
...,
respect
to the
Xn,
other and
x's.
in _laee of by z z. With
x3, xq, respect
..., to
to Ut°+zUI
where
= o.
than
where We
the U_
shall
Uo= z'uo';
-_-z_U b
quantity
and
U_
are
therefore
which two
U I becomes other
when
quadratic
we
forms
cancel with
have
U,-:Uo+zUl+z_U[
and U0÷ zU, = z UtO+ z-_(U3 + U_)+
:_U'.
239
w
In order to calculate W 0 + zWl, I may neglect the last may be divided by z 2 and z S, and I shall simply have
which
W0+zW,= I shall and to
U_
for
demonstrate
z which
illustrate
is
is
a maximum
is very
small.
zW I, i.e.,
for
that
a negative
must
For
purpose,
two
prove we
squares
which
V has
and
we
of
that
for W 0 +
Finally,
a sum
fact
U_
shall
is
write
having
terms
_U_.
positive
this
this
the
two
the
for It
x I = x2
is
= 0 J239
sufficient
zU_. definite
quadratic
coefficients
form
whose
form. U 1 as
sign
follows
I shall
not
f[
predict.
U I depends
only
on
the _'3_
This is forms.
always
Let
us
possible,
consider
_
the
z is
which
depends
to a sum same
as
assumed
of
only
on
the
n - 2 squares
those
very small not change
to be
for
A3,
A 4,
2 variables
• • -_
according
_a.
to
the
general
properties
of
quadratic
form Uo+_Ut=
where
n -
positive
_U_+(Uo+zU_), and
very
n - 2 variables having ...,
small.
The
x 3, x_,
...,
coefficients A n , since
--
whose due
to
form Xn,
signs the
U0 + may
be
must
be
fact
that
zU_
,
equated the z is
-- this form differs very little from U 0. Therefore, they sign when z makes a transition from positive to negative.
do
According to our hypotheses, when z makes the transition from positive to negative, n - 2 of our coefficients do not vanish, and, on the contrary, two coefficients make the transition from negative to positive. These
last
Therefore, In
order
two U_
coefficients is
to have
the
sum
U 0I, it
of
can
only
two
squares
is necessary
X$_.,._X
a
be
to
the
having set
following
coefficients. in
U_
!
U I vanishes,
Therefore,
the
negative
of U I.
_0.
TT
Then
coefficients
U_
is
and
U 1 may
a negative
be
reduced
definite
to U I . form. q.e.d.
240
_ J
Therefore, z which
is
V,
regarded
positive
and
as
is
a function
very
One will find in the same same time -- that V is minimum and for x I = x 2 = 0.
small,
lated
As I have stated, in the preceding
presented
at
the
we have section,
beginning
Existence 333. defined
Let
the
us
of
of
return
function
way for
thus and this
the
Smp , which
and
for
x2,
is maximum
for
/24O
x I = x 2 = 0.
-or rather one will find at the z which is negative and very small,
returned to the conditions it may be assumed that the section
Solutions
to
of x I and
of
has
the
been
Second
substantiated.
Type
hypotheses
given
in No.
depends
_,
the
on
of
stiputheorem
330.
We
have
2n variables (_
I X,-+_,.... , X,.+_., Y1+_t, Y,+'_, ..., Y,+_.
Xi's
The
_i's
and
the
We
sould
and
the
_i's
are
Yi's
are
the
values
like
to study
the
the
values
of x i and
of x i and solutions
JS _ p
Yi of
_IS,. p
the
=o.
of period T is at the same time periodic having to show that there are others in addition.
employed variables
to
regard
Smp
like
as being
to
solutions know one
(i)
solution I propose
I would
illustrate
dependent
The
t = 0.
equations
to Nos. 321 and of period mpT.
however,
for
t = mpT.
According solutions
First,
322, these We already
for
Yl
only
correspond to of them, since the
the method on
_ and
periodic a periodic period
which
on
the
mpT.
may 2n
be
- 1
(s) For
this
purpose,
we
shall Xn
Let
us
now
consider
the
-I-
assume _n
that
_-=_ o.
equations
dS.,p dS,,,p J(X_+ _) = J(_t+ _,j =°" We assumed
shall to be
employ the a function
d's to of the
represent variables
(i')
the derivatives (_), and shall
of S which is employ the a's
241
to represent the derivatives to be
a function I plan
of
the
to show
Section
No.
that
322
of this samefunction S which is assumed
variables equations
has
(i)
provided
and
us with
(i') the
are
(i) may
therefore
-- (L--
and
equations
(i')
may
view
of
the
5i-- _ m,._) -: X,.-- _i = o n),
written
as
be
energy
of
the
to
Yi's
therefore write
.....
Vi)
equations
(except
=
n--,),
F(_i,
equal
as
follows.
or,
identity
in view
of
ni +
0 is
F with
Let periodic cludes
Yi
- 2mi_
factor
242
F'
The
_,-+ ar;"t2'r:.....
theorem
be
of
of
2 m,,=)F'[_,,,
between
in
the
_i's,
and
identity
all
may I shall
.-r,Y,,'_= F(Y,_).
following
form
!- 2,n,,_) --- o,
increases
÷ 2 ma ,-. + O(Y.-
0 and
the
abbreviation,
_n-x-i-2rnn-I
written
finite
equal
preceding
purposes
_-9.m,,r, + (Y,, -- _ -- 2 ,n,,=.)] -- F(_
n i0 be of
immediate _i
and
will of
2mi_.
Xi's
i,
_.-
and where
2 m_.
F'
is
)] = o, the
(2)
derivative
of
to Yn"
solution
Therefore,
the
F[%
included
0 and _i
the
2rni_).
of
For
also
therefore
the
respect
have
may
(Y_--r,i,where
Hi-4-
all
F(_,, _:, ..., _,,; _1-+-'/mtn, My
we
(1'),
one)
be written
follows
equation,
F(X,.,
According
Xi -,- _i)].
be written
(i:: 7, 2, •.,
(i=,,_
In
equal.
following
dS = Z [( X,. -- _;) d(Yi + T,i) -- ( Yi -- "t,i-- _ rnt_)d(
Equations
the
values
period
T.
vicinity
X i will
never
of The
of
never
deviate
relationship
/241
([3).
the
_i and
n i which
region
under
point
deviate
greatly
(2) will
never
consideration 0 _ = _0, $i = _i' _i
greatly
from
correspond
n_. deviate
from
$o,
Therefore, greatly
and the from
to
the
only
in-
= n_. ni
or
second its
value
0 0 for _i = _i' Hi = _i' and in general this value will Therefore, the first we
factor of relationship
not be zero.
(2) must vanish, and
/242
have
In other words, equations (i') entail equations (i). We may therefore regard S as a function of the variables (_). When it is mp a maximum, considered as a function of the variables (_), it will also be a maximum as a function of the variables (a).
I have which
employed
correspond
values
of X i +
solution
Smp.
_i
of period
hypotheses
$_
to
and T
us
n_
periodic
Yi
to designate solution
+ ni will
changes
formulated
Let
and
the
in No.
Yi
into
322).
2_ Yi
Let
values
and +
2n_
2mi_,
S O be
in
the
of
T. +
$i
The
2mimP_
and
of
ni
corresponding (if
conformance corresponding
the
periodic
with
the
value
of
set 1_= :_o-;-l*';
V = Stop--S0;
and let us consider V as a function The function V will be governed by of the preceding section.
to
be
the
of period
No matter what _' the _''s and to the
of the
X_-_-_i_=._0
+ _i,_'
p', of the g',s, same conditions
may be, V and its first _''s will vanish when
and of as the
derivatives
with
the n''s. function
V
respect
If we consider the group of second degree terms of V with respect to the _''s and the n' 's, and if we regard it as one quadratic form which is decomposed into a sum of square terms, it may be seen that two of these coefficients of these square terms both make a transition from negative to positive, or both make a transition from positive to negative, when _ changes sign. The other coefficients do not vanish. The
expression __M,__sin t>m at T
changes
sign,
and
the
other
expressions BT_ sin pmal_T
243
kJ
do not no 2n
vanish.
The
coefficient
which
longer vanishes, and there is not - 1 variables, the variables (_). The
and
we
conditions
may
state
presented
that
the
in
the
the
other same
real
solutions
thing,
equations
other
solution
real
of
period
The
maxima
equations
(i),
We
solutions
must
for
of
the
in
addition
If find
therefore
maximum We
the
of
section
we
in No. have
therefore
323
only
hold,
to
_
= n'i =
0 or,
which
means
--o
(1)
than
those
corresponding
Smp,
or more
to the
periodic
to periodic
conclude
close
to
is
our
the
of period
differential
solutions
of
mpT.
equations
have
mpT, which differ from the solution of to that for _ = _0, and which differ only
_0-
drawn
periodic
that
generally
solutions
to
solution
the
preceding
of period
line
T need
of
reasoning,
we
shall
not
correspond
to
a
Smp.
shall
It is sufficient
other
of period identical
attention
that
dS"r
function
correspond
_
preceding
as D
because
T.
periodic solutions period T, which is slightly
one
dV
dS,_,p
have
designated
equations dV
have
I have another
therefore
set
m = I.
not necessary that the solution that one of the characteristic
of period exponents
It
T be stable. al equals
is
2k_Vv_ pT for We
such
_ = _0" therefore
If the equations that one of the
they will
244
obtain
also
have
the
following
of dynamics characteristic
periodic
result.
have a periodic exponents is
solutions
of period
solution close to
pT which
of period
differ
very
T,
/244
little latter
from the solution of period T, and which are identical when the characteristic exponent equals
to the
2k_I.LZ_ pT These
are
solutions
of
the
second
type.
Remarks
334. This entire line of function of X i + $i, Yi + hi"
reasoning assumes that Smp is a uniform Under this condition alone may it be
stated that all the maxima of Sm correspond (see No. 321). This fact cannot be stressed which will be encountered frequently when we of the theorem presented in No. 321.
Let
us
determine
assume
that
We
may
Sp
is
be
a uniform
the
clearly
to
m =
function
of
the
_i's
and
of the
sideration. Due to the fact diate vicinity of the values
is
we
function
determinant
the
Smp
i, which
a uniform
functional
respect
whether
of
a uniform just
illustrated.
the
_i's
and
$i's
and
the
Xi +
$i's
does
that
function
have
Xi +
ni's
to a periodic solution enough. It is an obstacle wish 6o derive the results
not
this
the
the
Yi
the
vanish
in
may
be
these In
ni's.
+ hi'S,
and
region
of
Yi the
+
variables.
addition,
It will provided
ni's
with
region
reduced
also that
under to
the
conimme-
it will be sufficient that the functional determinant is not zero at this point. This functional determinant may be written as follows (assuming that n = 2, tO formulate our ideas more clearly)
It must
therefore
dX,
dXt
dXt
dX,
-;t_.i -*-'
_;,;
dr;,
d--.c,-, dVI
dYt
dYt
dYt
7i_--,
d._-_,+ '
-d_
dX, d_t
dX, dT_'_t
dg, dgt
d_ dqt
be
verified
d-,,--;
dX, clot =I-t dY_ d_,
that
dX_ d_ dYt d"]t
the
equation
in
S
/245
245
d_.X. __ S d! I
dXi d qI
dYi
dYl
T[!T
d-_,
dx,
S
does
not
have
a root
According equation
dry2
dY,
,&,
dX,
d_, t
dXt
dXi d_
_0
(!-_2 -
d_ t
dX2
s
dq2
{&:
dY,
dY_
dY,
dY,
d;i
"d_
d;1
_lrj=
which
to the
is equal
statements
to
-i.
presented
in No.
60,
the
roots
of
this
equal e_PT_
where the
the fact
_'s
are
that
characteristic
we do not
:{ ......... where
k is
an
integer
exponents.
We
must
therefore
verify
have
_>T--
number.
By
-"
hypothesis,
the
exponent
al equals
7
pT
where
k
is
able
with
an
integer
number,
and
the
other
exponents
we
are
concerned
are
not
commensur-
= #227. , in general. T
The
difficulty
with
which
will
not
therefore
occur.
In
and
order
to avoid
this,
in No.
I assumed
that
(k integer
number)
(k integer
number)
not _t " :
pT
Special 335. assume
Let
only Let
246
330=
two
us
say
a few
degrees
us assume
that
of the
Cases
words
about
the
simplest
cases,
and
let
us
freedom. form
which
is
similar
to
that
which
I have
designated as U0, in the analysis of No. 331, is homogeneousof the third degree only in xI and x2. The
equation duo
dUi
d_', d::_ always
/246
has The
real
r_ots,
theorem
is Xl
as we
have
self-evident
duo
dU,
dx,
dx,
=o
(1)
seen. here,
since
this
or
three
real
to
clarify
equation
is of
the
--e
third
degree
assume
that If we
choosing _p2, the
will
in
x2
It
may
have
it has
only
one
in
order
w, =
a,?
cos?
-_ bt?
sin 9
x_=
a_?
cos?
+
sin?,
a
and
b
in
u0 :¢ U_
considered
then
the ratio
only
a maximum
This maximum and this sign, will correspond We
will
roots.
our
ideas.
that
U 1 is
Let
us
first
set
coefficients
have
one
then
and
b2?
such
a minimum
a way
in
No.
when
to
331
¢ varies
minimum, which are equal to values of _ which are
reduced
and far
from
0
to
2_.
which have opposite removed from _.
have Uo ! zU, = p_f(?)--zp,.
which
The function f(¢) has have opposite sign.
a maximum and a minimum which are The function U 0 + zU 1 then has:
For
z > 0,
a maximum
for
0 = 0 and
two
For
z
a minimum
for
p = 0
two maxima.
< 0,
and
Employing the English term, I shall use nate a point for which the first derivatives is neither a maximum or a minimum.
very
The same will hold small -- the terms Therefore,
have:
no
matter
true U0 + what
be,
the
and
minima.
the word minima to desigvanish, and where there
for the function V, zU 1 alone will have z may
equal
since -- if z is an influence.
differential
equations
will /247
247
v.)
z
<
A solution
of period
T,
A
of
period
pT,
for
z > 0.
solution
0 and
unstable
Let
us
The
function
equal
now
pairwise In
assume f(_)
and
this
the
of
first
the
that
equation
will
have
have
case
of
U0 +
ZUl,
second
which
type,
(i) has
three
opposite
type,
and
and
consequently,
a maximum
for
p = 0,
and
six
For
z
a minimum
for
p = 0,
six
maxima.
be,
the
matter
what
z may
is stable
real
roots.
three
minima
for
which
are
V have:
z > 0,
No
stable;
signs.
For
< 0,
which
three
maxima
is
minima;
differential
equations
will
therefore
have : A
solution
Three below
of period
solutions
that,
from
of
T,
of
period
a certain
the pT,
point
of
first of
type_
the
which
second
view,
none
is
type. of
stable; We
these
shall
see
solutions
are
different.
let
Let us proceed us assume that In
this
case,
to a case which is U 0 is of the fourth equation
(I)
is
of
a little degree. the
fourth
always has at least two real roots according two or four. We then no longer have
but
varies
us
first
assume
The function from 0 to
this
maximum
First The
that
f(_) will 7, as well
A distinction
248
complicated,
degree,
to No.
331,
and,
and
since
it will
it
have
rather
Let
of
more
may and
case. functions
be
this
The
are
only
two
real
roots.
then have a maximum and a minimum as when _ varies from _ to 27.
drawn
between
two
cases,
depending
minimum.
maximum
U0 +
there
ZUl
and and
the
minimum
V have:
are
positive.
when
on
the
signs
For
z
>
For
z
< 0,
In the
0,
a maximum
a minimum
addition
to
differential
z > 0, stable
and and
case.
The
constants
For
z >
For
z
The
differential
0,
type,
Third
in
The
differential
For
z
For
z
> 0,
a
< 0,
a
for
to
examine
the
The
equations
then
have:
For
z
two
0,
type
type of
these
and
two
maxima.
which the
always
second
exists,
type
two
solutions,
the
minimum
for one
is
/248
is
negative.
minima;
two
an
unstable
the
first
solution type
of
which
the
is
negative.
have:
first
the
first
which
case
of
is
the
of
minima.
have
solution
then
of
now
must
=
itself
second
first
Of
always
of
solution
0,
p
equations
solution
the
positive,
=
the
maximum
and
V have:
p
for
minima
solutions
< 0.
is
and
the
We
z
equations
addition
The
of
zU 1
a minimuma
case.
solutions
of two
for
two
0.
have
a maximum
< 0,
P =
maximum
U 0 +
0,
solution
have any unstable.
The
second stable.
the
p =
for
equations
do not one is
Second
for
in
type
which
is
stable;
type
which
is
stable,
one
is
which
stable
equation
and
and
one
is
(i)
has
four
two
unstable.
real
roots.
of
the
which
are
For tions
of
second
upon
> 0,
second
a
solution
type
which
of are
the
first
unstable,
type and
which k
is
solutions
stable, of
the
h
solutions
second
type
stable;
z
< 0,
the
type
a
solution
second which
The
integer
the
signs
type are
the
h=k-=_;
the
first
are
type
stable,
which and
2
-
is
stable,
k
solutions
2
- h of
solu-
the
unstable.
numbers of
of which
h
maxima h=2,
and
k may
and A-=I; h--k=o;
the
take
the
minima h=_,
of k=o;
following
values,
depending
f(_): h==,,
k=o;
h=k-=l.
249
CHAPTER DIFFERENT
336.
FORMS
OF
THE
XXIX
PRINCIPLE
a double
these
ACTION
Let
y_, Y_, ..-, be
OF LEAST
series
variables.
of variables, Let
us
and
consider
J :
y,_ let
the
F be
an arbitrary
function
of
/249
integral
, t, ( -- F -t- Zyi dr;;,} -dr- / dr. o
The
variation
of
this
integral
may
( In
order
that
this
be
written
dri
variation
as
follows.
d g ri _ dt
may
vanish,
it
is necessary
that
we
have dx'i dt which provides not sufficient.
dF dyi '
us with the canonical If it is fulfilled,
ctyi dt
dF dxl
(i)
equations, we have
but
this
condition
is
::_[)_O.rt]lzto
and it is still necessary that the second term of this equation be zero. This is what occurs if we assume that the 6xi's are zero at the two limits -- i.e., that the initial and final values of the xi's are given. Under these conditions, action is minimum. Let ables,
us
and
perform let
us
the
the assume
integral
change that
J which
I have
in variables. they
have
Let
been
Ey_dx_-- Eyidxi=
chosen dS
designated
' xi, in
Yi' be such
the
as
the
new
a way
vari-
that (2)
is an exact differential. In this case, we have seen that the change 1250 in variables does not change the canonical form of the equations, and this result is an immediate consequence of different propositions which will be
presented
below.
Let J'= ft"(--F+Zy_)dt.
250
Wehave j,
where
S O and We
S 1 are
therefore
the
values
j =fas
dt = st-
of
the
So,
function
S for
t -
t O and
t = t 1.
have
_J'= _J + [_s]_:. If
the
canonical
equations
(i) are
(3)
satisfied,
we
have
t=t I _J = + [Ey_ _xl],=t.,
and,
consequently,
in view
of
(2) and _J'=
In the
the
same
way
relationship
that
(4')
is
(3),
[zy_
relationship equivalent
to
a-7 = a-_,' However, we have just seen that tions (i) are equivalent to equations knew -- that the change in variables of
the
(4)
'_t,
(4')
(4)
is
the
equations
at -
equivalent
to
equations
a_
(4) is equivalent to (4'). Equa(i'), which means -- as we already does not change the canonical form
equations.
The
action
final values the principle variables. The
J' will
be minimum
of the variables of least action
equations
(i)
entail
when
we
assume
that
the
initial
h
is
and
x i are given. Therefore, a new form of corresponds to each system of canonical
the
energy
integral
F = h where
(i),
(5)
a constant.
Up to the present, we have assumed that the two limits t O and t I are given. What would take place if these limits are regarded as /251 variables? Since F does not depend explicitly on time, we do not limit the conditions of generality by assuming that t o is constant, and by only increasing t I by _t I. For example, let us assume that t O = 0 and that,
after
values
at
the
the
variation, the variables, x i and Yi' have the same t time _ 1 (t I + 6t I) that they had at the time t before
the
variation.
251
Before the variation,
we shall have J :
--hh-bZ
ididt. dxl
fy
However,
does
not
simply
The
depend
and
its
variation
when
t o and
If we
the
action
equals
J with
the
equal if we
respect
energy
we
do not
t I are
change
assume
to
constant
that
We
zero.
the
upper
h whose
integration
sign
is
equations
the
initial
and
final
F into
F - h,
J changes
(i) do
example, let (i) must be
which
it
not
change,
follows
us impose satisfied,
this
(6) expression
(6) is
still
7(
F -_-
Z yi
dF
/ (It
=
f
te
minimum.
constant which (6) is minimum,
the condition that the first i.e., the following must be
•
252
the
t_).
that J ==-
setting
of
into
However, if we change F into F - h, the energy to h becomes zero. Consequently, expression do not assume that t I and t O are given.
For equations
by
changed.
values
No matter what the variables x i and Yi may be, the action minimum. It will therefore be minimum afortiori if we impose condition upon it which is compatible with equations (i).
from
therefore
given.
,1+ h(t,Since
is therefore
this constant is zero, the action J is still minimum, if we that the initial and final values of the variables x i are given,
even
time
of
t I therefore If
assume
time;
d,ri
have
derivative
limit
on
=f y,
}I dt,
was even
J is a new
series of satisfied
/252
dF II :
The
action
J,
This
is
which the
is
thus
-- F -+- Ey,.
defined, of
principle
_y.
is
minimum.
least
action
written
the
variables
in
its Hamiltonian
form,
Let
us now
Therefore, but
we
we
assume
no
impose
that
longer
the
regard
following
condition
x i and
upon
them
with
equations
Yi
as
independents,
F=o.
This
restriction,
pede
the
action
We
then
J
which
is
from
being
compatible
Eyl
_d.rl
and, since h is zero, this integral assume that t I and t O are given. us
will
not
im-
have
J = f
Let
(1),
minimum.
then
impose
the
dt
is minimum
following d.t_.
even
when
we
do not
conditions
dF
dx"
from
which
we
may
derive
yi=?,
the
Yi'S
x,,x_
....
as
,z_,
a
function
d_r, #f'-dY'd.r,
of
the
---1,dt s
"'"_-/ d.r_'_
or
y_ -_ ,_, x,, _:,, ...,
In and
in
the the
place
of
the
d.o d.r_ -dr' dr, _,,, -7l?' ?tT,
Yi'S,
let
us
_dx3
dx, -ZIF....
substitute
' d:e,_ J_;
their
shall
(7)
values
(7)
in
J
/253
equation
dx, We
d__xj_ dt /"1
derive-- dt
as
a function
of
the
Xk'S
and
dxk, the-r--axl
s from
this
253
dxI equation. Weshall then substitute this value of -_-- in expressions (7) and in J. This last integral will take the following form
¢j '_ d_:l,
dxl dxl-_t_7
E yi
dx k where
_ is
integral, minimum.
a function
of
the
Xk'S
which is thus written This is the principle
If
h were
337. Let
not
Let us
zero,
us
first
assume
that
we
in of
would
examine we
Yi'
T is homogeneous while We
U is then
of
independent
of
the
derivatives
dx I.
This
a form independent of time, is still least action in its Maupertuis form. only
the
have
most
to
change
important
F into
F - h.
particular
case.
have F-
where
and
the
T--U,
second
of
these
degree
with
respect
to
the
variables
variables.
have dF Eyt _j_ = _.T,
According
to
the
principle
IÁ= T -i- U.
of Hamilton,
the
integral
Ct'(T't. + U) dt must
be minimum. Let
energy
us determine
equation
may
what be
the
action
of Maupertuis
becomes.
The
written T
The Maupertuis
principle
then
has
U ---h. the
following
expression
( T + U -÷-h ) dt. The
equations
dxi
dF
dT
;_ii- = 7i_ =
have
254
their
second
terms
which
are
linear
and homogeneous
with
respect
to
k_/
the Yi'S.
Therefore, T is homogeneousof the second degree with
respect
dxi the -_7-'s.
to
dxi . d--_-is
replaced
by
Let
dxi;
dT 2 represent
we
shall
that
which
T becomes
/254
when
have 71
_
__
ctt _
and
d_ 2 will
the
n
be
a form
differentials
which
dx i.
is
We
may
dt
The
Maupertuis
action
will
linear
=
deduce
---
then
=
For purposes of brevity, cases, let us set
and
derive
us
the
Yi'S
of
the ,,
so
as
to
take
the
xi's
and
the ordinary xi's and to
d's to designate the Yi'S, and let
tives
with
taken We
may
respect
readily
to
obtain
following
the
following
order
respect
from
to
this
expression
to be
able
to
study
other
equations
for
new
variables.
the derivatives us employ round
the
with
dF
xi's
and
the
the well-known Oil
yi=--:,
the
in
x' i 's
the
homogeneous
..........
have
338. particular
let
and
OH --- =__ Oxt
O,'ci
Let
us
employ
taken with respect to the 3's to designate the derivax_ 's.
relationships ,IF _
'
F = _x } 01I j_ -- II and we tions
will
see
that
equations
(i) are
d
OIl
equivalent
to
the
Lagrange
equa-
OII
dt Ox_ = bxi"
/255 Under
this
assumption,
let
us
examine
the
case
in which
H has
the
255
following
form H --
where
H0,
H I , H2
are
respect
to
We
have
with
then
the
homogeneous, variables
He+
of
lit,
degree
0,
i, 2,
respectively,
x'..1
_-x_ "a-'w = ax_ F =
and
tit+
the
_Ht
+
III,
lit--lto
0Ht
OH,
yl = 7_7z_ + o_
are
linear
functions,
but
they
are
not
homogeneous
with
respect
to
the
! TS,
xi
The
Hamiltonian
action
retains
the
same
form
.flldt.
Let
us determine
Let
h be
following
the
what
energy
the
Maupertuis
constant.
The
Maupertuis
becomes. action
will
have
the
expression lt +
but
action
it must For
be
this
written purpose,
in
the
let
us
form
h)dt
which
is
independent
of
time.
set d_, Hi---_,
and
d7 lit ---- d--t"
H 2 is
nothing
comes
when
H I becomes linear
other than energy, and dT 2 is that which this energy be! x i is replaced by dx i. In the same way, do is that which by dx i. It is therefore a form which is when x ' i is replaced
and homogeneous If we
take
the
with energy
respect equation
to
into
Ht --- lto+
256
the
h,
differentials
account
dx i.
from
the
which
we
Maupertuis
have
action
will
become
The Maupertuis principle may therefore be applied to the case in which we are interested, as well as to that of absolute motion. However, there is one essential difference from the point of view of the following statements.
In all the problems which will be encountered, is essentially positive; it is a quadratic, positive the case of absolute motion (No. 337), the action
,;_ is essentially positive. changed. On the contrary, terms. The first
d_ V/U + h
It does not change in actuality, the
f is always The it may
positive, second
therefore
and
f do be
does
changes positive
the energy T or H 2 definite form. In
when the limits are interaction is composed of two
',.d: 1¢'N7o+h not
change
sign
when
when
the
the
limits
limits are
are
interchanged.
interchanged,
and
or negative.
If we also note that in certain cases, the first term vanishes without the second term vanishing, we will find that the action is not always positive. This fact will cause a great deal of difficulty later on. 339. In order to show how applied to relative motion, let of a system. Therefore, let
the preceding considerations may be us first consider the absolute motion
H=T+U
and let us variables
where Xl, different in space.
assume
that
the
position
x2, ..., X n are sufficient points of the system, and
of
this
to find m is the
system
is
defined
the relative orientation
by
n +
1
position of of the system
257
If the system is isolated, U will dependonly on Xl, x2, ..., xn. T will be a form which is quadratic and homogeneouswith respect to x_, x2,' ..., X'n, _' whose coefficients dependonly on x 1, x2, ..., xnWewill
/257
then have the equation dT
d_ =P' where
p is
a
Under
constant.
this
This
assumption,
is
the
let
area
J be
integral.
the Hamiltonian
action
_t_
J =Jr We
shall
have
the
following,
if
_l-The
action
zero)
if
the
to be
given Let
are
--
us
given,
will
and
i.e.,
now but
if
assume not
the
final
;w
that
the
of _.
of motion
(or
rather of
the
xi's
= 0 for
t =
t o and
initial We
t=t
its
and
shall
I
are
satisfied
. ,_:,.
values
6x i = 6_
those
equations
_xi_-_5,
be minimum
initial
It dr;
first
final
variation
and
of
for
values
will
m are
be
assumed
t = t I. of
these
xi's
have
_
t_t
I
_j = [p _ ],:,. = p[ oo_],:,.. Then
let H'=
H --pt_'
and J' =/II'
and
we
shall
obviously
dr,
have ._J' = o.
We
may
,, the x i s, quadrati,c H'
258
derive
m' , which
is a
linear,
nonhomogeneous
function
of
dT from
the
function
therefore
equation_, which has
the
is not form
= p.
It may
homogeneous H 0 + HI
+ H2
also with which
be
seen
respect was
that to
studied
H'
the
is V
xi
t
a s
in No.
• 338.
final
The integral J' will thus be minimum, even values of w are not assumed to be given. We
m0
the
initial
and
have
/258 J'=: J--p(_o,--
where
though
and
_i
are
the
values
of
m for
_oo), t =
t O and
t = t I.
340. Let us now assume that we have a system referred to moving axes and subjected to forces which depend only on the relative situation of the system with respect to the moving axes. In addition, let us assume that the axes rotate uniformly with a constant angular velocity
This problem may be directly related to the preceding one. We need only assign a very large moment of inertia to the moving axes, in such a way that its angular velocity remains constant. For
the
absolute
motion,
we
then
have
II== T ÷ U : Tt-+T,-+ U. The function of the define the position
forces of the
U depends only on the variables x i which system with respect to the moving axes.
which
is
of
the
system,
ratic
form
respect
to
of
the
energy
with
the moving
axes,
the
depends
on
and
to m'.
x_'s
the
xi's , and T2,
which
is is
TI,
a quadthe
energy
equals I 2
and
the moment We
then
of
inertia
I is very
large.
have
p=_',
dT_"+'I_'
and It'=
I!--p_'=(T1÷
T,÷U)
" _-_ tiT,_'--I_-
'f
or It'=
TI ÷ U -- dTj _-j,
O_r
lto qt -_-
However, dT
! _o'=p--
I
_.
259
k_s
dT 1 Since gives
us
I
and
p are
very
approximately
large
the
with
respect
to _7 , , this
equation 1259
following ;
and more
exactly I dT, --_o I dto'
¢#_=p_ 1 In
addition,
we
have
dT
I_'* _ p' 2 al We
thus
1
P dto' ._,_/uT, V I +'21\d_'/ "
obtain
pt In last
the
term
second
member,
the
is negligible,
Since
we
Hamiltonian
may
add
term
because an
principle,
may
before
I is
arbitrary
we
, (tiT,
_'
the
very
last
is
a
constant.
The
large.
constant
to H'
without
changing
the
of
not
set It"= Tt _ U
and
we
know
that
the
integral J"= f
must
be
minimum
(even
though
the
Wdt
initial
and
final
values
w are
given).
H"
is
In the expression then a quadratic
of H", m' must function, which
be regarded as a given constant. is not homogeneous with respect
!
the
xi's , having For
example,
the
form
let
H 0 + HI
a material
plane_ whose coordinates We shall have
with
Tt=
We
therefore
+ H2. point
respect
(_'-- _'_)'+(_'+
having to
the
the moving
mass
1 move
axes
are
_'_)'
have _Orl
., = _"+ _'_', 51
260
H, = _'(t_'-- t'.,),
tlo=
-y (_'*
_')+ U.
in
a
_ and
n.
to
The integral tt
J =_t
(lit+
fl,+
llo)dt
o
is then as well
minimum, when we assume that as the initial and the final
The
energy
integral
may
then
the limits values of
t o and t I are $ and n.
/26O
given,
be written
H_=,,lto + h and we
have
seen
that
the
integral
J'=/(lt_ is minimum We
by
even
then
though
_- II,_-
we
do
not
llo+h)dt
_- J + h( tt_
assume
that
to)
t O and
t I are
given.
obtain
setting dst = d_ + dr,'. This
and
is
the
generalized
principle
of Maupertuis.
In the problems which we shall discuss, consequently J will always be essentially
U will always positive.
be
positive,
This will not always hold true for J'. If h is negative, we assume that the point E, n is divided into sections in the region fined by the inequality
must de-
Ho+h>o. The first term of the quantity under the sign S which is ds_ is essentially positive. This will not be true for the second term, which changes sign when we reverse the direction in which the trajectory is assumed to be traversed. If the point _, n is very close to the border of the which it is confined, and if, consequently, H 0 + h is very first term will be very small, and the second term is the will give the term its sign.
by
J' is therefore not means of the following
essentially equation
positive.
This
can
region in small, the one which
also
be
seen
261
k.J
J'=J If h
is negative,
the
+ h(t_--to).
first
term
J
is positive
and
the
second
is
negative.
Kinetic
Focus
/261
341. Up to the present, when I have stated that a certain integral is minimum, I was employing abridged terminology which was incorrect and could not deceive anyone. I should say the first variation of this integral is zero; but it is not We
this condition sufficient.
shall
now
try
is necessary
to determine
the
in
order
condition
that
for
there
which
be
the
a minimum,
integrals
J and J', which we studied in the preceding sections and whose first variations are zero, are effectively minimum. This investigation is related to the difficult question of second variations and the excellent theory
of
kinetic
Let
us
focus.
recall
Let xl, x 2, their derivatives.
the
principles
of
these
theories.
..., x n be the functions of t; Let us consider the integral
let
x'I, _2,
"'',
the
initial
and
' be Xn
tt
J-_o whose
first
values
shall
of
variation
the xi's
_J is
are
f(xi, zDdt,
zero,
assuming
In order that this integral may be minimum, a condition call condition (A) is necessary, but not sufficient. The
condition
is
regarded
as
a
Condition
function (A) is
of
order
that
the
not
not very close. Except condition which I shall I must first recall the In
final
which
I
that
f(xi, z_+ _i)- _
262
that
given.
ei's,
sufficient,
is
d/ _ ',' minimum. unless
the
integration
for this case, it is necessary call condition (B). In order definition of kinetic focus.
limits
are
to add another to explain this,
k_J
it is necessary and sufficient that the xi's satisfy n differential equations of the second order, which I shall call equations (C).
/262
Let x,= ?,.(t) be
a solution Let
us
of
these
set
the
equations.
following
for
an
infinitely
close
solution
m = ?/t) + _,, and let us formulate the variational equations, the which satisfy the _iVs and which I shall call (D). The
general
solution
of
these
equations
linear
(D) will
have
equations
the
of
following
form k-----_n
(i----t,
a,
...,
n).
k----t
The Ak's are 2n integration constants, and the _ik'S are 2n 2 functions of t, which are determined perfectly and which correspond to 2n particular solutions of the linear equations (D). Under this given times
two
between We
assumption, t = t' , and
which
we
may
shall
thus
let us state that t = t" . We shall eliminate the 2n unknowns
obtain
the
the
$i's
have A k.
all
2n linear
vanish
for
equations
equation
_(t',t')=o, where
The when
A
is
the
quantities t is
determinant
_ik
replaced
and by
Elk t'
oo,
ooo
¢
I
o,o
ooo
o,,
i
represent
and
by
ooo.
,o.
,.,o
that
which
the
function
$ik becomes
t".
263
X.J
If that in
the
these space
times are
t'
and
t"
two
conjugate
n
dimensions,
having
satisfy
In is
the
may
time In
the
if
of
to
now
that
the
direct
(B)
consequence
be
tl,
t 2,
t 3 be
four
Let
M0,
M I,
M 2,
M3
the
assume
If
that
condition
M I (A)
?,(1), is is
may _
say and
M" /263
conjugate
points.
conjugate that
to
M"
is
t'
the
condition
it
be
to,
us
we
after
t',
focus
(B):
of
There
which
M'.
is
no
conju-
is
necessary
and
sufficient
fulfilled.
may
be
O,
points
t I.
Let
x,= Let
time
a minimum,
and
=
two
?_(e), ?_(t'),
two
state
and
A
the
have
following
to
J be
(A)
the
may
t O between
order
are
is
we
state
conditions
A
t"
t'
that
v_(r) ..... ?,(t'), ...,
coordinates,
addition, closest
We gate
as
equation
and
which
_(t'), ?,(t'), respectively
the
times
inferred
this.
times. corresponding
z, = ,_,(O,
the
from
focus
of
fulfilled,
..., M 0 and
we
to "< tt <_ t, <
may
points _
of
the
curve
= ?_(O-
M 3 that
of
M 2.
have
t,
or to<
tt<
tt < ta
or tt < t_ < to<
But
Otherwise,
must
264
be
we
cannot
the
minimum
tt.
have
integral
since
condition
(B)
is
fulfilled,
and
the
integral
that
tl--E
will this
and
not be term.
minimum
since
the
condition
(B) will
This is impossible, since we may vary the t I - E without varying them between t O and
The geometric readily seen. A
curve
in
significance
space
having
representing a solution which I shall call (T). The
will
of
of
the
not
be
fulfilled
functions t 2.
preceding
for
x i between
statements
may
t2
be
1264
n dimensions
the
equations
(c)
can
he
called
a trajectory,
curve
represent
an
infinitely
close
trajectory.
If we draw one of these trajectories (T') which are infinitely close to (T) through the point M', and if this trajectory again intersects the trajectory (T) at M" (more precisely, the distance from M" to this trajectory will be an infinitely small quantity of higher order), the points M' and M" will be conjugate if, in addition, the point which follows (T') passes through M' and infinitely close to M" at the times t' and t". 342. always
In
the
fulfilled.
case
of
the Hamiltonian
In effect,
we
principle,
condition
(A)
is
have
lI =IIo-_H,-_H,, and
H 2 is
a quadratic
and
In all of positive.
these
form
problems
T
If we
change
which
x i into
of
is homogeneous
with
dynamics,
quadratic
this
respect
to
form
the is
x_'s.
definite
!
xi +
ei'
HI
will
change
into
dlt,
and
H 2 will
change
into
265
lll(x_) and
in
addition
we
Therefore,
which
we
we
dill. E_-t d-Zt '
+
have
have
ll(,r',+
from
+ H_(tl)
d(Ilo4-
_,') .---Iio + Ill q- II_ -+ v_g
finally
It,-_- It2)
have dl!
The
first
term
corresponds
to
the
function
J/ .f(_ + _,-)-- _,. y_. since that is
the the
quadratic
form
expression
is
H2(ci)
is
minimum
positive
for
ei
=
definite,
0
--
i.e.,
and that
we
may
see
condition
(A)
principle
in
fulfilled. 343.
Let
absolute
where
d_ 2 is
differentials For The
the
_2
_dXl
]
us
The
a
to
the
integral
positive
case to
definite,
the
Maupertuis
examined
quadratic
time
being,
let
us
select
(but
is
a polynomial
of
the
may
form
x I as
second
then
with
be
written
respect
to
the
independent
the
variable.
order
P which
is
not
dx i
__!
essentially
positive)
with
respect
set
d,_,
dz'---; - V
266
of
be
becomes
/dT geneous let
proceed
dx i •
integral
where
us
motion.
\d.,., /
to
the
dx I s.
homoTherefore,
We
must
determine
whether
{J_i is minimum second
for
e i = 0.
derivative,
d,4_
In other
with
respect
words, to
t,
we of
must
the
determine
whether
the
radical
_/jl{,_x, \ _t77-_-_'I) is
positive.
No
matter
dxi the d-_l'S
what
and
P\dxt
where then
a, b, equals
c
are
the
)
+E_t
independent
of
may
be,
we
shall
have
/266
=at_-+-2bt-+-c,
t.
_C
ei's
The
--
b t
2bt
+
second
derivative
of
the
radical
__°
(at
Since always
the polynomial positive, and
P is essentially condition (A)
344. Let us proceed We must then consider the
to the integral
choosing
_
as
the
independent
is positive.
determine
This
c) 1
positive, always
Maupertuis
whether
this fulfilled. principle
we
in
relative
derivative
with
is
also
motion.
have
+ .,,,)+ _,(._.,,_.,)]. the
second
i/( Ilo + h)(, + _")-:-
m'($r,'--
derivative
expression
+ _,(_ d_ - _ d_)],
variable,
f_[¢_:,){, We must therefore to _' of
is
VqT0+ h
7[d, or,
2+
respect
D)
is oq_,+h (l+ ,,',),
Condition Thus,
(A)
is
therefore
condition
(A)
always is
itself
fulfilled. fulfilled
in
every
case
which
we
shall 267
examine. Maupertuis
Focus
345. The kinetic focuses are not always the same, depending on whether Hamiltonian action or Maupertuis action is being considered. In order to clarify this point, let us assume only two dom, and let x and y be the two variables which define the system, and which we may regard as the coordinates
degrees of freethe position of 1267 of a point in
a plane. Let
=/,(t), be us
the set
equations
of
a trajectory
y = f, Ct) (T) which
x :/,(t)+L
will
y =/,(0+
be
a plane
Let
curve.
_,,
and, neglecting the squares of _ and of n, let us formulate tional equations. Since they are linear and of the fourth shall have
the variaorder, we
where
ni's
the
ai's
functions The
of
are
integration
constants,
and
the
_i's
and
are
t.
equation
given
in No.
341
_.(t', t') = o, may
then
be
written
(i)
=o.
It
is this It
equation
indicates
which that
(T), and the point x + trajectory (T'), occur t",
268
separated
by
an
the
defines point
_, y + at two
infinitely
the Hamiltonian x,
y, which
n, which different small
focus.
describes
describes the times, i.e.,
distance
the
trajectory
infinitely close at the times t' and
of higher
order.
However, these are not the conditions which the
Maupertuis
focuses
must fulfill. Two points of the trajectory (T) -- i.e., the two points M' and M" which correspond to the times t' and t" -- must be separated by an infinitely small distance of higher order from the trajectory (T'). However, it is not necessary that the moving point which traverses (T') passes precisely at the time t" -- for example, infinitely close to M". On the other hand, the energy constant must have the same value for Hamiltonian
One
We
may
(T) and for focuses.
of
the
(T').
solutions
therefore
assume
In
two
this
the
(*'),
functions
addition,
(T) and the is obviously al, a2, a3,
of
_1
the
the
last
condition
variational
is not
equations
,,'. =/',(t'), and
nl
difference
are
_; =/',
on
/26S
is
thus
_; =/',(,').
defined.
between
the
energy
constant
relative
to
energy constant relative to (T') is infinitely small. This a linear function of the four infinitely small constants a 4.
Without limiting the conditions of generality, difference is precisely equal to a 4. The same
imposed
that
_', =/; The
This
condition for T and
stipulating that the (T') is then a 4 = 0,
value or
we
of the
may
assume
energy
that
constant
be
= a, $, + a, ,'-_+ a, G,
For
t = t',
_ and
n must
be
zero,
from
which
we
have
equations
al _', + a, _', + a, $', -- o, a_ ,,'1 + a,, 4 + a, 7,'_= o. In addition, the value of x + _, y + n for t same (up to quantities which are infinitely small as that of x and y for t -- t", which may be written
= t" of a
+ e must be the higher degree)
(_+a,)_;+a,_;+a_;=o, (_ + a_ ),,; + a,,,; + a_,_; = o, from
which
we
have,
by
elimination,
269
kj
(2) _0.
By
developing
the
determinant,
we
obtain
-'0
I _'r/_-._'_'_,, . _"'3--_3'" I. . o .
and,
/269
setting
equation
(2)
becomes _(t')
Application 346.
If
functions the
period
we
are
=
to
dealing
_(t').
Periodic
with
fl(t)
and
f2(t)
of
2_.
The
same
holds
the
a
In
addition, other
according
to
particular
periodic
true
IV,
the
which
these
exponents,
equations, and
the
B is _'s
have
the
= +,(t)+
the
eaergy
equation.
the
_'s
F x,y,
_/_,
We dF
dF
d_ "_+ _
_ are
must
and
-_
are
the
periodic
= const.
have ,IF d_
dF
d_
_' + d--_ a_ + -d--_ d_ d d-t"
270
equations following
will form
?_tf',(t).
Let
be
2_,
periodic
r, = e-at'b3(t); (t),
constant,
and
period be
= e_%,.(t):
_tf',
a
variational
will
== e-a'?_(t),
In
of will
for
= e=t?_(t),
= ?l(t)+
solution section
_,=/_(t).
Chapter
solutions
Solutions
preceding
L=/,(t),
have
(3)
d -'dr
=A,
the
characteristic
functions.
the of
k_Y
where
A is
a constant.
equation, the by e at and -zero. We
first since
shall
If we
replace
term becomes a periodic it must be constant --
therefore
This indicates the following
q by
eatS2 , eatS2
in
this
function of t multiplied it is necessary that it be
have A
have
_ and
that the equations
two
_o
.
infinitely
x=_(t),
close
trajectories
which
true
trajectory
/270
y=_(t)
and x=f,(t)+e_tg,(t), correspond
which
to
the
same
In the same way, has the equation
value we
y=f_(t)+c_t+2(t of the
find
that
energy the
z=f.(t),-e--_t,?3(t), Nothing
prevents
us
from
_(t)
has
the
constant. holds
for
the
y=_(t)+e-_t+_(t).
setting
_3 = e-st?,, Then
same
)
following
r,_--_e-_t+,. form
_(t)=e_tG(t), where
G(t)
is
a periodic
function.
Case 347.
We
must
now
of
Stable
distinguish
Solutions between
two
cases:
i. The solution is stable and _2 is negative. In this case _2 and _3, and n2 and q3 are imaginary and conjugate. The modulus of _ and G is one. We shall formulate three hypotheses which we shall justify at a later point. I.
Let
us
first
2.
The
function
assume
that
G(t)
never
becomes
either
zero
or
infinite;
I t-b
--IogG(t)=, 2',
271
which is essentially 3.
real also constantly increases;
In addition,
let us assumethat log G(t) is a periodic
function. Equation (3) may then be written as follows, employing T' and _" to designate two values of • which correspond to t' and to t": , One
single
value
of
kit
(k is
• corresponds
single value of t corresponds have k = 0 without t' = t". k be
,
to
an
each
integer value
to each value of _. If we desire t" > t'
number)
of
We it
t,
and
/271
one
therefore cannot is necessary that
positive. By
setting
k I i, we
shall
give
the
smallest
value
to t"
- t'.
We
have
and
the One In
point
M"
factor order
is necessary all that we
is
then
must
be
that
the
the
focus
pointed
out.
preceding
that log G(t) be know is that G(t)
of M'.
line
of
reasoning
may
be
applicable,
a periodic function. However, is a periodic function, and as
it
in general, a result
logG(t) is
increased
by
2_.
by
a multiple
of
2i_,
for
example,
Then logG (t)--ikt
is
a periodic Let
us
function. then
set
G'(t)
= G( t)e -ikt, ik 2
we
have
We
_( t) = e_'_tG( t) = ela'tG'(
shall
then
no
longer
set !
272
t).
of
2ki_,
when
t increases
v
but rather !
"_= t + -logG'(t) 9. _r
Since log G(t) will be periodic, and equation (3) will be written
the
preceding
conclusions
remain
valid,
n,i_ (m is
¢I'
and,
in
addition,
M"
will
be
the
focus
an
of M'
integer
number)
if
/272
348. One of our three hypotheses stating that log G(t) must periodic has thus been proven. I may now state that the function must be constantly increasing, as we assumed. Let then
may •h
and
such to
us assume that find two times
T[
that
one
of
the
T2 '
=
another
Then such
I may
this function has a maximum T O for t = t O . We t_ and t_ such that the corresponding values ' function T are equal, and two other times t 2 and t " 2 T_
and
satisfy
t"I will
inequalities now
be
the
the
are state
such
that
following
focus
G(t)
five
times
which
are
all
of
our
when cannot
hypotheses
have
Unstable 349. positive;
Let in
us
this
now case
assume _2,
that n2,
_3,
For the same reason as given stantly increasing. However, two
very
close
inequalities
' tl,
of
impossible that
the
" that t2
of
condition vanish.
t_. A
We
is
We
saw
been
above
that
fulfilled.
have
The numerator and the denominator of g(t) are imaginary gate. If one of them vanishes, the other also vanishes, so function g(t) cannot become either zero or infinite. Thus,
be T
and that
conjuthe
proven.
Solutions the n3,
unstable _, _,
above, the hypotheses
solution G are
and
_2
are
real.
function • will are possible:
be
con-
273
i.
The quantity constantly
increases It
then
Maupertuis
happens
_(t) cannot from 0 to +_ that
no
The
quantity
_(t)
may
for t = t O + 2_, and since it must become infinite in infinite, In
point
order
infinite
to
vanish
for
it must
also
clarify
our
be
values
which
differ
assume
When
t increases
a
for
t = t O.
It will
also
either In the
a maximum same way,
vanish
_273
or if
a minimum _(t) can
_(t)
becomes
to vanish. let
us
assume
that
tl,
g(t)
will
The closed trajectory therefore be divided
pond
to
the
Each
increases
following
of
the
following I may
these t_,
add
t O to
by
a multiple
of
2_,
and
vanishes
t_+2_.
with t" be t" n be
tl,
constantly
values
points
of
or
from
from -_
t I to
of
t
to,
It,
one
of
to
"+"
the
t2,
or
from
t 2 to
to +_o.
(T) which represents into two arcs, whose
our periodic solution extremities will corres-
2,'_.
arcs
will
corresponding
to
have
its
the
values
first
focus
on
arc. that
the
points to,
and let shall have
t0-1-21_
t:,< t_< t'_< t++ 2_:.
from
2_,
their
two
t_,
t,,
of
t
t_
focuses.
a value of t corresponding to an the value of t which corresponds
"
lira t;_--
274
has
that
tO +
Let
able
from
to<
coincide
solution
thoughts,
r
the
periodic
for
t = tu,
I shall
our
it cannot have the interval.
/=tO,
and for
of
infinite, and from -co to +_.
focus.
2.
become
vanish nor become when t increases
t r
2
= --.
arbitrary point of (T), to its hn_h focus. We
However,
this
is
not
all;
we
shall
have
e:_C,G (t,_)= If
n
is
large
very
and
small,
large
since
so
and
we
that
if
G(t")
assume
if
t"
is
that
is,
for
e_-_,"G(t"). not
e
is
infinite,
positive,
example,
t n" -
since G(tn)
included
will
between
t"
be
tO
is
very
very
and
tl,
the
difference t "_a -- "l ll "_
will
strive If
n
toward or
strives
tl,
t_and
_in2
periodic
will
function
a must
the
always
be
even.
From
the
periodic
it
could
is
therefore
be
periodic
asked
2_
into
p
and
equations
p =
m
of
whether
these
i,
the
let
us
the
to
to polar may
two
cite
assume
and a
this
solution
have
will
-
2n_
t'0 is
n.
be arcs,
are
e st . in
However, one
period.
divided and
by
the
this
interested,
two
number
the
categroies.
exist
in
a
un-
However,
actuality.
It
examples. of
a moving
point
in
a
plane.
written
that
we
have
dU
d_ U
--._ =__[,
_?j = o, will
we
with
times
of
into
t O and
t" 2n
by
will
categories
some
dU
(i)
which
t O or
where
of
(T)
coordinates be
points
number
divided
between
decreasing
number
in
be
toward
difference
multiplied
even
U_o,
Equations
the
strive
included
constantly
view
may
motion
be
that
certain
solutions
be
of
will
trajectory a
will
t"
an
closed
advantageous
difference
function
vanish
point
indefinitely.
add or
is
tO +
For
should
-- _2nl
tl,
Let
whether
I
' correspond t_ , t 1
350.
The
this
values
to,
stable,
increases
increasing
Consequently, points
n -_,
on
tI •
constantly
The
but
toward
depending
between
either
' when tO
toward
the
d?
d? 2 .- ?fro)
.
solutions
correspond
to
a
closed
trajectory
which
will
be
circumference.
275
kj
Let
and
us
let
us
J_275
set
formulate
the
variational
d_ _
The
second
may
equations.
dv
be
d' v
integrated
They
may
be
written
dE
immediately
dv dt _ 2_ = const.; but
this
constant
the
same
value
must
for
be
the
zero
if we want
trajectory
(T)
and
the
energy
for
the
constant
infinitely
to have close
tra-
jectory. Therefore, become
will
if we
dv _by
replace
-2_,
d,_ _ _[ ?(t)-dt _ Equation having
(2)
which
a periodic
remains
to
the
first
is
be
(2)
integrated
is
known
G and
that
they
have
two
solutions
shall
of
A
G 1 are
periodic
shall
have
a stable,
periodic
I must
also
The
now
and
the
189
(see,
following
in
form:
that
we
may
have
case mentioned constant A (case
above. of
solution.
focus on the first
periodic,
(T), and we category.
unstable
solutions
category
if
category.
solution in
show
29
of
every to a
a Maupertuis solution of
vanishes
276
< 3, we
second
equation
functions.
If A > 3, there will not be have an unstable, periodic
the
linear
_ = e-=tG,(t)
We are going to present examples for Let us first assume that ¢ may be reduced central forces). If
a
coefficient.
= e=tG(t), where
equation
3].
These equations were discussed in Sections addition, Chapter IV, in various places). It
variational
such
will a way
be
unstable
that
the
and
ratio
of
the
second
G
G GI
which corresponds to the vanish, and consequently
the
function _(t) of the can become infinite.
We may obviously formulate following conditions : I.
It
has
2.
These
two
simple
zeros
will
a periodic
zeros also
and
every
_ 2•
time
its
second
will
vanish
dl
in
such
a way
that
the
_
finite.
One could obviously conditions. The periodic tion G will category. As
also
d_t- "
C _! G
derivative
_,
remains
two;
satisfies
that _
vanishes, ratio
only
G which
/276
can
d(;
dtT
a result,
function
sections
vanish
d_G
As
preceding
correspond
an example
of
formulate function
to
an
This function another zero if
unstable,
function
G :
periodic
G satisfying
sh, t--
vanishes
a function G which satisfies these _ formulated by means of this func-
for
solution
this
of
condition,
the
second
we may
set
_(cost--cos3t).
t = 0 and
t = _,
and
it
does
not
have
I
For
order
that
for
the
t =
_, we
ratio
have
,I_ G
dO
di;
+ 2x _{7 = o,
G _
may
vanish,
it
is not
sufficient
that
I
In
t = 0 and
G vanish;
it
is
still
necessary
that
G 1 does
not
vanish.
277
kj
However, this same time, the
the
could
only
differ
differential 351. that
fact
the
a
constant of
the
because
if G
and
G I vanished
at 1277
factor
second
(since
order),
they
and
satisfy
this
is
the
same
absurd.
One point to which I would like to draw attention is the the unstable solutions of the first category and of the
stable
Let the
by
equation
second category one to another of
is what occurs, two solutions
us
form two separate groups, so that we cannot pass from continuously without passing through the intermediary solutions.
first
preceding
confine
section,
ourselves
and
let
to the
us
particular
reconsider
the
case
given
in
equation
d _
-_ = _(? --3) dt _ Let
us vary
the
function
we can pass directly unstable solution of
(2)
_ continuously,
and
from an unstable solution the second category. For
let
us
determine
of the first this purpose,
sary that the function G, which is real, be first incapable and then be capable of vanishing. We would thus pass from which the equation G = 0 has all its imaginary roots to the it has real roots. At the time of passage, it would have a or, more
generally,
This order
of
a multiple
root
zero, 2m
which would be dG - 1 for _, on the
on
on the
order
the
order
order of
of
2m -
of
whether
category to an it is necesof vanishing, the case in case in which double root
2m.
2m
for G, would be on the d2G 2 for d-_, so that the ex-
pression ¢t_G
dG G
would
be
On solution For
come the of
infinite,
which
is
impossible
other hand, we may pass from one or the other categories.
a stable
solution,
G is
since
a stable
imaginary.
At
it equals solution
the
time
¢ - 3. to
when
an
the
unstable
solution
becomes at this solution
unstable, the imaginary part of G becomes identically zero. If /278 time the real part of G has zeros, we shall pass to an unstable of the second type; if this real part never vanishes, we shall
pass
to
an unstable
No
difficulty
equation
278
solution is
of
encountered
the in
first
type.
passing
from
the
case
in which
the
real part of G = 0 has all imaginary roots to that in which this equation has real roots, provided that at the time of passage the imaginary part of G is not zero. 352. In order to clarify the preceding statements, I shall return to an example which is already familiar to us. Let us return to the equation of Glyden, i.e., to equation (i) given in Number178 (Volume II). Weshall assign the number (3) to this equation, and we shall write it as follows d:.z. dt t
It
can
Just integrals
be
seen
that
_ 7(-
it has
q, + _, co_2t). the
like equation (2), we have having the following form e=tG,
which
we
have
written
in
the
The
case
of h
real
the
case
of h
We
also
considered
the
second
is
corresponds
unusual
we
have
obtained
this
(2).
equation
has
in No.
178
as
two
follows
to to
the that
integrals.
case
of
stable
of unstable The
first
solutions,
solutions. is
even
V'(o) =o]
the
=o,
/'(o)=
following
F(t)
i]
/279
conditions
F(_)=/'(_)=
where,
equation
uneven If(o)
and
that
given
corresponds
two
as
e-iht?t(t).
[F(o)=,, and
seen
notation
then
imaginary
form
e-:'tGt,
eiht?l(t),
and
same
(3)
cosh_.
I shall now return to the figure presented on page 243 (Volume II) assuming that q and ql were the rectangular coordinates of one
point,we separated those corresponding shaded.
the regions to unstable
These different regions are analytic curves, whose equations
corresponding solutions.
to stable solutions These latter regions
separated from each I have presented on
other by page 241
and are
four (Volume
II).
279
Following
are
these
equations: F(=)=
,,
F'(_)'-o,
(_)
F(=)=
,,
f(=)
=o,
(_)
F(=)=--,,
F'(=)
= o,
(¥)
F(=)=--,,
f(_)
=o.
(6)
To what category do the unstable solutions belong which correspond to our shaded regions? It is apparent that the unstable solutions corresponding to one of these regions are all of the same category. This is a direct result of the preceding statements. At reduced fore, if (6), the
of
a point of one of to f(t), and this
the curves (8) and (_), the function G may be function may vanish, since it is odd. There-
a region is bounded by corresponding solutions
However, this is not our unstable solutions
Our solutions such
example of two
a way Our
that
an arc of one of the curves, (_) and will belong to the second category.
the case in all of our regions. belong to the second category.
may be readily transformed in categories. It is sufficient this
equation
coefficient
(3) may
may
then
be
become
Therefore,
such a way to replace
that we have q2 by q, in
negative.
written
__d'x = x(-- q + q, cos2t). _It_ Let
us always
take
q
and
compile a figure similar to figure located to the right q's will be similar to the of the ql axis, at the side region which is bounded by
ql
all
as
(3')
rectangular
coordinates,
and
let
us
that shown on page 241. The portion of the of the ql axis on the side of the positive figure shown on page 241. But to the left of the negative q's, we shall have a shaded a kind of parabola tangent to the axis of the
ql's. The
shaded
regions
on
the
right
will
second category, as we have just seen. for the shaded region on the left. To
demonstrate
this,
it
is
correspond However,
sufficient
to set
have x=el¢-$;
280
==_/--q;,
G =,.
to
this
solutions
will
ql = 0,
not
from
of
the
hold
true
which
we
/280
case.
353. I have still only presented a discussion for a particular In order to extend it to the general case, I shall show that
always arrive the preceding
at an equation section.
having
the
form
as
equation
(2)
Let us first consider the case of absolute motion. If potential and if x and y are the Cartesian coordinates in a plane, the equations of motion may be written
force point
dU
and
same
the
variational
equations
U
we
in
is the of a
dU
may
t
(i)
be written
d' U
d'- U
(2) d" U
For the
purposes
derivations
longer
of with
represents The
energy
the
greater
brevity,
respect value
integral
cl_U
to of
may
t.
I shall Thus,
_ for
the
corresponding
integral
dU x'$'+y'n'=7_+d_ To
so
that
apply
we
the
shall
Maupertuis
accents
_" represents t" , as was
_{2
the
to designate here,
case
and
in No.
no
341.
be written _U+h,
.....
and
t =
employ
of
(2)
dU n+_h
(6h
principle,
is
we
a constant).
must
assume
that
have dU
x'_'+y'_'=
_
dU
+ _
_,
or
(3) Our solutions
equations which we
(2) and (3) will then have have called in No. 345
I _,=z',
_t----y,
three
independent
linear
(4)
281
Let
us
set O= _y'--r_z'.
If we three
then call solutions
called
01, (4),
the three values of O corresponding to the shall have O I = 0, and the function which we
02,
0 3
we
_(t)
in No.
345 will
may
derive
the
We
(s)
be
nothing
following
from
else
than
equation
(5) (6)
0'--$'y'- 7,%'+ _.y'-_,x" and 0"= _y"--
However,
x'
and
y'
satisfy
_.v" + _'y'--
the
x
the expression values which
by
their
of have
values
d _-U d:_
-
0", let us thus been
(2).
We shall
(2),
shall
designate
AU (or
more The
the
briefly
sum by
following
into
is
the
differential
We
shall
the and
the
two
282
our
may
be
easily
becomes
x'"
and y'" by _" and n"
/2S2
(7) derivatives
--dx2 d2U + d_ d2U
by
verified
(5),
r_x'+ _'y'-- r/x') ,}'y'--_x"-7,y"),
(6), 0_)-
equation
(7)
and
_(x'x'+y'y')O'-+ which
set
equation
derivatives the derivatives
,,'z').
second
0 = ? tx/_+)', and
have
_'x')
(x" +y")(_'-This
we
A).
identity
account
that
d: U y,.
replace found,
-- 2(x'x'+y'y')(_y'-= 2(ytx'--y'x')(_'x'-+
taking
so
have
of
2(x" + y")<_'y'-
or,
"_"r').
d -_U + _-U._Tj"y''
0"-- 0 _U = 2(,_'y'-I
2(_'y"--
equations
d_ U
In the
r/x'-+
+ :_(x'1 +y':)(_r'
(3),
_x')
we have ).(x "_+ y")O
defines
the
= o.
unknown
(8) function
O.
o" '-- = _ --
2'T
.
y ' .y"+ 3 _r'_ + 3 v "! x'-' + y'* '
t'+
(9)
an equation having the same form as equation (2) of the preceding section. The conclusions of the preceding section therefore remain in force. One periodic unstable solution is of the second category, or of the first category, depending on whether the function _ can vanish or not. We cannot pass directly from an unstable solution of the first category to an unstable solution of the second category, but can only pass through stable solutions. 354. motion? The
Do
the
equations
same
results
of motion
still
then
remain
valid
_ designates The
the
variational
speed
of
will
t
of moving
that
the
(i')
axes.
/283
d2U
energy
relative
be
d IU
I T,'+2_,'-- d'U
Due to the fact hold true for
of
dU
rotation
equations
case
become dU
where
in the
(2')
d'U
equation
is
still
valid,
the
same
will
(3) Let
us
and
equations In
set
(5)
addition,
and since
(6) will x'
and
continue y'
must
to hold. satisfy
equations
(2'),
we
shall
have ,z'--2o_y'=
d_U , d'U 37/, :r + d-7-d-_y.r', d' U
Taking taking and we
these equations into account, into account equation (3), we again obtain the equation
,
d 2U
as well as equations (2'), may simplify the expression
and of
also 8",
283
k.2
O"-OaU
we
(7)
Since the identity given in the preceding shall obtain equations (8) and (9) again.
to be
changed 355. The to
tried an
= 2(_'y"--_'z').
the
However,
which
end
conclusions one
trajectory determine
action
same
in
new
given
question
preceding
section.
arises.
is
smaller
than
any
infinitely
adjacent
arc
with
the
points. whether smaller
Let us first assume that a point B on the curve (T), so that the
closed
the
valid, needs
(T) is a closed curve. Up to the present, we have whether an arc AB of this curve would correspond to
However, we may also question corresponds to an action which is closed curve.
focus
in
section is always Therefore, nothing
this entire closed curve than every infinitely small
A of the curve (T) has arc AB is smaller than
its the
first entire
solutions
category.
/284
curve. This
is what
occurs
for
unstable
of
the
first
We have seen that the curve (T) may be divided into a certain even number of arcs for these solutions, and that every point on one of these arcs has its first focus on the following arc, so that -- starting from an arbitrary point -- its first focus will be encountered before the entire curve (T) has been traversed. This stable
also
occurs
solutions,
we
for have
certain set t÷
and
we
differ will
than
have seen that the iT by--. Therefore,
be
encountered
If
this
it
is
is
for
the any
solutions.
the
the
case
of
!
-- IogG(t) = 2:
(T) is
infinitely
In
347)
r of a point, and that _ i if T is larger than _,
before case,
stable
(No.
completely
action adjacent
cannot
of the
its
first
focus
of
focus, a point
traversed. be
less
for
the
curve
(T)
curve.
Let ABCDEA be the curve (T), and let us assume that D is the focus of C. Since E is outs_e the focus of C, we may attach C to E by an arc CME which is very close to CDE, and which corresponds to a smaller action.
we
284
If I represent shall have
the
action
corresponding
to
the
arc
CME by
(CME),
(CME) <(CDE)
and, consequently, (ABCMEA) ..<(ABCDEA).
Let
us
now
It may be it is for
consider
solution,
stated that the action any infinitely adjacent
In order a T ranges
that
a stable
to
clarify these i i between _ and _,
is encountered (T) twice.
before
such
will no closed
ideas,
longer curve.
I have
in such
traversing
that
(T)
less
for
(T)
a figure,
that
the
focus
of
times,
and
after
traversing
AB,
Let ABCDA be the curve (T). The focus F will be located and it will be encountered after traversing (T) twice.
arc
Since B is located beyond AEHNKHMEB, such that
this
(AEIINK[IMEB) Since
the
focus
of
A is
not
(T),
we
shall
encountered
have
in
we
may
attach
A
assuming a point /285
between
to B by
an
< (ABCABCAB).
Figure
traversing
focus,
than
compiled
a way
three
be
by
describing
the
arc
AB without
i0
addition
(AE + ES)>(AB),
285
k_J
from
which
we
have
the
following
by
subtraction
(EIINKItME)
<(ABCABCA)
or
(EmXIE) + (HNK;I)<
We
must
therefore
have
_.(ABC:X).
either
or
In (T) and
every case, corresponds
there is a closed curve to a smaller action.
which
differs
little
Therefore_ in order that a closed curve may correspond which is less than any infinitly adjacent closed curve, it that this closed curve correspond to an unstable_ periodic the
first
let
us study
to an action is necessary /286 solution of
cateRory.
356. stable,
from
Is
this the
periodic
condition
asymptotic
sufficient? solutions
In
order
to
corresponding
determine
this,
to a similar
un-
solution.
Let x = ?0(t , be
the
equations
of
y = +0(t)
the
periodic
solution,
x=
9o(t)+Ae=t?t(t)+A2e==t?_(t)+
y = +o(t)+Ae='+,(t)+ be
the
_i(t)
equations will
setting
be
of
the
periodic
of
t.
We
If
U is which
In
will
286
The
may
functions
also
write
the
_i(t)
and
following
Ae _t = u, x=?o(t)+u?l(t)+.
u,
....
solutions.
.....
)'='_o(t)-_-u,_,(t)_
and
let
A':=t,_,(t)+...
asymptotic functions
and
not
sufficiently
small,
are
with
addition,
vanish.
periodic the
For
x
and
respect
functional
I,(t,u),
......
_F(t,u).
y will to
t
be of
functions 2_.
determinant
d(x,
y_
d_
dW
dW
d,l,
O(t,
u) =
"_
du
dt
du
u = 0,
uniform
period
this
determinant
may
be
reduced
to
of
t
?;(t),',(t)-However,
this
given
in
No.
unstable
345
divided
solution
Due for
expression
to
u =
O,
If
u
functions
The
is
the it
fact
will
of
is
sufficiently
of
x
and
is
none
by
e _t.
Therefore,
first
category.
the
that not
+; (t)?l(t).
the
other
than
functional
vanish
for
small,
the
expression
it
will
determinant
sufficiently
u,
not
cos
t
does
small
and
sin
vanish
u
not
if
the
vanish
either.
t will
be
uniform
y.
equations
of
the
asymptotic
solutions
may
be
written
(l) +(t, y,_ = _F(t, and
it
can
be
seen
that
the
functional
,)(x,y)
point, (T)
vanish. do [this
This
not is
means
intersect the
if
o(t, _)
that
each case
the
curves
other,
it
is
t
and assumed
t
_
.....
_/287
determinant
_ O(
o(t, A) cannot
Ae_O Ae _l)
do
(1)
do
not
intersect
that
u
is
not
have
sufficiently
a the
double trajectory small.
!
_-_
," ,_-
,,.;,
\
i"
]
Figure
This
would
not
be
the
case
if
the
ii
curves
(i)
were
extended
indefinitely
287
k.j
in
such
a way
The therefore is sho_n
that
u becomes
very
large].
curves (i) corresponding have the appearance of in the figure (2). The
by a solid line,but a solid llne in the inside of the other The
spiral
to the asymptotic solutions will spirals passing around (T). This form closed trajectory (T) is represented
I must point out that figure. Of these two represents (T).
curves
(i)
are
there are two curves sho_m by curves, that which is located
represented
by
a dashed
I may note that there are two systems of ponding to two characteristic exponents which opposite
curves
asymptotic are equal
solutions corresand have the
sign.
These asymptotic solutions of which are similar to curves
different In
line
direction. the
case
They
of
an
the second system (i), except that
are not
unstable
shown
solution
_288
will be spiral they turn in a
on
the
figure.
of
the
second
category,
curves
(I) would have an entirely different form. They would intersect the closed trajectory (T) an infinite number of times, and the intersection points would form an infinite group having a finite number (even number) of boundary points. These boundary points would correspond to the values to, t I considered in No. 349. 357.
Let
us
return
to
unstable
solutions
of
the
first
category
and to asymptotic solutions of the first system which are shown figure (2). I shall establish the fact that the action is less than it is for an infinitely adjacent closed curve. I shall by an shown of
(T) and Let
case,
consider
infinitely in figure
we
passing
us
arbitrary
amount. a closed
through
confine
have
an
small (2) by
the
the
ourselves following
Let
AIB I, A2B 2,
The
end
points
...,
of
points to
thesetwo
orthogonally,
curves we
AnB n be
these
intersect
shall
arcs
the
curve
case
from
(T)
(T'), is outside
of
absolute
motion.
In
this
theorem.
located
BIBI...B_
series
on
two
of
trajectory
curves
•
trajectories
.....
differs
(T)
B 3.
a continuous
are
which
which I shall call with a solid line,
C and
have (AtB,)=(A,B_)
288
the
well-known
AIA_...A,, If
closed
This curve, curve drawn
in for
(A_Bn),
AIBI,
A2B 2,
...,
AnB n
arcs.
where the (AIBI) •
action
corresponding
to
the
arc
AIB l is
Let us therefore compile the orthogonal (i). These trajectories, which I shall call following differential equation \ dr= +
always
designated
by
trajectories of the curves curves (2), will have the
dt t ] dt
(3) + k u, ,_ + _
_4,;)(u,,+ _,,,,,,_+ t_,
+ ,¢,,,, _,, a,, = o.
One curve (2), and only one, passes through each point of the plane, provided that u is small enough. This could only not be true if the coefficients dt and du were zero at the same time, which could only occur if the functional determinant of @ and _ with respect to t and u vanished. We have seen that this was not the case. The
curves
Let
AIA2,
(2)
are
shown
on
...,
A5,
BIB2,
..., B 5 be
curves. They intersect AsB5, on the curves (I) For the
my
action In
purpose, of
it
(CB3)
is
we
have
effect,
the and is
arc the
the
figure
two
of
than
for me
for
the
to
a dot-dash
the
A2B 2 on (T), the arc CB 3 on (T').
sufficient
larger
(2) by
line
infinitely
arcs
AIBI,
establish
corresponding
adjacent
A3B3,
the arc
fact
A4B4,
that
A2B 2 of
(T).
( A, B,) = (A3 Bs) and, in we have
the
infinitely
small,
curvilinear,
rectangular
triangle
A3CB3,
(CB3) > (A3B3). We
therefore
have (CB,) > (A=BI),
and,
consequently, action
of
(T')
> action
of
(T). q.e.d.
for
358• We must now relative _otion. The
irreversibility
determine
of
the
whether
equations
the
same
result
constitutes
is
still
a great
obtained
difference
289
kJ
from the preceding case. The action for an arbitrary arc AB is no longer the same as for the same arc traversed in a different direction. If an arbitrary curve satisfies differential equations, this will not hold true for the same curve traversed in a different direction.
Finally,
the
orthogonal
trajectories
of
the
longer have the basic property which I discussed section. However, there are other curves which which
have
preceding
this
property.
section
In
No.
340,
to we
This
is
remain
valid.
obtained
the
sufficient
following
curves
(i) will
in the I shall
for
for
the
the
no
preceding define, and
result
of
expression
the
of
/290
the
action J For
purposes
of simplification,
I shall
set
VH0
+ h
= F.
I shall
no
longer designate the coordinates by _ and n, but rather by x and y, in order to approximate the notation employed in the preceding sections. And I shall no longer designate the angular velocity by m', but rather by
_,
removing
the
accent
which
has
J'=:[F from
which
we
become
useless.
then
have
v:d-x'--+ dr'-+-o_(xd: --y dx)],
have _J' _--f
[_v as + V ax _d.,"ds -_-,(r _'.r + _o(_z dy - _y dx)+
or,
I shall
integrating
by
_o(z Zdy
- y _d..)]
parts,
,
ds
+
The definitive expression of 5J' therefore includes two parts: a definite integral which must be taken between the same limits as the integral J', and a known part which I have placed between two brackets (according to common usage) with the indices 0 and i. This notation indicates that we must calculate the expression between the brackets for
the
two
the
second
Let
us
integration now term
tial equations which will be curves
limits,
assume
that
of
is set
(4)
and
the
must
expression
equal
to
which will be precisely satisfied by all of our
then
take
included
zero.
We
the
under
shall
the equations trajectories,
difference.
of motion, particularly
sign
f in
differenand the
(i).
These equations may be obtained in an infinite number because 8x and 6y are two entirely arbitrary functions.
290
the
obtain
of ways,
dF Wemay first assumethat _x = O, from which we have 6F =d_v _y" Dividing by _y ds, we may then write our equation as follows dF
cl_
dF
d.y
d t v
(6)
If,
on
the
contrary,
we
had
assumed
dF _
These two beforehand, dy ds
and
into
-_- 2to
dy ds
dz ds
dx d_s' respectively,
and
i r the
would
obtain
/291
d_x _- F -ds_f
readily be multiplied
following
determined them by
relationships
are
taken
account
arrive
at
an
-- \,is J
;_j'=
Let
AIBI,
the
curves
curves
A2B2,
Let
...,
action
end
r_,,r
i-d.r_Y
+(o(x_),_y_x
satisfy equation (4) becomes
) ]1 .
¢ls
o
a continuous
series
of
points
AIA2...An,
BIB2...B
two
these
which
arcs n
pertaining
form
two
to
continuous
C'.
AiBi, Ai+iBi+
by an infinitely Ai, x + 6x, and point
ds ,ts_
the curves (I), they will into account, relationship
[Fd,
AnB n be
(i), whose
C and
-ds 5_'s-_+
l;
identity.
If we therefore consider If we take this equation
(6).
1 be
of
small amount. let y + _y be
arcs
differ
from
each
Let x, y be the coordinates of the coordinates of the infinitely
other
the point adjacent
Ai+ 1 .
Let J' be relative If
a Is
the to
the
action relative the arc Ai+iBi+
angle
which
the
curve (I)] makes with the axis of satisfy the differential equation I" (cos
we
dF ds
_
_y = 0, we
equations are equivalent, as could If they are added after having
d_/
we
that
shall
a 3z
+
to I.
the
tangent the
sin _t _y)
+
arc
to
x's,
to (x
AiB i and
the and
_av -- y
curve if
8x ) =o,
the
J'
AiB I two
+
_J'
be
[which curves
the
is
a
C and
C'
(7)
have
291
v
and,
consequently, (A, B,) :_ (A, B,) .....
The curves defined by equation the orthogonal trajectories of section. We may therefore the curves shown
that
orthogonal There will
(A_ B,,).
(7) may therefore play the role which the curves (i) played in the preceding
consider figure by the dot-dash
(2) again, and we may assume line no longer represent these
trajectories, but rather the curves be nothing to change in the proof.
However, rectangular
one
point
triangle
is no
longer
clear.
defined
In
the
by
equation
infinitely
J292 (7).
small,
A3CB 3, I have (CBa)>(AaBa).
The the
triangle is no longer rectangular, definition of the action. Does It may
No. 341, equality
be
readily
seen
and we have seen therefore holds,
that
and in addition I have changed the inequality still exist?
this
equality
in No. 344 that they and our proof remains
equals
conditions
are fulfilled. valid.
(a) The
of
in-
To sum up, in order that a closed curve corresponds to an action which is less than any infinitely adjacent closed curve, it is necessary and sufficient that this closed curve corresponds to an unstable, periodic solution of the first cate_orM. 359. We must unstable solutions
make into
a few remarks regarding two categories.
the
classification
of
From another point of view, the unstable, periodic solutions may be divided into two classes. Those of the first class are those for which the characteristic exponents e is real, so that e at is real and positive, The a has
of
in -_
where
solutions as
of
be
divided
We may
the the
an imaginary
In the preceding the first class.
also
292
T is
set
into
period. second part,
class
are
those
that
e aT
is
so
for which
real
and
statements, we only considered Let us see whether those of the two
categories.
.
= _ =' + 'i"'
this
exponent
negative.
unstable solutions second class may
where _' is real.
where
_2,
changes We
_2,
_3
into
t +
then
have
Wemaythen set, just
and T.
as in No. 346"
_ = e_'e?_,
_, = e_'e_2 '
_ == e-_'e93 ,
_3 _- e-°ct+3,
_3 are functions These functions
of t which change will be real.
sign
when
/293
t
_,_ _ _, _ = _,_,G(t_.
change
The numerator and the sign when t changes
It quently
is therefore certain that the same holds
denominator into t + T. that these true for
of
G are
two
functions
functions
of
vanish,
t which
and
conse-
These last two functions satisfy the same linear differential equation of the second order, whose coefficients are periodic functions of t which have not become infinite. The coefficient of the second derivative may be reduced to a constant. These two functions cannot become zero at the same time, because if two integrals of tion become zero at the same time, they could only factor. However, _(t) is not a constant. The numerator and the denominator of _(t) zero, and do not become zero at the same time. sequently G(t)] may vanish and become infinite, All of the unstable second category. Apart preceding statements.
solutions from this,
the same linear equadiffer by a constant
therefore Therefore
both become _(t)[and con-
in question therefore belong to the there is nothing to be changed in the
293
CHAPTER XXX FORMULATION OF SOLUTIONS OF THESECOND TYPE 360. Weshall now demonstrate the manner in which periodic solutions
of
the
second
type
may
be
effectively
f294
formulated.
Let d:ri
dF
-)7 = ;if,'
be
a system
periodic
of
canonical
solution
of
first
We plan to study periodic from the solution of the The
analysis
may
if equations (i) in the variables. We
shall
increases
where
I
by
k I and
are
assume one
be
that
period,
k 2 are
at
least
then
assume
are
and
if
presented
this
were
in No.
208.
that
not
have
Y2
only will
for
two
purposes
by
which
of
a series
degrees
increase,
type
of
274,
361. the
or Let
canonical
to the us
the I =
true,
analysis
recall
changes
freedom.
respectively,
When
t
by
periodic O_
X!
_
solution
0 t
I could
if this variables
_1
_
were not the case, given in No. 202.
(2) may
be
the
presented results
to
01
perform
the
in No.
obtained
reduced
change
in variables
of periodic given /295
44.
in Nos.
273
to
277.
Let
equations d.v_ dF dt -- _i'
294
pro-
discussion,
of
Under this assumption, we shall see how the determination solutions of the second type is related either to the analysis in No.
a
numbers.
X
because,
they
second
form
I may first assume that k I = 0, because, could cause k I to vanish by the change in I may
that
(2)
a suitable
there
integer
assume
solutions of the first type (2).
to
Yl
us
(1)
y,- = +,(t).
simplified,
reduced
,t:_,,
type
x,-= ?,.(t),
ceed
dF
at Let
equations.
the
dv¢_
dyt _
--
dF dx_
(i-,,
_, ...;_)
(1)
include a parameter X, and let us assumethat they have one periodic solution c/= of period
TO,
corresponding
?i(t), to
yi = q_/(t),
the
corresponding to X = 0. Equations series having the following form; of the quantities ._,
Ake=_t,
value
(2)
C O of
(i) will be these series
h',e-=_t
(k -= ,, 2 ....
the
energy
constant
and
formally satisfied by will develop in powers
, n--
,).
The coefficients will be periodic functions of t + h, depending on the energy constant C. The period T will also depend on C and the products AkA _. It may be reduced to T O for C := Co,
), =o.
The exponents _k are constants which may be developed X and the products AkA_, and in addition they depend on reduced to the characteristic exponents of the solution
of be
The
Ak'S , the
real,
of ak
AkA_- = o,
When and
studying we made
In order the second are purely
remove
and
Ax.A2 = o,
h's
integration
asymptotic one of the
the
order
are
that
exponent
we _ be
may
two has
)_ = o. constants.
solutions, we assumed two constants A vanish.
to apply the same results type, we shall assume, on imaginary.
I shall assume only the index k which In
that
A_'s
C = Co,
degrees become
obtain
in powers C. They may (2) for
that
the ak'S
to a study of periodic the contrary, that the
of freedom, useless.
which
allows
were
solutions exponents
me
to
/296
periodic
commensurable
solutions, it is necessary 27 with-_. If our series were con-
vergent, this condition would be sufficient. However, they are divergent, and only satisfy equations (2) from the formal point of view. A more detailed discussion is therefore necessary. A method similar to that employed in Nos. 211 and 218 could be applied. We would thus obtain series which would have the and 277 as the series of M. 127. By solutions different
same relationship Bohlin have with
with those
those given
given in Nos. 273 in Nos. 125 and
an indirect method, we would thus fall back on the of the second type. However, I prefer to proceed manner.
periodic in a
295
kj
Effective
Formulation
of
the
Solutions
362. By performing the changes in variables presented in Nos. 209, 210, 273, 274, which are always applicable when we have a system of canonical equations having a periodic solution, we may change our equations to the form of the equations presented in No. 274. In this section, we have formulated the following equations (page 95)
.... dt
_z',
dx, v=,
-dF ....
(3)
F' = F; + _v; + ,, F; + .... where
F' p is
a whole
polynomial
of
degree p + 2 if it is assumed ' is of the second order. and x 2
periodic
functions
Just become
as
whose
in Section
useless We
of y_
then
and
No.
shall
assume
!
x2,
which
that x_ and y_ The coefficients period
274,
write
the
!
in x I, y_
we
F,
following
is
remove
xi,
(see
be
homogeneous
of the first order, this polynomial are
2_.
shall
Fp,
are of
will
Yi
pages
the
accents
have
F' , F; , xi, '
instead
of
q7,
99)
98,
which
Yi" '
Fo=Ilx2+2B_tY,, where
H
and
B
are
constants.
I could
also
set
H
= i, but
I shall
not
do
this. Just
The
as
on page
equations
will
99,
let
retain
us
the
other
respect
which
terms to i
Our we
v
u is
This
296
...,
will
respect
and we
shall
have
2Bu;
be
periodic
of period
2_,
both
with
to Y2.
where the parameter e plays we may employ the procedure
However, and 44.
F2,
with
IIxt+
form,
equations will then have the form which is similar to that have studied several times, and in particular, in Nos. 13,
125, etc., Therefore, tions.
x2 No.
FI,
and
set
canonical Fo=
The
then
there zero,
fact
is
one
which
compels
me
obstacle.
is
to
precisely
assume
the role of the parameter given in No. 44 for these
The
Hessian
one
of
that
the
F depends
of F 0 with exceptions
on
respect given
a certain
42,
_. equa-
to
in
parameter
%, and we shall carry out the development at the same time in powers of X and of e. Wesaw in Chapter XXVlII that when studying periodic solutions of the second type it is always convenient to introduce a similar parameter, because the property of being reduced to a solution of the first type for X = 0, and of differing from it for X _ 0, characterizes solutions of the second type. To facilitate the discussion, instead of an arbitrary parameter, I shall introduce two parameters, which I shall call X and _. Weshall therefore assumethat the different coefficients of F may be developed in powers of two parameters % and p, and that for _ = %= 0, H and 2B may real number.
of
c,
I shall in the
where %1, X2, termined, but which follow.
No.
be
reduced
to -i
assume that X and following form
..., are I reserve
and
to -in,
_ may
be
where
developed
constants which I shall the right to determine
Under this assumption, let us 44 step by step. We shall set
n
follow
the
is
a commensurable,
in
increasing
provisionally them in the
computation
powers
leave undecomputations
presented
in
/298
x, = _o -+c-t, + _'_,÷ .... u = Uo -_-"u, + _' u,+ ....
These
of
t;
where below.
formulas
are
similar
The _k'S, nk'S , Uk'S , and $0 and u 0 are constants,
_ is
an
integration
to the
formulas
the Vk'S are and we have
constant
(4)
which
(2)
therefore
I shall
of No.
44.
periodic
determine
more
functions
completely
Instead of %, p, x2, Y2, u and _ let us substitute their expansions in powers of E in F. Then, F may be equally developed in powers of E, and we shall have
if we
I would
like
to point
assume
that
_p and
Up
out are
that of
_k
is homogeneous
degree
p +
2,
_p
of and
degree Vp
are
k + of
2,
the
297
degree
p, It
Xp
is
and
_p
is
therefore
of
degree
a whole
p.
polynomial
with
respect
to
(p >o), and
with
respect
These
last
Finally,
the
q0 whose
period
In
to
two
quantities
coefficients is
addition,
may
of
be
this
assumed
to be
polynomial
are
on
the
order
periodic
of
functions
i. of
2_. we
shall
obtain
,l,m= O_--- Sk-- f*tuk-i- ),k Ilo_o+ "2Bog*kuo, where
H0
assume only
and
B0
that
we
are
the
values
dH dB -= 0 d> dX
have
dH _-_
of for
dB _
and X = _
for
X = _
= 0.)
In
= 0.
addition,
Ok
depends
on _p, %, Our
differential
For
k = 0,
they
up, %,
equations
d_.k d'l'l¢ _-l-[= d_-_' may
be
demonstrate
the
d,l',, -d;_7'
duo
_ is
fact
a constant
be
which
must
v_
be
:
dt -- in. u 0 are
inl
+
constants,
and
us
develop
xl
and
Yl
determined.
in powers
of
+q', + i
The pansions
expansions
(4')
y,
may
:=
P 2 t 4-e_' t _- _ _'-_ i t r,o
be
that
by,
We may advantageously add other equations having equations (4) and (5), which are only transformations Let
(5)
d'l'k du-_"
dvo
=t;
_0 and t,
dr j, _ dt ----
to
_
that
/299
written
du_ d,l,_ --0{ = -d('o'
d_jo
=o;
"_o -=
where
(pSk--J).
then
reduced
dt -- dt They
Z_, ILp
may
dr,_. _ dt
cl_u
directly
E, and
a similar of them.
form
+..-, '
•
then
find
that
Ck is
a whole
to
let
(4')
.-
concluded
from
the
two
last
(4). We
298
(We may
polynomial
with
respect
to the
ex-
quantities _p, %, _, _, lp, _p (writing and that that
this
polynomial
is homogeneous
of
degree
Sp is of degree _,_ is of degree %, Ip,ep is of degree We
then
have
the
following
which
are
We
form
as
equivalent
may
note
to
d+,
dq,_ d-_f'
that
_k with
respect
-- d_' o '
the
last
two
the
a_4_
d+_
dt
a_o '
are
(6).
having
Using
the
the
same
same
convention
find that they are homogeneous, of order k, and the last two of
have
us
replace
replace
nO
that
we
have
set
We
thus
have
by
t,
_0,
in
_0 by
equations
k = i,
the
and
six
d'_,
(5)
d;]'
1
Let us first a homogeneous,
dot
and
+=L
and
(5'),
at
the
in which
them
to
same
it must
determine
time be _i,
let
us
assumed nl,
Ul,
equations
dO,
d$1
dr, dO1 -dr = -- duo
2 _1 80,
du, in dr_--_o +_B°llt _[tZl
values,
us employ
following _lq_ 1
d_', dO, dt -- dr/o
these
let
dui dO, -di- = -_o'
is
assume
(5).
_ _o= _oe-¢nit Let
we
(5')
polynomials
quantities
employed above regarding the degrees, we the first of the order k + 2, the second the order k + i. We
2, if
p+% p+,, p.
equations
d,_k d,I,# d,_ d_"-;'-'d--_d_0
to
k +
(6)
equations
a_. d[
nO separately),
duo dO, dr_'o -- d7l_ duo
consider the second whole polynomial of
dO,
. ,, tn_t+_Bo
(7) _,_o,
dO_
of these equations. The second term the third degree with respect to
299
/300
01
whose
coefficients Since
n
is
are
periodic
The
period
multiples
of
Our the
will
2n
second
following
functions
commensurable,
function of t on which equals t, and by means
as
our
it depends of £'0 and
be
are
can
of
second
nO
term
in two ways: n_ which are
a multiple
there
term
"_0,
will
period
also
he
by means functions
of
2_,
i.e.,
in
the
denominator
be
developed
units
therefore
= % of
a periodic
of no which of nt + _.
it will
in
2_.
equal
as many
of n. Fourier
series
in
form
z ,'__/'J'___',_'+=I', where p and lute value,
This
q are since
integer numbers. our second term
is
(8)
However, q does not exceed a polynomial of the third
3 in absodegree.
As a result, in general the mean value of the second term is mean value will be obtained by retaining the terms which are
pendent
of
t in
the
series
(8),
zero. inde-
i.e., p+qn---o.
due
I have stated that JqJ can not exceed 3. I would like to to the fact that our second term is a whole and homogeneous
of degree 3 in _0, _0 and second term cannot contain
n_, it _'0 and
q must
take
be
odd
and
Therefore,
can
we
only
can
only
is @0
assumed that except in an
one
of
1 or
3.
the
values
add that, polynomial
_0 is of degree 2 or a uneven power -- i.e., +_i or !3.
have p_qn_o
if
the
denominator
We number,
the
i. fact
shall but
of n
exclude
two
equals
the
cases
first
remain
hypothesis
for
our
which
would
make
n a whole
consideration:
The denominator of n does not equal 3. that the mean value of the second term
In this is zero,
case, due to the equation
will immediately provide us with _I by simple quadrature. Then $i is determined up to a constant which I shall call YI, and this constant remains undetermined up to a new order. It should be pointed out that the
same 2.
be
30O
holds The
integrated,
true
for_.
denominator it
of n
is necessary
equals that
3. the
In
order
value
of
that the
the
second
equation term
be
may zero.
For this purpose, we shall employ the constant _. have
Let [ @I] be the meanvalue of @ 2. _
r__(_l
,z[o,l
nLd.,,o.l= and
we
shall
therefore
determine
Weshould point out that we
d_
_
by
;
the
equation
d[O,l (9) and
a
quadrature
Let may
be
us now
pursued
polynomial n
is not
d@l
is
of an
will
take for
the
then
the
this
first
us with
equation
equation.
third
integer
provide
degree,
number,
(7).
However, but
we
rather
shall
be
_I,
up
The
of
a constant
same dO 1
since
_0
the
certain
to
line is no
first
that
of
reasoning
longer
degree,
the
YI.
mean
and value
a since of
/302
zero.
d_0 It is therefore sufficient for us to take second term may have a mean value of zero, and determined to a constant 61 . Let us now pass written as follows
on
to
-
the
last
a-7
_
d'_ 'j
that
The second integration
firstequation term of the
They
may
be
+ =Bq_,_;,
does not include a term second equation does not
In order that the
containing e int, include a term
e -int .
dut
be discussed more readily by considering (7) which are equivalent to the last two,
dot,
dr,
at = -_ "
With
(7).
the be
dO,
This double condition could these third and fourth equations and which may be written
It
equations
terms are the known periodic functions of t. may be possible, it is therefore sufficient
second term of the and that the second containing
two
%1 = 0 in order that in order that nl may
is necessary respect
to
that the
the
mean
first
of
-_
dot
= - _ values
these
-- _ _, Bo. of
the
equations,
second the
terms
be
condition
zero. is
301
k.J
fulfilled
by
itself,
and
we
have
[e0 ] a[0,] dr,| = This latter expression denominator of n equals 3, identically The
is and
d---_"
zero because of in the opposite
zero. second
condition
may
be
written
e[o,] = duo
If
the
denominator
of n
On
the
other
if
vide
equation (9), if the case because [01] is
us with
_I
hand,
--_tBo-
equals
the
= 0, because
3,
it will
denominator
[01]
is
provide
does
not
identically
us with
equal
3,
_l" it will
pro-
zero. /303
and
Thus, of _.
we may see that _I, nl, _'I, @I They may therefore be developed
following
are periodic in Fourier
functions of series in the
t
form A _iIpt+qnt+q_)
However, having
the
we
may
following
d_
shall
a few words
more.
We
must
deal
with
equations
form
_
dt and we
add
derive
_r Aei(pt+qnt+q_)
the
d_ d-_ + ittN = Y = Y,Beapt+q.t+q_!,
1
following A t(lat+qnt÷qml
=E_(p+qn)
E _(p+qn+n)
_= where
y and
¥'
are
Therefore, respect
integration
if X and
Y
zero,
will
y and _ and Let
computed.
302
+q_)
.Jr_ ._' _-Int
whole
and
homogeneous
polynomials
with
to
same
stants
+qnt
_s
constants.
are
_/_o, the
el(pt
q-
us
hold y'
n will apply Due
true
are
for
zero. still
these
to the
_o eilnt+=_, g and If
be
whole
principles fact
n,
it is
that
¢ ge-t_nt+=' unless
not
it
assumed
polynomials, to
the
is
assumed
that but
quantities
these
not
that
the
constants
conare
homogeneous.
which
we
have
just
are polynomials ployed regarding
which, degrees,
dO1
dOi
dOj
d_t
a_o'
_S,,0' 7if'
a_;
according are
to the
of I,
the
same
will
hold
true
the convention which following degrees,
3,
a,
we have respectively
em-
2,
for
_,, _,, _',,_',. When we substitute the values of these quantities which are, respectively, of degrees i, 3, 2, 2, instead of these quantities in 92, it may be seen that @2 becomes a polynomial of the _urth degree, and that dO,
will
be
polynomials
of
the
dO_
following
dO,
/3O4
dOl
degrees,
respectively
2, 5, 3, 3. We
may
therefore
formulate
Equations (5) and (5') q_ from place to place.
g_, value zero.
of
Let
will
be
the
us
second
assume
term
that
polynomials
of
one
this
respect
is
Let us assume less than k.
know
this
result.
not
k,
equations
occur.
(5) were
It may
be
gk' qk' the mean
different
stated
from
that
degrees
k+l,
k+l
to
the coefficients of t of period
We
the
following
_' where tions
of
does
k+_,
with
of
enable us to compute the unknowns This would only be prevented if
of
the
a generalization
that
of 2_.
that
@k
is
_ei{nt+_', V/_o¢-i(nt÷_',
these
this
is
a whole _q,
polynomials
valid
for
polynomial _q,
_,
_
are
every
of (q
(10)
themselves
value
degree
Qf
k +
periodic
the
index
2 with
func-
which
respect
to
(ii)
3O3
assuming that these quantities are of degree q + 2, q, q + i, q + i, respectively. If we substitute polynomials whose degree, with respect to the quantities (i0), is precisely q + 2, q, q + i, q + I, in place of these quantities (ii), it is apparent that the result of the substitution will be a polynomial of degree k + 2 with respect the the quantities
(i0). Therefore,
quantities
Ok
(i0),
is
and
a polynomial for
the
same
dO_
will
be
polynomials
of
the
of
respect The
to
same
k +
dOk
dO_
following
same
quantities.
up
to
the
holds
true
for
following
the
the
four
equations us with
/305
terms
of
shall
retain
only
by
means
this
of
q.e.d.
,['_e+i_,,t+=_,
arbitrary
one
of
(5)
_',e-i_,,t-,-_J,
constants _",,
these
_.
constants
us
try
h,
_,,
_,,'
_,,'
and
by
(5')
we
shall
set
_', = --_;.
let
and
and
to
determine
setting
It is necessary that the second term of the a mean value of zero. This mean value equals
ao,1. d_o.I
304
second,
terms
assumption,
equations
first,
repeating the previous same holds true for
_.
_,= 81= o, Under
the
by the
(7) has introduced four new integration information concerning _I, nl, _I, n'l,
_,, _, We
the
k+i
second
_h-, _,
_,, _, containing
to
degrees
fifth, and sixth equations (7). Consequently, line of reasoning, we should readily see that
The integration of constants. They provide
respect
dO_
k+l,
the
7,k,
2 with
reason
k, k+a,
with
degree
k
=
first
2. equation
(5) has
and we always employ the brackets to represent the meanvalue of a function. Wemust therefore have dOll _,o j = o. Let
us
assume
that
is
@2
developed
(9) in
Fourier
series
in
the
following
form A ei(pt+qnt+ql_).
Since 0 2 is a polynomial of the fourth degree, q could not exceed 4 in absolute value. Consequently, if the denominator of n is larger than 4, [@2] will be identically zero, and the condition (9') will be fulfilled by itself. The constant U will remain undetermined. If the termine _.
denominator
of n
If the denominator determined by condition mine
the
constant
Let
us
We
2 or
4,
of n equals 3, the (9), and condition
the
condition
(9')
will
constant _ has already (9') will enable us to
de-
been deter-
y_.
calculate
obviously
equals
1306
the
will
terms
depending
on
this
XVl in
constant
@2.
obtain
(ao, T; \ _lf_
ao,.-,,.,,+.,A eiU_t+=J-
drt' *
,],
i,e,,
The
The
mean
value
condition
l
[
(9')
of
may
this
will
therefore
be
be
written
{ if
it
is noted
that
d'[o,il
n L_7_,T_dC_ J = -_-_,-
•fi_ d, fo,] +rl
where
(9')
H
depends
on _,
If
the
is
independent
denominator
but
not
of n
of Y'I"
=o,
on _i" does
not
Therefore,
equal if
3,
this
[@i]
is
zero
denominator
and
condition
equals
2 or
4,
305
equation (9') will
depend on _ and not on _i and will
If the denominator equals 3, condition determine Y'I (it
will
determine _.
(9') depends on Y'I and will
provide us with y_ = 0).
In any case, having thus determined _2, let us try to calculate _2 by meansof the second equation (5). Weshall employ %2in such a way that the second term has a meanvalue of zero. Weshould point out that
%2
will
not
be
zero
in
due
to
the
fact
general,
and
d[O,] d_o will
not
nomial
be
of
pendent
zero
in
general,
degree
4,
it will
of
periodic
1 I _k s and
the
function
of
! nk
because, include
!
s,
a term
The
t of period
containing
coefficient
2_,
and
that @ 2 is a poly2 _0 which is inde-
the
of mean
this
term
value
will
will
be
a
not
be
zero
in general. Let us proceed to equations (5') or, two equations (5). The second terms a mean value of zero.
last have
We
must
which is of these
the same last two
thing, to the equations must
includes
terms
have _/o_]= _ 2_,Bo, duo J
which
determines
_2"
However, dO.
is a polynomial of the 2 2 xlYl, and consequently
The
coefficient
is
not
zero
_2
are
not
of in
this
fourth dO2 u 2 -duo
term
general.
zero.
This
order.
the
F 2 therefore
includes
is
a term
a periodic
Therefore,
is
dO_
d_,
same
containing
function
in
general
line
of
containing
of ] °u2 ] [du0J
reasoning
t,
whose
and,
as
mean
value
consequently,
is employed
for
%2. We
must
then
have (12)
306
However, it may be stated that this condition is fulfilled Wehave the energy integral series of equations _o
Let
us
consider
the
_
COnSt.,
third _=
qbl
of
F = const., from which we may deduce the _
COnSt.,
these
_
_
COnSt.
will
have
thus
02--_2--inu2+
_2tlo$o+
determined yl,
depending on the these constants,
82,
....
2Bo_tUo=
$2,
_2,
T;elfnt+m),
_2,
8,=o,
const.
(5) and, when _2, _2, $2, /208 first three equations (5), We may therefore be assured consequently, that the con-
_2
up
to
the
We
shall
retain
1 The
last
two
consequence to condition
conditions of the (12).
Expression in
Fourier
r<< l
rdo l
LdT;J =o;
Ld_oJ+_&Ito=,,,
L_u_j+2_kB0=o.
determine
one
of
and
_k"
The
of
order
k +
2.
second
learned
If
it
will
with
is
ability
be
a
respect
developed
series A
the integer is smaller
%k
first, according to what we have We must then study the first.
d@k d-_0 is a polynomial
only
way. The conditions
r<,o,l will
terms
_ = __.
363. The calculation will be continued in the same of equations (5) to be integrated requires the following
a_,oJ =°'
following
_;e-i(nt÷=)
four arbitrary constants. and we shall set _,=
1
equations
This equation may replace the fourth equations n2 and v 9 have been determined by means of the it will _etermine u 2 without any integration. that the determination of u 2 is possible, and, dition (12) is fulfilled. We
by itself.
eifpl+qtat4-qt_
I
numher q cannot exceed k + 2 in absolute value. than the denominator of n, we could not have
If
k +
2
p -¢-- _/n = o
and
the
mean
value
of
our
expression
d_,
will
J = O
be
zero.
The
condition (13)
307
will
therefore be fulfilled
by itself.
Wehave introduced the following
arhitrary
constants: (14)
and @kmay dependon r
_, Let
us
determine
considering
the
the
form
t
T,,
of
T,,
this
t
,
Tk-,. Let
dependence.
us
assume
that
we
are
expansion F = _0+ _%+_'_÷--.
and
that
in
n' 's by
this
their
expansion
values.
we
The
pend on the constants (14). constants y', retaining only
In
the
expansion
replace
the
different
(16),
terms
(15) _'s, of
In _.
this expansion We will thus
_+
_
let
us now
the
the
n's,
the
expansion
(15), obtain
_ 's, will
and
then
the de-
let us cancel all a new expansion
+ _,_;+ ....
the
(16)
replace
the
constant
_
by
the
expan-
sion
where _i, developed anew
_2 in
are its
in powers
new constants. turn in powers
of
c, we
obtain
Each of E. a new
+_+ _ This that
expansion must be identical the constants _ are replaced
term in the expansion When this expansion expansion
+ _,_;+ ....
(17)
to the expansion by the suitably
(15), under the chosen functions
I
constants
Tk"
It may
and
that
be
readily
_k depends
seen
only
may
and
3O8
v Yk
conclude
on
from
this
only
on '_.
We
" depends Ck
that
(16) may be is ordered
T_,
"'"
T_-,"
that
_k
depends
•f,,
-_;,
...,
._
only
on
on
condition of the
It
may
be
where with
and
readily
seen
A is
a numerical
respect
to _.
we
then
that
coefficient
The
order
of
and where this
2:_2 q--...q-
is at least equal to i, and since that _k is zero, which we already
Let us consider where _h is not
but
is
a derivative
of
equals
k_k.
_0 does knew.
an arbitrary term where zero. We must have
_k'
not
depend
_k-l,
''',
on _$
_h+l
value
and
of
important _,
result
and
may
consequently
be
concluded
the mean
of Ok.
2,
[O k ] will
denominator
of
If
the
denominator
of n
equals
k+
I,
If
the
denominator
of n equals
k,
[@k ] will
If _3.
the
denominator
of
k -
which
n equals
I have
just
concerning
value
the
statements
k +
this
If
The
n equals
from
i,
made
[@k ] will
[Ok]
depend depend
depend will
concerning
on
depend
[Ok]
zero,
of D_ m _h have
the mean
only on _ _,
it may
are
If the denominator of n is larger than k - h + 2, the mean value II will be zero. This means that those terms of _k which depend on a mean value of zero. An
_' m
have _" ---- In "+"_t+
Since m be seen
D_' m derivative
and _I.
ml
and
on _,
also
on _.
_2"
_I,
_2
apply
to
" which
Therefore, will only
If and _i. ship
if the include
denominator of n equals _, will determine _.
the denominator However, _ will
k +
2,
relationship
(13),
equals k + i, relationship (13) will contain _have been previously determined by the relationdo,_,]= dr,,
j
o.
309
Relationship (13) will
therefore determine _and,
consequently, _I-
If the denominator equals k, relationship (13) will contain m, _i and _2. However, _ and _I will have been previously determined by rela- 1311 tionships having the sameform as (13). Therefore, (13) will determine and
_2
consequently
_2.
This
process
will
then
be
continued.
Discussion
364.
The
arbitrary
solution
which
we
have
obtained
still
includes
the
following
constants
With respect to the parameters I and _, we have obtained them from their expansions in increasing powers of E, and we have successively calculated the coefficients of these expansions. These coefficients Ik and _k depend by
on
means
the of
two
the
constants
following
$0 and
u0;
these
aok] + _,no=
where
d@ k d$--_and
Ok,
dOk u0 _
are
whole
_o,
Let
where
us
P
set
is
a whole
polynomial
coefficients We
then
us
then
calculated
[ao_]
polynomials
_/_o
in
e±_nt+mL
are
with
...,
periodic
respect
to
_ .....
_,
_;,
functions
(18)
_ ....
of no.
have dO, = E h_ Q,
Let
were
__ Ox---- E P_oS,%'h, __,h_ -EQ,
_,, _,, whose
coefficients
equations
replace
the
quantities
u,
= E
(18) hy
their
Q.
expansions,
and
let
set _ n ,'b,
where
B is
following
310
a periodic from
this
function
'bs_b s
of t of
the
period
2_.
We
obtain
the
us
dOk
dok
_° d_ = _h_n, We
shall
hi
n.
obtain
/312
raoq
rdo_l
while retaining the The different terms
terms which of R contain
e
In
h t+
_° 2_ = _ _
order
that
this
term
may
ip!
be
are independent the following
X
el!nt+_hb,+li,-bs--h,
which illustrates the denominator
of
the n.
fact that Therefore
h,--
of b,--
b1 + h1 we have
t, it
is necessary
indicates that R is divisible 1 exponent _ (b I + h I + b2 + h2).
There
would
only
be
an
by
exception
b2
u0,
to
that
h2)=o, -
h 2 must
b,+ hj-_b2 _ h2 > bl+ hi--bt--h2> denominator which
expansions. as factors
].
independent
p + n(bt÷
of t in the exponentials
since
this
be
of
u 0 is
if we
divisible
by
n _ 2, included
with
the
had
bj+ h,= b,+h_ but
we
would
then
have
either bl+ht_t, bj-+'h,+bt+h_
in
such
a way
that
R would
be
_,
divisible
by
u 0,
or
bl--hl=bl--hl=o,
from
which
we
have ht+ht 2
However,
In
the
the
corresponding
same
way
terms
R will
always
would
be
not
then
divisible
appear
in
by
unless
_0,
u 0 Fd0kl
LduoJ" h 3 = 0,
in
F_d k_l which
case
the
term
would
not
be
included
in
_0
[_0J"
311
V
Therefore,
to sum up we have
[de,. 3_u_ j1' and, consequently,
_k and _k are whole
fore X and _ are series which
but these three constants Let iary
let
We
have
may be developed
do not enter
in powers
the
method
which
we
employed
to
only
served
to
simplify
the
again
consider
the
/313
of
arbitrarily.
which
us
There-
of _Q and _.
c,
notation
introduce
given
the
discussion.
in
No.
274,
auxilFor
and
on
this page
set
x,=_xt, Therefore,
polynomials
recall
constant
purpose, 95.
us
i Okl L_J
y,=_
our equations
,
x,=_'x;,
do not cease
E_
Xi)
'
y,=y;.
to be satisfied
when we change
t
y',,
x,
into
_k-l,
and when
, xi"
the parameters
x'tk,
y;k,
x_k2
X and _ retain
their initial
values.
We then remove the accents which have become useless, and we develop ' ' ' which we shall hereafter designate by the letters YI'
X2'
Y2'
Xl' YI' x2' Y2' in powers
of e.
We
thus obtained
the expansions
_o +_,_,+_-'_:+.+ _'_, +.:., , no+ •
i
We shall not cease to satisfy and if we multiply
2
k2,
r
the equations
the four expansions ],
k,
312
if we change
(19), respectively k,
or, which is the same thing, if we change
into
(19)
[pk,-p,_pk-p, ['pk,-p,,$kt-p.
by
e into _,
By means
of
this
change,
identical to the expansions stants _0 and u . However, into k250 and k_u0 by means
we
must
again
(19), but with it may be seen of this change.
obtain
expansions
which
are
different values of the conthat _0 and u 0 are changed
/314
Therefore
change
when
into
_0
and
u 0 change
In other words, tively by E 2, i, E, in powers of
The
same
must
_0,
Uo were
be
E2_0 which
be
and
e_.
we
may
k2_0
of % and
into
_,
k2u0 .
us
assume
It
is
apparent
_250
_, which
k250,
let
derive
and
if the four expansions (19) are multiplied respecE, the four products thus obtained may be developed
true
changed
Therefore,
into
and
not
have
to
change
when
e,
k2u0 .
that
% and
that
g_0
365. Let k + 2 be the determined by the equation
did
we
_ are shall
inversely
denominator
expressed thus
as
as
have
The
of
relationships
functions
of n.
functions
of
constant
% and _ will
from _. then
dOk] There is only determined by
an
exception
to
this
in
the
case
of k +
2 =
2, where
is
[ to, 1 _o.J
The
d@k -dqo
expression
is
a whole
:o,
polynomial
of
degree
k +
2 with
respect
to
Therefore,
each
Only
which
terms
of
these
are
terms
independent
contains
of
factors
t will
remain
having
in
the
the
form
mean
value
Fd0k],
[d 70J
313
and we have seen that q must be divisible
hy the denominator of n, i.e.,
/315
byk+2. Therefore,
our
expression
has
ae. im(k
I shall For
now
this
show
that
purpose,
the
+ l_ "4"-b
the
following
-q- ce -i_(k+
z),
coefficient
I shall
form
b is
employ
the
t0,
_,,
....
_-,,
_o,
_1,
...,
_k-J,
zero.
following
method.
Let
us
cal-
culate
_, _',, .... _,_,, _;, _;, ..., __,, by
the
procedure
retain
an
presented
arbitrary
cels[ d@k]
to _.
above.
value Then
for
the
However,
g,
instead
following
when of
computing
assigning
_k'
a value
I shall which
can-
equation
Ldn oJ d_k dOk dt -- d_o will
allow
me
tion term
of
_k will
t,
to
compute be
_k"
However,
a periodic
instead
function
of
of being t in
a periodic
func-
addition
to
a non-periodic
consists
of
again
l-dO_l
t L2_j" We
have
and
,
another
consequently,
forming
the
No shall
encounter
the
of
calculating
term
to,
_,....
70,
_t,
S 0,
_,
•
v_k-l,
t[Ld--_03. S1,
will
be
some
difficulty
second
', ..-
presented
determine
difficulty
In effect, following form
314
this
calculation
We shall
we
method
This
in
No.
...,
by
entailed
in
dS_.
4r---;,+_B
_
term
per-
274. means
of
equations
calculating
when
dS#
method
calculating
SO,
(2) S1,
...,
Sk by
the
on
page
Sk_ 1,
100. but
equation
= _+C_.
represents
a group
of
terms
having
the /316
A
where m R and integration,
m 2 are integer numbers. provided that we do not irtli
Since
2B
eImlY_+_svt
equals
in,
where
-_-
n is
equals k + 2, the second term this condition. As a result, v, but may equal
Nothing have
2n_2B
impedes
us
from
performing
_0.
a commensurahle
number
of our equation Sk will not be
whose
denominator
will include terms a periodic function
satisfying of Y2 and
Tk -_- Y2 Uk,
where
Tk
and
Uk
are
periodic.
Having thus determined the f_nction S and having obtained mation to terms of the order c k+ , we may employ the procedure No. 275 and may thus determine Xl, Yl, x2, Y2" These two computational let us set
fore,
methods
v
Let
us
compile
the
must
S0+¢St+
equations
to
the
same
result.
There-
.-4- skSk.
(see
dE
lead
the approxigiven in
page
102)
dE
dZ
dE
7Y;z,' ,,= dC
n, = and let is thus
us derive x 2 from them obtained must equal
dC
_o_o'
as
n, =
a_o
a function
of
t.
The
value
of
x 2 which
$0+ El,+...+ ZkSk up
to
ular
terms We are term
of
the
order
interested
in
ck+l. calculating
Sk,
particularly
that
of
the
sec_
[ ok] This secular Y2Uk •
We the
thus
secular
term
can
only
come
from
have
the
following,
up
terms
in
the
equation
the
to
secular
terms
of
term
the
of
order
Sk,
which
ek+l
equals
(equating
/317
x 2 = dE ) dy2
315
MJ
_kt [aokl dU, I__fGo.i = _*y' u-yT" In have
the
(see
first
page
approximation--
i.e.,
u=_o=uo,
hi=i;
term
therefore
of
(20),
hi=in;
commit
of
the
order
c
--we
n_t-_gi=v=vo=i(nt+_).
of
the
order
¢k+1
if,
in
the
second
we replace
_o,
dUk
terms
nit+r_t=yi=_o=t,
an error
by
We
to
102) x,==o=_o,
We shall
up
(20)
shall
therefore
However
Uk
=0, _0,
y,,
Uo,
i(nt+w).
pdOkl kd-_n0J
obtain
only
t,
includes
v
by
terms
making
the
same
substitution
in
containing
dy2 imiyl
lint + "_mlB
where
We
-4- rn I v,
therefore
= o.
have dUk dy,
_ -- _ B
dU, d--'_" dU k
However, cannot
Uk
contain
contain
a
a
term
is
a
term
which
periodic which is
function is
of
independent
independent
of
Therefore,
Y2
and
iv.
of
v.
Therefore
dv
[_n0]
does
not
_r. q.e.d.
In one
more
order
to
remark•
clarify The
the
mean
preceding
motions
calculation,
n I and
dC nt=--_-_ In for
E =
general,
316
depend
on
I would given
like
to
make
by
dC
0, e,
are
n,=--_-_
o"
they
are
and
only
reduced
to
1
and
6n
O,
However, replaced
they
n 2
by
we the
are
here
arbitrary
employing
two
functions
of
parameters c, .or,
if
X it
and is
_ which
may
preferred,
we
be may
/318
employ an infinite number of constants %1, %2, then employ these constants in such a way that 1 and to _n, no matter what c may be. 366. following
In order form
to determine
ae
where
a and
zero,
otherwise The
c are
conjugate
$
could
equation
will
_,
we
therefore
{k+%)l_
-4- C_ -{k+l)i_"
and
imaginary.
only
be
provide
--., BI, B2, .... n I and n 2 remain
_
an equation
of
the
o_
determined
us with
have
We may equal to
the
In
general,
to
the
a and
c are
following
following
series
not
approximation. of
real
values
for •_ _0+ _,
_0, w0+ _--_,
3_ _0+ }-+
It is apparent that we do not have different when we change _ into _ + 27, may be stated that the two values
, ....
two values which but we have more
are actually than this.
It
2_
do
not
correspond
to
two
periodic
solutions
which
are
actually
Since t is not explicitly included in our equations, into t + h we may transform an arbitrary periodic solution solution which is not essentially different. Therefore, Then
q0
let changes
us
change
into
no + i(nt+
iv, of
t into 2h_ _nh_
and
become
nO
and
v 0 will
change
2h_,
where
v 0 = i(nt
h is
+ _)
by changing t into another
an
integer
number.
into
+ w).
Since all of our functions are we shall not change our solution vQ 27 from no and _-, respectively,
again
t +
different.
periodic, of the period 2_, "in any way by subtracting
in two
for
Then
example
2h_
and
2W_
.
no and multiples nO will
into
i(nt÷ _nh= + _--2h'=), In other
words,
we
will
have
changed
w ÷ 2=(n/_--
way
However, that
we
may
always
choose
the
_
/319
into
h'). integer
numbers
h
and
h'
in such
a
317
kj
!
nh -- h'-We therefore changing
_
do
not obtain 2_ -G + k +------2"
into
a
k-+-a
solution
which
is
actually
new
by
q.e.d.
We
therefore
corresponding
have
to
the
only
two
two
solutions
following
_0_
We we
must
shall
now
employ
determine equations
_0
which
values
-_
_
are
actually
different,
of-_
"
the
constants
which
relate
e2_0 these
and two
e2u0 •
For
constants
this
to
the questions which are customarily discussed, there is only parameter, and we have introduced two in order to facilitate It is therefore convenient to assume that I and U are related tionship
--
for
example,
purpose,
I and one the by
_.
In
arbitrary discussion. one rela-
% = _.
The expansion of % and that of _ in _owers of _250 and e V_ begins in general with terms containing e _0 and e u 0 (if we disregard the case in which the denominator of n equals 3). If from of
we
therefore
this
the
which
may
expansion
coefficients
of
assume be
developed
in powers the
that
_ = %, we
shall
in powers
of _
will
be
real,
or,
powers
of
_/_
will
of Vr_
expansion
in
derive
_2_0
Either on be
and
the
the the
_V_
coefficients
contrary, ones
the
which
are
real.
and hold
In the first will not have true. In order
the
case, the problem will have two real solutions for % > 0 any for % < 0. In the second case, the opposite will
equation
taining
s2.
to
determine
which We
which
relates
shall
_
of
these
two
cases
to u 0, restricting
is valid,
ourselves
to
let terms
_ IdOl]. 2B0 [auo]'
_, [dO_']. X ----- Hoo[ d_oJ
rd02] z mayfirst observe thatt I andtd
of [.
There
is
only
one k +
318
con-
have t -- _--
but
us examine
exception 2 =
2,
3 or
for 4.
(21)
of
t
This is due to the fact that, for k + 2 > 4, terms having the following form ei_pt
,qtzt
* q_l
which may be included in the second can only be independent of t if
term
since
qn must
lq[ cannot
exceed
4 and
since
Thus, the second terms of equations functions of _0 and u 0. The coefficients absolute constants which are independent
However, u 0 must be positive; equations (21) added to inequality
in
one
be
of
the
equations
an integer
(21)
number.
(21) are linear and homogeneous of these linear functions are of _.
otherwise V_would u 0 > 0 will determine
be imaginary. the sign of
I need only point out that this sign does not depend on _, tions (21) do not depend on it. We have seen that the equation termines _ has two solutions which are actually different
The %.
since equawhich de-
7_
In
conformance
which will be of them. The
It of
_2_ 0 and E V_.
E V_
of
generality,
our
formulas
Out positive.
of
are
we
the
may
assume
any
way
our
two
systems
real,
a periodic
by
the
will
both
relationships
however.
Without
that V_o changing of
solution
% is suitably chosen corresponds does not depend on _, and these
% > 0 and
from
case,
in
we
statements,
be
imaginary
that two periodic solutions correspond for _, since two systems of values for
obtained
is not
Therefore,
Two
preceding
both be real for will hold true.
first appears the equation
This
the
real if the sign of choice of this sign
solutions will or the opposite
tion
with
between
values,
into there
_,
is
_,
the
because
we
and
_ into
only
one
for
type
for
% > 0
-_,
for
%
< 0,
to each soluthe unknowns /321
restricting
is positive, V_
to each two
e2_Q
and--
conditions
do
not
change
_ + 7.
which
_
is
have:
periodic
solutions
of
the
second
(or
for
_<0).
No
solution
of
the
second
type
for
% < 0
(or
for
% > 0).
319
k.2
Let us again employ the notation given in Chapter XXVIII and, in particular, that given in No. 331. U1 may be reduced to p2, and corresponds to the term containing xlY 1 which appears in 0Q. U0 may be reduced to a constant factor multiplied
by p4, corresponding
[d2 and l
[d_q
to
terms
The
first
following
form
LT oJ
fromL_u0j
coming
term
of W which
may
not
be
reduced
to
a power
U1 has
the
which
must
p,_ _=[A. cos(k -.- _)? ÷ B]
and
comes
from
Ok+ 2.
The
function
the
role
play
whose
of
the
maxima
and
minima
we
on page
247,
study,
and
function Uo__ =U 1 = p,/(?
studied
must
will
have
the
zp'
following
form
Apk+' cos(k _-:_)?+ P P_--zp', where
P We
is a whole have
n equals
2,
polynomial
disregarded 3 or
in p2 with
the
particular
they
Let
Then
[O2],
will
include
The
which only
us
assume
that
of this
F_ %l' F_
equation
will
cases
coefficients.
in which
the
denominator
of
4.
Discussion 367.
constant
provide
terms
containing
for _-
wiil
us with
the
sign
of
I may
Two
real
solutions
One
real
solution
two
depend of
the
Particular
Cases
denominator
equals
no longer e+4i_ •
always
yield
be
two
independent
different
periodic
solutions.
on _-, the
following
second
type
for
4.
and
solutions
Due cases
X > 0;
of-_,
to may
zero
the
fact
that
occur: solution
for
x
320
of
the
second
type
for
X > 0;
one
solution
for
%<0; Zero real solution of the second type < O.
for _ > Q; two solutions
for
The function U0 + zU1 given on page 247 becomes p_(Acos4?÷ B)--_p,. Let us now assumethat the denominator of n equals 3. The expansion of _ in powers of e then begins with a term containing E_0, so that if we set _ = %, we shall obtain c2_0 and e_/_- in series which may be developed in powers of %, and no longer of V_. The
sign
of
negative
V_will for
will
be
mainly
Therefore, positive,
The
function
on
_,
if
if
of the % < 0.
U0 +
the
it
is positive
for_=
D 0 it
= WO + _.
second
zU I given
type
on
denominator
for
page
Ap_cos3$-Finally,
and
if it is always convenient for we shall readily find that we
A real solution second type for
the
_
depend
of n
us to have:
assume
% > 0 and
a real
that V_
solution
is
of
247 becomes zpL
equals
2,
[02]
_|d02|
Fd02]
include
'Ld_0j ' Ldu0J' terms
containing The
equation
e -+4i_, e +2i_. for _
/323
takes
the
AcosC4w÷ and
it has
eight
form
B)_-A'cos(=_ ÷ B')=o
solutions 3_ 2
2
m_, _,÷_,
_-_-_,
--, m_-t- 3_ 2
Of
the
two
terms
-_0 and
_I,
at
least
one
is
real.
The following hypotheses are (0, 4), (2, 0), (1, l), (0, 2).
possible:
(4_
The periodic
parenthesis the second
represents is the same
first number between solutions for % > 0,
the and
0),
(3,
i),
(2,
2),
(i,
3),
the number of number for I < 0.
321
The function given on page 247 becomes A_cos_c?-'-B?_cos2_-+-,
Application 368.
Let
us
return
to
C_s;n2?+D?
to Equations the
dxl
canonical dF
Just periodic meter _
as in
No.
13,
No.
function of the in the following
42,
y's, form
t"
of No.
13
equations
dyt _
,_-7--_z,'
_-z_
of
dynamics:
dF
(1)
-a_----u_"
No.
125,
which
may
etc., be
I shall
developed
assume
that
F is
of
a para-
in powers
a
F=Fo+MFI+... and
that We
tions
F 0 depends saw
of
only
in No.
the
42
first
on
that
the
x's.
these
equations
zi = ?i(t),
where
the Let
There and
functions us
have We
the two
opposite saw
_i
consider
Let T be will be
in
and
one
an
infinite
number
of
solu-
_i may
of
yt = +i(t)
be
these
(2)
developed
solutions
in
signs,
where IV
that
When _ varies let us assume
we
may
assume
a depends
on
increasing
powers
of
_. /324
(2).
period, and let _ be one of of them, which are different
Chapter
powers of _/_. _. For _ = _0, to 2ni_.
have
type
the characteristic from zero, which two
_,
degrees
and
may
continuously, the same that aT is commensurable
of be
exponents. are equal
freedom.
developed
will hold true with 2i_ and
in for equal
We may conclude from this that, for _ which is close to _0, there are solutions of the second type, which are derived from (2) and whose period is
(k +
seen sign,
2)T,
where
k +
2 designates
the
denominator
If we put aside the cases in which k + that two of these solutions exist when and
that
they
do not
exist
when
I
of n.
2 equals 2, 3, or 4, we have I (here _ - _0) has a certain
(here
_ - _0)
has
the
opposite
sign. I have regarded,
322
stated
and
I may
that do
the this
cases without
in which causing
k + any
2 = 2,
3, 4 have
inconvenience.
been The
dis-
following aT _i_
may be values
developed in powers of _, n is therefore
larger
than We
itself
therefore the
occur
for
Which
of
have
two
solutions _
of
the
second
in
order
and
these
to
the
two
hypotheses
the
canonical
_
> _0,
or
this
Q, which
depends
LduoJ
sign,
we
considerations take
term
shall
will
a simple
case,
F = x2 _xT+
_cosyt
not
need
to
formulate
this
suffice. which
will
be
that
presented
/325
dyi dt
dF dx_
_
yields
The
function
dY_ --d-'['=--',
o,
S of
Jacobi
dz, d---_-=
may
S = z°y, two
constants
x02 and
l
where
for
equations
dz, d--7--=
with
only
is valid?
da"i dF dt -- dyi ' which
occur
the sign of a certain of u 0 and _0 in
determine
following
369. Let us first No. 199. Let us set
with
type
< _0"
Everything depends on on the coefficients
In
small definitely
hypotheses:
d_oJ'
term,
For is
4.
Either they
of _/_, and vanishes with _ very small, and its denominator
A and
yO
are
C.
x,=¢C-_cosy,,
two
new
be
x_ = x °,
integration
dyl --_--------2x,.
(i)
written
÷f_C
We may
. _ s,ny,,
-- _ cosy, derive
dr,
the
/
following
dy,
y, = -- t+y °, A--t= 2(C--_cosyi
from
this
(2)
constants.
323
i
It may
be
seen
that
the
elliptical
integral
is
introduced
(3) --;_cosyl This integral has a real period, which is the integral and 2_, if ICI > I_], and two times the integral taken
if
ICl
<
us
call
A periodic with 2_. If
0
I_]"
Let
able
taken between between
ICl
period.
>
The
_
this
solution However, I_], Yl
real
period.
corresponds to each value we must distinguish between
and
Y2
corresponding
increase
periodic
by
a multiple
solutions
are
of _ which is two cases. of
2_ during
solutions
of
commensur-
one the
first
type. If
ICI
<
IDI, Y2
increases
by
and Yl returns to its original solutions of the second type. This tions
discussion
which
> 0,
and
must these
must
be
be
regarded
solutions
of
I have stated the first type,
be
regarded Let
us
us
to
2_ during
by
two
unusual
of
the
first
solutions then
the
C a value
one
period,
solutions
are
periodic
solu-
type.
Let
us
set
be
y,=--t#y;, y,=--t_y;,
of
of
corresponding
as
C= s, C=--S,
that it must be assumed and that the solutions
as solutions assign
The
supplemented
will
I x,=xl, _x,=x_,
a multiple
value.
second which
x,=o, x,=o,
that these corresponding
y,=o, y,=_.
(4)
latter solutions are to [el < lUl must
type. is
a little
higher
than
-_,
and
let
set C =(,--Q_,
where We
e is very
shall
small.
approximately
Yl will
C--y.
and
the
period
m will
not
be
be
equal
cosyl----_
to
¢_'
324
able
to
deviate
have _
-
'
'
very
greatly
from
_.
from which we may draw the following conclusions. Let a be an arbitrary numberwhich is commensurablewith 2_. There is a series of periodic solutions such that IcI < Ipl and that m= _. If _-is very close to C will be very close to -B, and for
these periodic solutions will coincide with the second solution (4) which is of the first type. Wemay now recognize the characteristic property of solutions of the second type. It may be seen that the second solution (4) -- i.e., that of the two solutions (4) which is stable -- gives rise to solutions of the second type, as was explained in Chapter XXVIII. If the other solutions of the first type -- those which are such that ICI > I_l -- do not produce solutions of the second type, this is due to /327 the very particular form characteristic exponents Let
us
first
of the equations are always zero.)
consider
solutions
of
(i).
the
(For
first
these
type,
solutions,
such
that
IcI
the
>
Ipl.
Let us set C = C O + s. The period _, i.e., the integral (3) taken between 0 and 27, may be developed in powers of e and of p, and the known terms may be reduced to
Let us assign a periodic solution
an arbitrary every time
commensurable that we have
value
to
V_.
We
shall
have
The equation is satisfied for c = p = 0, and we may derive s and, consequently, C from this equation, in series which develop in powers of p. The equations (2) will then give us x I and Yl developed in powers of _. These are the expansions of Chapter III. Let C = Ep.
us We
pass to the shall have
second
type,
can
be
seen
that
_/_is
only
that
ICI <
IPl.
Let
the
other
us
set
9,
=
It
such
2 _ -- cosy,
a
function
of
e.
On
hand,
325
kJ
_--=¢_-co5y,,
<,_,-t)¢_= f
dx,
,
x1 which and
indicates of
E,
to
which
us
are
that
sin
doubl
Yl,
periodic
also functions of (A- t)V_Therefore, to _, we
if shall
we assign obtain
be
developed
multiples
of
in
Fourier
of
function
which
I would
this
purpose,
For
We
may
vary
like we
e from
the
to
functions (A -
of
t)
(A
V_.
-
They
t)V_are
_inee _ is a function of _V_. which is solutions.
commensurable For these
with 27 solutions
Z328
and
according smallest
coefficient to
to
the
sines
and
common
multiple
of
expansion,
the
of
cosines w and
and
it
of the
2_. is
The this
study.
must
-i
siny,
are
respect
value periodic
series
2_____t, where T is T _ is an arbitrary
function
andv_
and of _,
a constant series of
a
Yl
with
cos y_,
may
cos
first
to +i.
study
For
the
e = -i,
relationship
we
between
e and
have
_¢_=_For +i,
_ = +i,
we
mV_-increases
The
of
and
are
real
imaginary
It result
326
there
m, which
coefficients
which
is
of
is
_.
to + only
when
e varies
from
-i
to
_.
a periodic
commensurable
the
Therefore,
solution
with
2_,
Fourier
expansion
the
line
corresponding
to
a given
if
are
therefore
functions
of _,
for
for
is apparent if,
_--=
from_2
Therefore, value
have
instead
that of
same
of
reasoning
would
lead
to
the
same
we had set
where F0 depends only on Xl and x2, and IF1] depends only on xl, x2 and Yl. The > PO"
solutions
of
the
second
type
would
still
have
been
real
/329
for
370. In the general case, the quantity Q, which was in question at the end of No. 368 and whose sign we shall try to determine, obviously depends on _. If _ is sufficiently small, the first term of the expansion will provide its sign. Let
us
determine
the
function
S by
the
S =So+V_S_+_S=+ If p
is small
enough,
it will
Bohlin
method,
and
let
us
set
....
obviously
be
the
first
two
terms
So+ _S, which
will
be
the
most
important. F = Fo+
If we
set
;_Ft+ M_Ft+...,
we have seen in Chapter XIX that S O and S I depend neither on [FI] , but only on F 0 and [FI] , where the mean value of F I is by IF1]. Let
us
again
take
the
quantity
its expansion will only depend IF1]. The same would hold true
Q
F = Fo+ which
exist
is,
consequently,
In the preceding only for
the
same
section
from
on S O and if we had
we
as
No. SI, set
368. and
The
F 2 or F I designated
first
consequently
term
of
on F 0 and
M[F,], in
found
the that
preceding solutions
section. of
the
second
type
I_ > Fo.
This conclusion ciently small. What
No.
Let 275.
is
the
still
value
holds
of
_0
in
the
for
us again consider the The exponent a which
general
which
notation appears
this
case,
provided
conclusion
that
would
no
_0
is
suffi-
longer
given in No. 361, which is that there may be developed in powers
hold? of of
327
k.J
the product AA'. It may be reduced to the characteristic
exponent for AA' = 0.
Since we assumethat the solution of the first type is stahle and ]330 is imaginary, A and A' are imaginary and conjugate, and the product AA' is positive. For small values of _, _ decreases when AA' increases. If the reverse were true, solutions of the second type would exist only for P < PO" The crease tive
328
desired
when of
AA'
_ with
value
of
increases. respect
_0
is It
to AA'.
therefore is
therefore
that
for
that
which
which
_ ceases cancels
the
to
de-
deriva-
CHAPTER XXXI PROPERTIES OF SOLUTIONS OF THESECOND TYPE
Solutions
tions
of
371. I cannot of the second
the
Second
pass type
Type
and
the
Principle
over the relationships and the principle of
of
Least
Action
between the theory of least action in silence.
wrote Chapter XXIX just for these relationships. However, in understand them some preliminary remarks are still necessary. Let us assume two variables of the first of a point in a plane. equations will comprise
/331
order
soluI
to
degrees of freedom. Let x I and x 2 be the two series, which may be regarded as the coordinates The plane curves which satisfy our differential what I have designated as trajectories.
Let M be an arbitrary point in the plane. Let us consider the group of trajectories emanating from the point M, and let E be their envelope. Let F be the n-_ kinetic focus of M on the trajectory (T). This trajectory will touch the envelope of kinetic focuses. I would focus of the order n, is the
E at the point F, accordin_ to the definition like to recall that the tr_ focus of M, or its tn_-h point of intersection of T with the in-
finitely
passing
adjacent
trajectory
of this contact may vary. the curve E, and that the general case.
It may contact
through
happen is of
M.
However,
the
conditions
that F is not a singular point of the first order. This is the most
Let x,= ?(x,)
be the equations of the trajectory (T) and very close, emanating from the point M.
of
a trajectory
Let z I and z 2 be the coordinates of the point be the coordinates of F. Since (T) passes through passes through M, we shall have =,= ?(=,), Due function jectories
ul= _u,),
which
is
M, and let u I and u 2 /332 M and F, and since (T')
_(_,) =o
to the fact that the trajectory (T') is very close to (T), the _ will be very small. I may call _ the angle at which two traintersect the point M. It is this angle which will define the
trajectory (T'), and the function _ will depend on the angle be very small if, as we have assumed, this angle _ is itself and it will vanish with _.
The
(T')
value
of
_'(z2)
(designating
the
derivative
of
_ by
_. It will very small,
4')
will
have
329
the samesign as _. With respect
to _'(u 2) [if we assume that _ is very coordinates has heen defined in such a way uniform, which is always possible], it has
small and if the that the function
system _(x2)
of is
the
_, if
F is
same
sign
sign
if
F is
One that
as
a focus
of odd
characteristic
_(u2) For If
is
of
the
same
let
sign
of
us
the
case
order
as
assume
_ is
of even
order,
in which _2,
that
such
that
and
we
_(u2)
we
pass
If
the
from sign
M of
at a point F' which touches E after F', F',
the
action
The
is
results
to
F'
_ is
proceeding such
sect
tact
(T) close
greater would
to F
In
this
case,
It
cannot
is
of a higher
that
along
be
and we
happen
the
opposite
the
may
say
that
F is than
_(x z)
in
the
fact
close that
the to u2
trajectory
the point F, and > z2)In this
is negative,
(T) intersects
(T)
than
it
is
along
(T').
if
_(u2)
were
negative.
How-
adjacent to (T) there are F, and others which inter-
of F.
that
order
is
sign.
from M than F. In this case, (T') E before F'. When we pass from M to
opposite
short
same
(T).
the trajectories (T') close to F and beyond just
the
(T) touches E after F'. According action is larger (at least in absoproceeding along (T') than it is
_'(u2)
(T)
just
interested
of
is positive,
along
is farther away and (T) touches
ever, in any case, among some which intersect (T)
it has
is positive.
_'(u2)
case, (T') touches E before F', while to a well-known line of reasoning, the lute motion) when we pass from M to F'
are
always
(T') will intersect (T) at a point F' which is not as far away from M as the point F (assuming
when
and
order.
of
the
example,
a focus
F is
an
an
ordinary
the
ordinary point
focus. of E,
and
that
the
con-
first. _333
would
Let
us
develop
The
condition
under
powers
which
there
of _,
would
and
be
let
us
set
a contact
of
higher
order
be +_(,,,) = o
But
we
already
have
and the function _l(X2) satisfies a linear, differential equation of the second order, whose coefficients are finite and given functions of x 2. The coefficient of the second derivative is reduced to unity.
330
If the integral _l(X2) vanishes, for x 2 = u2, it would be identically Therefore,
there
is never
a
as well as its first zero, which is absurd.
contact
of higher
derivative,
order.
However, it may happen that F is a cusp of the curve E. Either the cusp point is on the side of M, so that a moving point proceeding from M to F will encounter M with the cusp point directed at M, or the cusp point is turned the opposite way so that the moving point encounters M with the cusp point turned away from M. In the first case, I shall state that F is a pointed focus, and in the second case I shall state that F is a taloned focus.
the
In one pointed
has the true i_
and the other focus has the
opposite the case
case, _(u2) is on the order of _3. In this case, sign of _, if P is a focus of odd order, and it
sign of _ if of a taloned
F is a focus focus.
of
even
order.
The
In the case of a pointed focus, all the trajectories (T') (T) at a point F' which is close to F and beyond F. Proceeding to F', the action is greater along (T) than it is along (T').
opposite
is
intersect from M
In the case of a taloned focus, all the trajectories (T') intersect (T) at a point F' which is close to F and just short of F. Proceeding from M to F', the action is greater along (T') than it is along (T). Let F' be a point of (T) which is sufficiently close to F. In the case of a pointed focus, I may join M with F' by a trajectory (T'), if F' is beyond F. In the case of a taloned focus, I may join M with F' if F' is just short of F. It could finally more complicated than singular focus.
be an
the case ordinary
that F is a singular cusp. I would then
point of E which is state that it is a
I would only like to note that we cannot pass from a pointed focus to a taloned focus except through a singular focus, because at the time passage _(u2) must be of the order _4. 372. correspond ponent and successive
Let
us
of
Let us now consider an arbitrary periodic solution. It will to a closed trajectory (T). Let _ be the characteristic exT be the period. In Chapter XXIX we saw how to determine kinetic focuses (No. 347).
assume
that
whose numerator is p. in No. 347 shows that
_ equals
2in_ T , where
n
is a
commensurable
number
In this case, the application of the rule given each point of 6) coincides with its 2p -_ focus.
331
/334
If, just as in No. 347, we take a unit of time such that the period T equals 2_, we have _ = in. If we designate the value of the function are the values of this at the point M by t0, and if TI, _2, "''' T2p function • at the first, second, ..., up to the 2p_ focus of M, according to the rule given in No. 347, we shall have the following it.
If p ple
of
is
2_,
the
i.e.,
2it.
numerator that
M
of n,
and
its
it 2p_
_pir.
can
be
seen
focus
that
2p_
T2p
-
r0 is
a multi-
coincide.
The trajectory emanating from the point M which to (T) will therefore pass through the point M again around the closed trajectory (T) k + 2 times, if k ÷
is infinitely close after having gone 2 is the denominator
of n.
know
The point M is therefore what category of focuses
classification
presented
in
its 2p _ focus. However, we may wish it belongs to, from the point of view the
preceding
Let us coordinates,
adopt a system of coordinates so that the equation for the
and
_ varies
so
that
jectory.
The
from
curves
0 to
p = const,
2_ when are
the
section. which are similar closed trajectory
one
then
to of
passes
closed
around curves
to the polar (T) is
this which
closed form
an
traen-
/335
velope around each other in the same way as concentric circles. The curves _ = const, form a bundle of divergent curves which intersect all the curves p = const., in such a way that the curve m = a + 2_ coincides with
the Then
parture regarded
curve
_ =
let
_0 be
M. The as the
a. the
value
of m which
corresponds
to
the
point
value of _ which will correspond to this same 2_ focus of the point of departure, will be
point
of
deM,
Let
be the through in the the
equation M. The preceding
sign
of a trajectory (T') which is close to (T) and passes function _(m) will correspond to the function _(x2) given section. We shall have _(m0) = 0, and we must now discuss
of
_[_o÷
332
2_ + _)_].
Wemust therefore formulate the function _(_), and for this purpose we need only apply the principles of Chapter VII, or the principles given in No. 274. For example, if we apply the latter principles, we shall obtain the following. The function @(_) may be developed in powers of the two quantities Jt ¢=%
The
coefficients
of
the
A'e-_oJ
expansion
are
period 2_; A and A' are two integration it is a constant which may be developed _ :_o-_-:_I(AA')_-
equals
The term in.
If
(T')
_0 equals
differs
the
very
periodic
constants. in powers
of
the
respect product
to s, AA'
=_(AA'): ÷ ....
characteristic
little
functions With of the
from
exponent
(T),
the
of
two
(T),
i.e.,
constants
it
A and
A'
are
very small. They are on the order of the angle which I called _ in the /336 preceding section, and which must not be confused with the exponent which I have designated by the same symbol in the present section. If we take the approximation up to the third order inclusively with respect to A and A', _(_) will be reduced to a polynomial of the third order with respect to these two constants, and I may then write +(_)=
Ae_
+ A'e-_'+f(Ae=%
A'e-_)
where f is a whole polynomial with respect to Ae _, and A'e -s_ cludes terms of the second and third degree. The coefficients polynomial f, just the same as o and o', are periodic functions period 2_.
Under
this
assumption,
order, and since we may write the respect to A and
since
it equals following, A':
s equals
s0,
up
to terms
s 0 + s I (AA') up to terms of neglecting all terms of the
of
only inof the of the
the
second
the fourth order, fourth order with
or
_(_)
--- A e=0,0_ + A'e-=,o _' + at c0AA'(A e=0oe -- A'e-=_o,';
When o and k +
o',
2 for
_
increases
do the
not
by change.
denominator.
(2k
+ 4)_,
The
same
the
Ae=_,
A'e-=0_',
A'e-=0o).
coefficients
holds
Therefore,
+ f(Ae=o%
true
the
for
same
f(Ae_%
of e s0_,
still
f,
as
since
holds
well as s0 -_- = n has
true
for
A'e-_).
333
kj
We
finally
have + (to+ _,_ + _ _)- +(_) = (_k + _)_, _'(A e_0_ --:X'e-_,_:').
However,
_(w0)
is
zero.
The
term
whose
sign
we
must
determine
is
therefore (_£ + 2)_, I shall
employ
I should
o 0 and
first
point
according
to the
in general that this
be pointed term always
AA'(A e_o_'o_o-- A'e-_#o_'
o_0 to designate out
preceding
that
section,
focuses has the
this
the
term
values
is
indicates
or taloned same sign,
o).
of
the
to us
focuses. and that
of o and third
that
It its
o'
for
order
our
which,
focuses
may now be coefficient
w = m0. /337
will
stated cannot
vanish. The
two
constants
A
and
A'
are
related
by
the
following
relationship
q_(_Oo)=o which ties
may
be
written
as
follows,
since
A and
A e_o_0,0 + A'e-=o_o,g In
addition,
conjugate.
A'
a0
The
The cannot In
same
is
purely
holds
product be zero
AA' at
is the
addition,
we
imaginary,
true
for
and
A and
are
infinitely
small
--- o.
(1) o0
and
o' 0 are
imaginary
and
cannot
vanish,
since
the
However, trajectories
equations
these close
(i)
and
A and
have (2)
A e_o_o_0 -- A' e-_0_o _ ----" o, because
quanti-
A'.
therefore positive, same time.
cannot
A'
and
(2) would
entail
equations are impossible. to (T) would pass through
the
the
following
They would point M,
mean which
that is
all clearly
false. Therefore, our term _(_0 + 2k_ + 4_) focuses are therefore all pointed focuses_ Everything,
usual
373. case
k + 2 would
334
depends
on
the
sign
of
always has the same or are all taloned
sign. Our focuses.
_I"
We have disregarded the case in which _i would be zero, an unin which all the focuses would be singular, and that in which equal
2,
3,
or
4.
Following
is
the
reason
for
this.
Wesaw in the computations performed in Chapter VII that the following small divisors are introduced
(see No. 104, Volume I, page 338). The calculation divisors vanishes.
is finished,
and secular terms occur if
one of these
It may be readily stated that if k + 2 equals 2, 3, or 4, we are /338 thus finished with the calculation of terms of the first three orders, which are those which we had to take into account. If, on the other hand, k + 2 > 4, we will only be finished with the calculation of the terms of higher
order,
which
are
not
included
in
the
preceding
analysis.
374. For example, let us assume that all the focuses are. pointed. Let M be an arbitrary point of (T); this point will be the 2p -_ focus with respect to itself. Let M' be a point located a little beyond the point M in the direction in which the trajectory (T) and the trajectories close to (T') are traversed. I may draw a trajectory (T') emanating from point (M), which will deviate very little from (T), which will pass around (T) k + 2 times, which will finally end at the point M', and which will have 2p + 1 points of intersection with (T), counting the intersection points M and M' Due to the fact that the focus is a pointed focus, the trajectories (T') which are close to (T) will all intersect (T) again beyond the focus. We may therefore draw the trajectory (T') which satisfies the conditions I have just discussed, provided that the distance MM' is smaller than _. It is apparent that the upper limit, which must depends upon the position of M on (T). However, there is not a singular focus. It is therefore 6 equal to the smallest shall assume that 6 is
value which a constant.
this
upper
not exceed the distance MM', it never vanishes, since sufficient for me to set limit
can
take
on,
and
I
Therefore, if the distance MM' is smaller than 6, we may draw a trajectory (T') satisfying our conditions. We may even draw two of them, one intersecting (T) at M at a positive angle, and the other intersecting it at a negative angle. Under
this
assumption,
let
us
assume
that
our
differential
canonical
equations depend on the parameter X. For X = 0, the closed trajectory (T) has _0 = _n as the characteristic exponent. Let us assume that, for X > 0, the characteristic exponent divided by i is larger than n, and that for X < 0, on the other hand, it is smaller than n.
2p_
For X $ 0, the point M will no focus will be located a little
longer be its own 2p_ short of M for X > 0,
focus. Its and beyond M
335
for
%
< 0.
Let
F be
on the position this distance.
this
focus.
of M on (T). It is apparent
The
%, and that it will vanish with the focus F is always beyond M, to the principles teristic exponent. Let
F' be
given The
%. or
MF
will
naturally
We should point out, always a little short
in No. 347, distance MF
a point
distance
depend
I shall designate E as the largest value of that c will be a continuous function of
located
a
depending can never
little
that for % _ O, of it, according
on the value vanish.
beyond
F.
We
may
of
the
charac-
connect
M with
F' by a trajectory (T'), provided that the distance FF' is less than certain quantity 6'. It is apparent that 6' is a continuous function %,
and
that
Let
it may
us
play the provided
set
role that
be
reduced
% > O,
in
to 6 for
such
a way
a of
% = O.
that
M
is beyond
F.
We
may
have
of F', and we may connect M to itself by a trajectory the distance MF is smaller than 6', or provided that
M
(T'),
< 6'. For enough
% =
so We
deviating secting
0,
that
may
e is the
then
2p +
and
inequality
connect
a little (T)
zero,
from
_' = 6 > O. is
the
we
may
take
% small
satisfied.
point
(+),
Therefore,
M
passing
to
itself
around
through
(T) k +
a trajectory
2 times,
and
(T')
inter-
1 times.
.C,
111
pf'/
O,
D
Figure
In
the
figure,
BA
represents
MC is an arc of (T') trajectory bordering
starting upon M.
the
traversed.
trajectories
are
an
12
arc
from M and The arrows
of
(T)
on which
DM is another indicate the
M
is
located.
arc of this same direction in which
The point M may also be connected to itself not through one trajectory, but through two (T'). For one, as the figure indicates, the angle CMA is positive, so that CM is above MA. For the other, the angle CMA
336
would be negative. The trajectory (T') must not be regarded as a closed trajectory. It leaves the point M to return to the point M, but the direction of the tangent is not the sameat the point of departure as it is at the point of arrival, so that the arcs MCand DMdo not join each other. The trajectory (T'), thus proceeding from M to Mwith a hooked angle at M, will form what may be called a loop. If the same construction is followed for the points M of (T), we shall obtain a series of loops. We shall obtain two of them, the first corresponding to the case in which the angle CMA is positive, and the second corresponding to the case in which this angle is negative. These two series are separated from each other, and the passage from one to another may only be made if the angle CMA is infinitely small. The trajectory (T'), which is infinitely close to (T), would pass through the focus F, according to the definition of focuses. However, since it must end at the point M, the points M and F would coincide, and this cannot happen according to the principles presented in No. 347.
loops
Therefore, if all of the focuses for % > 0, and we have no more If
all
repeated. that there
the
focuses
were
are pointed, for % < O.
taloned,
We would find that there are no more for % > 0.
the are
same
two
we
line
series
have
two
series
of reasoning of
loops
for
of
could %
be
< 0,
and
375. Let us consider one of the series of loops defined in the preceding section. The action calculated along one of these loops will vary with the position of the point M; it will have at least one maximum or one minimum. If the action is maximum or minimum, it may be stated that the two arcs MC and CD coincide, so that the trajectory (T') is closed and corresponds to a periodic solution of the second type. For
example,
let
us
assume
that
the minimum of the action, and that BMD, just as in the figure. Let us
the
trajectory
the angle then take
(T')
corresponds
CMA is larger a point M 1 to
to
than the the left
angle of
M and infinitely close to M, and let us construct a loop (T_) which differs by an infinitely small amount from the loop (T'), having its hooked
point From
M
at M I. and
According point
M
up
to
from to
the
Let
MIC 1 and
M 1 I may
MID 1 be
draw
two
a well-known
theorem,
point
equal
Q will
two
normals
the
the
arcs MP
action
action
of and
this MIQ
along
along
/341
loop. on MIC 1 and
(T')
(Ti)
from
from
the
MD.
the point
337
4
P
to
M1.
We
shall
therefore
have
action (T',)=: action (T') ÷ action(M,P)--
action(MQ)
or action or
(T_ _ = action
(T')
_- action
(,_13lt)(cosC3lA--
cosBMQ),
finally action(T_)<
which
is
the
absurd,
since
(T')
was
action(T'),
assumed
If
we
correspond
to
the
minimum
of
set CMA <
we
to
action.
would
arrive
We
must
at
the
same
therefore
absurd
assume
result
that The
the
two
same
Each
arcs
line
series
of
of
placing
M I to
the
right
of
M.
that C3IA
i.e.,
13MD,
----BMD,
coincide. reasoning
loops
may
be
therefore
applied
to
contains
at
the
case
least
of
two
the
maximum.
closed
trajec-
tories. Each and to
of
these
intersects CMA
fact
is
that
one
the
curve
(T')
as
the
Therefore, types
of
of
intersection
thus
of
principles
338
to
fact the
the
assumed
of
there
preceding
presented
p
points,
it
that
these
(T)
k
points, it
is
must
intersect
this
closed
regard
point.
each
any
For
series,
series, be
only
are
not
the
there
reduced
hut
line in
of
+
2
the
times,
angle
negative. (T)
as
similar Due
many
to
the
times
in
p
trajectory
arbitrary of
and
these for
one
consists of our
types,
the
other
the p
2p loop
types,
of points thus it
series.
may 4p,
may
first
second
not that
we the
around
p
direction.
hooked
which
other
closed,
other
passes
For
the
is
to
loops
them,
obtains
The result
for
be
the
belong
the
2p of
may
points.
because as
would
Among
one
it
_,
belong
least
in
trajectories
2p
and
2p
would
at
positive,
direction
defined
closed
(T)
of
two more
to
different of
reasoning,
preceding
are closed
them but
chapter.
therefore
not
trajectories. closed is, may
in be
two,
but
at
However,
/342
trajectories. general, concluded
not
the from
the
The if k is
trajectory odd,
(T') +
2)p
defined
points
have
if k
] _(k
+
l)p
as two tra-
no matter how large % may be, as long as the exist, they will intersect at 2p points.
376. The entire line assumes that we are dealing
of reasoning presented with absolute motion.
If this line of reasoning is extended difficulties will be encountered which are shall not try to surmount at this point. with,
we
This
points,
small values of %, and it remains valid no matter how large % may be long as (T') exists. The number of double points could only vary if branches of the curve (T') were tangent to each other. However, two jectories cannot be tangent to each other without coinciding.
To begin
is even.
double
for
same reason, (T) and (T')
double
will
is valid
For the trajectories
and _(k
thus
must
modify
the
in
the
preceding
section
to the case of relative not insurmountable, but
construction
employed
in
two
motion, which I
the
preceding
section. Instead of drawing MP and MIQ normal to MIC 1 and MD, we must proceed as follows. In order to construct MP, for example, we should construct a circle which is infinitely small and which satisfies the following conditions. It intersects MIC 1 at P and touches the line MP at this point. The line connecting M to the center must have a given direction, and the ratio of the line length to the radius must be given. The line MP thus constructed has the same properties as the normal in absolute motion. Unfortunately, in certain cases this construction entails certain difficulties. In addition, the line of reasoning
this could could
be reached at occur without
action would
the the
(MMI) still
point M, necessity
is not always positive. If have a defect. The maximum
so that of the
it or
became zero, the minimum
the action (MM I) is zero, arcs MC and DM coinciding.
Our line of reasoning therefore only applies motion, if the action is positive along (T).
to
the
case
of
and
this /343
relative
In any case, one of the conclusions is still valid. The closed trajectory (T') always exists, since -- if the line of reasoning given in the preceding section is lacking -- the same does not hold true for the line of reasoning given in Chapters XXVIII and XXX. In addition, (IT') intersects (T)
at
2p
points,
and
has
_(k
+
i) or z_(k +
2) double
points.
This is valid for small values of %, but it cannot be concluded any longer that this is valid no matter what % may be, because two trajectories may be tangent without coinciding, provided that they are traversed in the opposite direction.
339
Stability
of
and
377. Let us assume that there the characteristic exponents are
have
opposite The
Instabilit_
are only two degrees of freedom, two zero, and the two others are equal and
signs.
equation
which
has
the
following
as
roots
c_=T
is
an equation
the
period Its
the
second
are
they are unstable.
one
therefore
real
and
the
they
are
imaginary
real
cannot
There
are
real
the or
coefficients
are
characteristic
imaginary the
_'s
and
are
(T represents
conjugate.
real,
the _'s are imaginary are purely imaginary,
real
exponents).
and
the
periodic
and conjugate. and the periodic
solu-
Since the solution
negative, the _'s are imaginary but complex, with iT equalling-_-. The periodic solution is still unstable.
part
They
whose
of
positive,
If they are imaginary, product equals +i, the _'s is stable. If
order
_ represents
roots
If is
tion
of
and
be
and
real
and
have
opposite
two
kinds
signs,
since
the
product
equals
+i.
the
following
therefore
of
The
passage
occurs
solutions,
corresponding
to /344
two hypotheses e _T_a_
type
unstable
for
from the
stable
e _T
solutions
to unstable
solutions
of
the
first
to unstable
solutions
of
the
second
value 3:-----0.
The type
occurs
378. type.
passage
At
for
Let
from the
us
stable
value
first
the moment
solutions
of
study
the
passage,
passage we
e _T
340
have -_-
i.
to
unstable
solutions
of
the
first
Let us again consider the terms _k and 4k defined in Chapter III, let us consider the equation d+, d+,
a+,
d+,
d+,
d_, _ s
d,_,
a+,
and
(1)
This
equation
has
the
d,_h
a+,
d+__, _ s
a,_,,
d+_
d+,
following O_
At
of
the
Before dynamics
time
studying with two
d+, a+,
roots
O_
of passage,
egT--
the
the simple degrees of
Ip
e--gT__
four
i.
roots
become
period
F be
T with
an
arbitrary
respect
to
function t.
Let
of x,
us
,_ dF 2_ =Tjy' These
are
the
equations
ever, due to the tion F = const.
Let
us
fact
assume
period T. The equation which
of that
that
dynamics F depends
these
characteristic is similar to
zero.
case in which we are dealing with equations freedom, and in which we assume that the
function F does not depend explicitly on time equations have the energy integral F = const., for a moment a case which is even simpler. Let
--:0°
dF _7;
with
only t,
canonical
one
they
will
be
degree
do
not
have
a periodic provided
following
roots
all
become
Let us assume that = 0, the two roots of will depend not only on
by
freedom. the
energy
solution the
Howequa-
of
following
roots exT
These
/345
(3)
-----0
the
of
equations
of
_e+,_ s has
periodic
a+,
_-s
which
is
the consider
(2)
(2) have
exponents (i)
that, consequently, is advantageous to
t, which
the
dy _7 =--
equations
d,_t
y and
consider
on
and it
zero
--
at
! ,
e -_tT
--
I.
the moment
of passage.
F depends on a certain parameter v and that, for the equation (3) are zero. The functions 41 and 42 B1 and B2, but also on _. We shall assume that F
341
k.J
may be developed in powers developed in powers of BI, The
periodic
of _, and _2 and _.
solutions
will
be
that
consequently
provided
by
the
_I and
_2 may
following
be
equations
(4) +,=o, For
_ = 0,
respect will
to
not
the
_I B's
vanish
=
_2
is
at
=
the
0,
zero.
the
% =o.
functional
However,
same
time.
in
For
determinant
general
the
example,
let
of
the
four us
_'s
with
derivatives
d_ dB k
assume
d+, Xo, and first
we
shall equation
derive (4),
B1 in series and we shall
developed substitute
in
powers of B2 and v from in the second equation
it
the (1).
/346
Let
•'(_,, z*) = o be
the
we
shall
result
of
we
i.
may
The
In
of
this
In
_I
and
case,
a plane, at
distinguish
the
the
_2
curve
the
is
with
if we
origin,
general,
between
d_ _
derivative
determinant
point
substitution.
Our
functional
two
cases:
determinant
being
zero,
have
However,
in
the
(5)
not
zero,
respect
assume
that
represented where
the
second
by
or,
to B2
in
B 1 and and
H
equation
tangent
other
will
_
are
words,
is not the
the
functional
zero.
coordinates
(5) will be
the
line
have _
an
of a point ordinary
= 0.
derivative "d! _1"
will
not
curve
be
zero,
i.e.,
the
origin
will
If we
intersect
the
line
H = _0,
we may have, two points of intersection the vicinity of the origin, or we may of _0.
342
not
be
a point
where
H0
is
of
inflection
of
the
(5). a rather
small
for this line and the not have any, depending
constant,
curve (5) in on the sign
For example, if this curve is above its tangent, we shall have two intersections for _0 > 0, and consequently two periodic solutions, and for _0 < 0 we shall not have any. Wehave thus seen two periodic solutions and then disappear.
approach each other,
coincide,
Let us consider the two points of intersection of the line p = P0 with the curve (5). They will correspond to two consecutive roots of the equation (5) and, consequently, to two values having opposite signs of the d_ derivative d--_2,and therefore to two values of opposite signs of the functional determinant of the _i's with respect to the B's, that is, of the /347 product ( e_'r-- I)(e -_ri.e.,
of
of
The
_i and
of the stable, dP _
derivative
_2 with
two periodic solutions which and the other is unstable.
respect
= 0, to
will
The curve (5) then has be an ordinary, double
will
Two branches always meet
periodic
of be
or
Bl
in
and
other
p,
is
a singular point.
no matter
what
the
the
OPI, OP2, OP3, Op 4 be the OP I will be the extension
at
tive
positive.
stable, unstable
The
that when
derivative
for example. when we pass
the we
at
the
of
origin
_0 may
to
determinant
which,
in general,
P0
be.
regions in the vicinty opposite the peak _ will negative.
four half-branches which converge at the of OP 3 and OP 2 will be the extension of > 0; OP 3 and oP 4 will for the angles PIOP2,
stability
depends
pass
OPI,
will
functional
then
the origin, and the line p = We shall therefore have two
sign
OP4. OPI and OP 2 will correspond to P0 uo < 0. The function I will be positive negative for the angles P2OP3, PiOP4. We have just seen d_ d--_2. For example,
coincides
zero.
point
of the curve intersect the curve at two points.
solutions,
words
The two branches of the curve determine four the origin. In two of these regions which are positive; in the other two regions, it will be
Let origin.
to
e-_T
_2.
Therefore, one disappear is always
2.
I)=: 2 -- eat-
be
over
positive,
It will also be stable over OP 2 or OP 3.
on
the
_ will and
when
sign
change
the we
correspond to P3OPq, and
of from
solution
pass
the
over
deriva-
negative
will OP4,
be and
343
k.J
The periodic solutions an analytical sequence with which
are
which
are We
correspond respect to
thus
have
and
two
at
analytical
this
series
instant
of
to OP 2 and which are unstable are those which correspond to OP4, and
(5) may However,
have no
at
the
matter
of
time
We have just studied the two of other cases corresponding
curve
periodic
the
two
encounters
in
which
exchange
simplest cases, but there to different singularities
coincide
their
may be which
stabil-
a multithe /348
origin.
what
them
solutions
series
these
singularities
may
an even p + q number of half-branches emanating for _ > 0 and q for _ < 0. Let us assume that origin
they form OP 3 and
stable.
_ = 0,
tude
and to
unstable.
Conversely, those which analytical sequence with
the
for
corresponding to OP 1 are stable, respect to those which correspond
the
following
be,
we
shall
observe
from the origin, i.e., p a small circle about the
order
Let (6)
be
those
which
correspond
to p > 0 and
let (7)
OPl,,._, be
those
which
Then stable may
the
correspond half-branches
solutions
state The
In
that same
and
holds
addition,
(6) will
OPp the
and
for
the
holds
Therefore, let p' and p" be of unstable half-branches
correspond
are
both
true
the for
alternately
For
purposes
alternately
half-branches
OPp+ 1 are
same
OPp+q
solutions.
half-branches true
...,
< 0.
to unstable
these
Consequently,
number
to _
01"¢,+,.,
number p > 0,
of
stable
periodic
brevity,
I
or unstable.
(7).
stable
for
to
or both
OPp+q
of so
and
unstable.
OP I.
stable half-branches that we have
and
the
p'+p"=p. Let q' and q" be the corresponding numbers for _ < 0, so There are therefore only three possible hypotheses
344
that
q' +
q" = q.
p'=
p",
q'= g',
p' ----I a" :,-I,
rl ' = q" -+ _,
/=/--I,
c/= q'--I.
In any case, we have p" _p" _--q'_
in
Let us assume such a way that
that p does not a certain number
q,.
equal q, and, for example, that p > q, of periodic solutions disappears when
we pass from p > 0 to _ < 0. It may be seen that this number is always even, and in addition as many stable solutions as unstable solutions would always disappear, according to the preceding equation. Let us now assume that w_ have an analytical series of periodic solutions and that, for p = 0, we pass from stability to instability, or viceversa (in such a way that the exponent _ vanishes). Then q' and p" (for example) are at least equal to i. Therefore, p' + q" is at least equal to 2. It follows from this that we shall have at least another analytical series of real, periodic solutions which intersect the first for _ = 0. Therefore, if, for a certain value of p, a periodic solution loses stability or acquires it (in such a way that the exponent _ is zero) it will coincide with another periodic solution, with which it will have exchanged its stability.
379. Let us now return to the case -- that in which the time does not enter consequently, we have the energy integral two degrees of freedom.
and
I shall pursue I shall assume
the solution which very little from T following equations
the that
which I was first going to discuss explicitly in the equations, where, F = C, where finally there are
same line of reasoning as was the case in No. 317_ the period of the periodic solution, which is T for
corresponds for adjacent
_, = o.
to p = 0, periodic
+, = o,
fli = 0, equals T + T, and solutions. I shall write
_., --o,
F = Co,
differs the
i% = o.
(i) which
include
the
following
variables _"
_"
_,, _,
s, _.
According to our hypotheses, the functional determinant of the _'s with respect to the _'s must vanish, as well as all its minors of the first order. However, the minors of the second order will not all be zero at the same time, in general. Therefore
let
us
set
fll = 0 in
equations
(i),
and
let
us
consider
the
345
functional
determinant A of
/3SO
with respect to
This general
determinant
the
Let
minors
us
_'s
2. No.
64
The
longer
at
the
The same
not
position
of
of
the
then
For
example,
_'s
two
first
zero
of
the
not
all
all
this
orders
at
shall
but
in
F
and to
could
of
same
of
two
two
of
only
the
of
the
the four
happen
if
determinants
of
time,
which
does
same
time.
We
the
not
assume.
zero
along
vanish,
vanish.
determinants
F were
of
The
the
_'s
following
to
at
the
the
periodic
solution.
saw We
in
shall
No.
assume always
eliminate
and
of
F with
respect
to
T were
all
zero
values
a periodic (see
longer
We may therefore A are not zero. us
not
and
this.
equilibrium
Let
will
_'s
determinants
we
zero
derivatives
no
the
of be
_'s,
to _, and with respect zero at the same time? of
which
must
time.
shall
order
T's were
and
correspond of
We
to
the
functional
derivatives they
assume
3.
the
minors
general,
that
no
would
the
respect
in
first
to the theory were true:
All
with
occur,
the
when
_ with respect Can they all be
According following i.
of
consider
four functions variables B.
the
vanishes
solution
strictly
speaking,
but
to a
68).
this. assume
four
that
of our
all
the
unknowns
minors
B and
of
the
• among
first
the
order
equations
(i).
equation
Due
to
of the arrive
346
of
the
the
fact
let
us eliminate
following
form
that
equation
this
preceding section, it will at the same results:
BI,
B3,
has
exactly
be
handled
B4,
r; we
the in
the
shall
same same
still
form way,
have
an
as equation and
we
shall
(5)
i.
When
number,
and
2.
periodic as
When
many
solutions stable
a periodic
continuously
(in
380.
Due
to
Let
the
us
fact
a way
none
after
having
solutions,
loses
that
passage it.
proceed
that
disappear unstable
solution
such
that at the moment of period coincides with
as
or
acquires
a vanishes),
another
real,
the
second
case,
may
of
the
characteristic
when
always solution
that
in
an
even
we
vary
disappear.
stability
we periodic
to
coincided,
always
he
certain
of
the
3_
same
which
exponents
vanishes
for
= o.
except T which
the two which are always coincides with the first
On there cide
shall
the
are with
the
What
may
have For
ber T.
have them, these
other
hand,
periodic
a U
we
of unstable Let q' and
Let the
say
p'
and
solutions q" be the
I may
p"
the
all the Applying
with gated
find the in
if that
we these
refer four
three hypotheses. No. 335.
to
the
numbers The
_
For
of
0.
_
> 0,
become
stable
Chapter
2T,
=
for
solutions
period
XXVIII,
which
coin-
example,
unstable
2T, without for _ < 0.
of
and
having
we
for
_
the
num-
the
< 0.
period
following
three
hypotheses
regarding
q'=q"+1, P",
a + p' = p",
However,
for
in
period
will
the period numbers
the
p'-'-l---p',
shall
of
T
T which number
solution
solutions of period 2T, whether they have the principles presented in No. 378 to
postulate
P'=
is
periodic
presented
type,
stability?
period
be
no
principles
period
which have corresponding
us then consider period T or not.
I find that four numbers:
of
is
second
their
solution
let
the
whose
regarding
there
to
of
solution
stable
> 0,
according
solutions
given
zero, for
q'=
q"-+- 2,
q" = q".
principles cannot simplest
given take and
all most
in
Chapter
values frequent
XXVIII,
which cases
are are
we compatible investi-
347
k_j
Application 381. tain
In Volume
periodic
XXI
solutions
No. 9, and considers which he attributes This fictitious small perturbed He
has
thus
M. planets Jupiter. unstable tic
Darwin
Acts
in
the
Orbits
He
M.
discusses
G.
Darwin
studied
hypotheses
cer-
given
planet which he calls Jupiter, is ten times smaller than that
in
and to of the Sun.
describes a circular orbit around the Sun, and a having zero mass moves in the plane of this orbit.
acknowledged
the
existence
of
certain
periodic
solutions
included in those which I have called solutions of the which he has studied in detail. These orbits are referred turning around the Sun with the same angular velocity as orbits are closed curves, in relative motion with respect axes. has
called
the
first
class
of
periodic
orbits
the
class
of
The orbit is a closed curve encircling the Sun, but not encircling The orbit is stable when the Jacobi constant is larger than 39, and in the opposite case. The instability corresponds to a characteris-
exponent For
The
having-_-as
values
of
periodic
the
the Jacobi
solutions
corresponding
imaginary constant of
orbit
the
will
be
part. which
second a
are type
closed
passing around the Sun twice. The two loops from each other, and both differ very little We a later M. are
shall
study
these
solutions
of
the
close whose
curve
39,
with
there is
second
type
are
double.
a double
of this curve differ from a circle.
Darwin also those which he
obtained oscillating we discussed in No.
in greater
obtained
satellites
satellites which he 52. They are always
which,
strictly
to the system of moving axes under consideration, circling Jupiter, but not encircling the Sun. For is
to
period
point very
little
detail
at
point.
Finally,
which
H.
the
A.
therefore
and
/352
of Darwin
Mathematics,
detail.
a perturbing a mass which
planet planet
which are again first type, and to moving axes, Jupiter. These to these moving
of
to
C = 40 stable.
(C is For
the Jacobi constant), C = 39.5,the satellite
called a and unstable.
speaking,
describe
with
closed
b,
respect
curves
en/353
we have only one satellite A A becomes unstable with a real
exponent _. However, we have two new satellites B and C, the second of which is stable, and the first of which is unstable with a real exponent _. For C = 39, we obtain the same result. For C = 38.5, the satellite C becomes unstable for
348
C =
with 38, we
a complex obtain
exponent the
same
_
(whose
result.
imaginary
part
is _).
Finally,
Wemust therefore consider three passages: i.
The passage of satellite
A from stability
2.
The appearance of the satellites
3.
The passage of satellite
B and C;
C from stability
The last two passages do not entail
to instability;
to instability;
any difficulties.
Two periodic solutions B and C will appear simultaneously which differ very little from each other. One is stable and the other is unstable; the exponent _ is real for the unstable solution. This conforms with the conclusions reached in No. 378. The passage of the satellite C from stability to instability no longer presents any difficulties, because the exponent _ is complex in the case of instability. The conditions presented in No. 380 therefore hold. We therefore have periodic solutions of the second type corresponding to closed curves which pass around Jupiter twice. 382. On the other hand, the passage of satellite A from stability to instability entails great difficulty, because the exponent _ is real in the case of instability. According to No. 378, we should therefore have exchange of stability, with other periodic passing around Jupiter only once. calculations of Darwin.
by A.
We are naturally led Darwin do not represent
Other
The unstable
considerations
to think that the unstable the analytical extension
lead
stable satellites satellites A have
solutions corresponding This would not seem to
to
A have orbits
the
same
to closed curves result from the
satellites A discovered of the stable satellites
result.
ordinary closed curves for orbits; in the form of a figure eight.
the
/354
How may we pass from one case to another? This may only be done by a curve having a cusp, but the velocity must be zero at the cusp and, for reasons of symmetry, this cusp could only be located on the axis of the x's. It could not be between the Sun and Jupiter. In Figure i, Darwin gives the curves of zero velocity. For C > 40, 18, these curves intersect the axis of the x's between 18, and the passage
but
the Sun occurs
We are left with this is no longer
ponding
to
C =
40
and
and Jupiter, but this no longer between C = 40 and C = 39.5.
the hypothesis satisfactory. to
C = 39.5.
that the cusp Let us compare The
first
holds
for
C < 40,
is located beyond Jupiter, the two orbits corres-
intersects
the
axis
of
the
x's
349
k_J
twice at a right angle, once beyond Jupiter and once just short of it. Let P and Q be the two intersection points. In the same way, the second orbit (if we disregard the double point) intersects the axis of the x's twice at a right angle, once beyond Jupiter, and once just short of Jupiter. Let P' and Q' be the two intersection section point P or P' which is beyond sign
of dd-_t • We
other.
shall
However,
see
that
this
dd-_t would
have
to
points. Jupiter, sign
Let and
us consider the let us determine
is positive
change
sign
when
for
one
passing
interthe
orbit
or
through
the
the
cusp. The
point
P,
the
hypothetical
cusp,
and
the
point
P'
cannot
therefore
be regarded as the analytical extension of each other. We must then assume that at a given moment an exchange has occurred between the two intersection points of the orbit of the satellite A' and of the x-axis, that which is located on the right passing to the left, and vice versa. Nothing in the behavior of the curves constructed by M. Darwin justifies such an assumption. Therefore, analytical lites
I may
extension
A become
conclude of
the
that
stable
the
unstable
satellites
satellites
A.
But
A are
when
do
the
not
the
satel-
stable?
I can only formulate hypotheses on this point and, in order to do /355 otherwise, it would be necessary to reconsider the mechanical quadratures of M. Darwin. However, if we examine the behavior of the curves, it appears that at a certain time the orbit of the satellite A must pass through Jupiter, and that it then becomes what M. Darwin has called an oscillatin_ satellite. 383. these
Let
planets The
us
study
from
orbits
the
planets
stability
of
these
to
A in
greater
detail,
and
the
passage
of
instability.
planets
correspond
to what
we
have
designated
as
periodic solutions of the first_The orbit with a double point, which passes around the Sun twice and which differs very little from that of the planet A at the moment when the orbit of this planet has just become unstable' corresponds to which we have designated as periodic solutions of the
second If we
type apply
(47). the
procedure
second type from those shall obtain solutions In
solutions
of
of of
the
the the second
by
which
we
deduced
periodic
first type to solutions second type exactly. type,
the
mean
of
anomalistic
solutions
the
first
motions,
of
type,
the we
which
differ ver_ little from the mean motions strictly speaking, are in a commensurable ratio, We must therefore consider the case in which, for our solution of the second type (and, consequently, for the planet A at the time of
350
passing from stability to unstability), the ratio of the close to a simple commensurablenumber. Since the orbit the Sun twice, this ratio will be close to a multiple of
meanmotions is must pass around i _.
In other words, at the momentof passage, the term which M. Darwin has called nT must be close to a multiple of 7. In effect, this is what occurs. with the following c=
of
be is
The mean Jupiter.
A stable,
nT .= r54°,
A stable,
nT = 165°,
39
A unstable,nT
= ,77%
A unstable_nT
---w9_o
seen that the passage close to 180 ° . motion
of
of M.
_o
C = 38,5
It can number
tables
C=39,5 C=
this
The
the
planet
must
A is
be
made
therefore
Darwin
provide
around
nT
=
170 °,
almost
three
times
us
and
that
We could consider applying the principles presented in Chapter XXX to a study of these solutions of the second type, but several difficulties would be encountered because we would be dealing with an exception. It would be better to resume this study directly.
set
384. Let us the following,
again just
consider the notation as in this section z,=
L--G,
Fo =
The term L must have eccentricity must be of the eccentricity,
their
shall
313,
and
let
us
_Y2---- l'+g--t,
R+G=
Fo+/tF,+...,
_
x, -- x,
the same sign as G (see page 201, in fgne), and the very small. Since x I is on the order of the square this variable will also be very small.
Since we only wish to determine stability, we shall be content We
in No.
x,=L+G,
2yi---- l--g4-t, F'=
given
therefore
neglect
the number of periodic with an approximation.
_2F 2 and
_FI, we only take into account secular period, and we shall neglect the powers shall have
the
following
terms.
solutions
and
In
term
terms and terms with a very which are higher than x I.
the
long We
351
F" I
where
a, b,
c are
functions
term
which
has
period
with
The
very
2y2
- Yl-
We
long
then
been
we
may
-F"
CXi
COsuJt
of x2,
and where
CXlCOS_
is
the
very
long
retained. terms
therefore
are
terms
with
_ +
3g
- 3t,
i.e.,
terms
have
have F'=
and
b.z'l
only
period
We
_-= _t-+-
apply
(xt+
a
the method
2 -+ :'1,, ......
x,)'
Xt' I
+_.(a_a_bxt+cxlcosw
",
)
of Delaunay. _357
The
from
canonical
which
we
equations
the
the
integral
have F'=
With
have
approximation
+ - -- -- + 1_(a + bx,+ • a
(/, = xl)' which
has
been
adopted,
cxtcos_).
we
may
replace
a,
b,
c by
t ao--
designating
that
place
k.
x 2 by
which
a,
the
coordinates
us
t
constants
which
_ = bo-- 2ao, depend
assume of
that
a point
C designates This
double
352
curve
point,
ao,
a'o, bo,
co when
we
re-
k is in
also
this
we
have
a constant, and
,!x_cos!_,_!_ sin_ let
us
compile
the
are
rectangular
curve
C_
constant.
depends
double
and
3,r, + _I(_+ _xt + 7x, cos_). ".
k ' 2
a plane,
a second
-_ = co
on k,
F r =
where
by
Thus,
_. F' = ....... tk--x,)' Let
Cot
bo_
da b c, become dx---_' '
_t--_ao, designate
2xlao_
on
point
the will
two
constants
correspond
k to
and
C.
a periodic
If
it has solution,
a
which will be stable if the two tangents to the double point are imaginary, and unstable if the two tangents are real. Weshould note that the curve is symmetrical with respect to the two axes of the coordinates and that the two douhle points, which are symmetrical to each other with respect to the origin, do not correspond to two periodic solutions which are actually different. The double points may only be located on one of the axes of the coordinates, so that they will be obtained by setting _0_0_
If we set
W=_.
k
the curve F' gents to the
= C passes through the origin double point are given by the 3 .....
f,':_
Therefore,
a
:,-
l_
-q--
and has equation
_y costo
_
a double
point.
The
tan-
/358
O.
if 3
the
tangents
are
imaginary.
If 3+_.
the
tangents
are
real.
Finally,
(2)
__y,
if 3
--sT>_ the
tangents
are
again
(3)
---._÷s_,
imaginary.
The coefficient B is positive. also assuming that y is positive. If change _ into m + _.
I wrote the preceding inequalities y were negative, we would only have
to
The double point at the origin corresponds to the solution of the first type, i.e., to planet A of M. Darwin. It may be seen that this solution is stable when the inequalities (i) or (3) hold, and is unstable when the inequalities Let = 0.
(2) hold. us now
study
the
double
points
which
may
be
located
on
the
line
353
kJ
If
we
set
_
=
0,
the
function
F'
becomes
F' = --_'--+ ....._, +,_ ( h -- x, )' "_ 2 Keeping
k
minima
of
constant, F'
are
if given
we
vary
by
the
x I
from
0
to
(4)
+ I1x,(_ ÷ _) = C.
k,
we
find
that
the
maxima
and
equation 3 .... ( k -- x l )'J
which
has
tion
in
a the
solution
if
opposite
case.
Therefore, stantly
decreasing
a maximum,
and
This or
if
if then
maximum
rather
to
the
the
inequality
inequality it
(3)
holds.
The
(3)
holds,
does
not
and
does
hold,
function
F'
not
the
first
have
function
a
F'
increases,
solu-
is
con-
reaching
decreases. corresponds
two
(5) + _(_ +-()-- o,
",
double
to
a
points
which
determine
how
double are
point
located
symmetrical
on
with
the
line
respect
_ = to
0,/359
the
origln. However, given tion
value of
k.
we
the
constant
We
must
deduce
However,
from
which
must
of
equations
we
C.
x I from
(4)
we
may
Equation
and
it
obtain (5)
as
a
function
(5)
may
be
written
F'=
C,
dF' dxl
-- o
dC
dF'
dk
_
dk
dxL
dF' +
- _ d _1r'
d IF'
terms
containing
B,
dxl
we
we
dF'
d' F'
354
+
_-=
--|
dx,
have
dkdxl
and
O.
have
_--
which
7ix,'
-_!
dF' _
d],"
_
dk --
(lar--_l÷tlk
from
of
dF'
d_ F' -- d--_
ct_ F' =°;
dx_
_
t '_. -- (k--x,)
double us
have dx,
Neglecting
these
provides
_
C.
with
points x I as
for a
func-
a
d/c dxj It
results
from
this
that
x]
dC
is a constantly
decreasing
function
of
C.
For a value of C, we have only a maximum at the most, i.e., we have at the most two double points which are symmetrical to each other with respect to the origin on the line _ = 0. Let
C O be
the
value
of
C which
satisfies
the
double
equality
point
2 k Co=_+--+iI_t, ._ ___ /c _
We
shall
see
= 0 and
on
that,
that,
for
for
C > CO,
C < CO,
The same discussion the line _ = _. The
3 _ -_- i._(_ -__7)_
there
there
has If
a solution
C 1 is
the
if
value
be
not
be
a double
two
of
them.
on
the of
(2) or
satisfies
the
line
located
3 :_ + ,_(_ - "c)= o
inequalities C which
the
f360
may be applied to the case of double points values of x I will be given by the equation
(h---_ x,--3_which
will
will
o
(5') (3) hold. double
equality
k c L= %, + 7 + sx' /._ -- _.+ s(_ --7)=0, the
is
for the
condition
for which
there
are
two
double
points
on
the
line
C < C1 . We would like to point out that C 1 > CQ, that C O is the value of C which one passes from inequality (2) to inequality (3), and that C1 is one for which we may pass from inequality (i) to inequality (2).
When compiling the curves, we would readily real for the double points located on m = O, and for the double points located on m = _.
find that
that they
the are
tangents imaginary
are
355
Wemay therefore sum up our results as follows: First
case C > Cl.
The
inequality
The
solution
There
of
is no
Second
(i) holds. the
first
solution
of
type
the
(planet
second
type
A)
is stable.
(orbit
with
double
point).
case ci> C > C_,.
The
inequalities
The
solution
There
is
Third
case
(2)
of
the
hold. first
a solution
of
type
the
becomes
second
type
unstable. which
is stable. /36l
C < Co. Inequality The
(3) holds.
solution
There
are
of
two
the
first
solutions
one of which is unstable. located on the line m = _, located
on
the
These the
value
adopted
38.
356
not
by
are M.
verified
It is therefore had continued
of The and
the
is stable. second
type,
one
of which
is
stable
and
first corresponds to the two double points the second corresponds to the two double points
m : 0.
conclusions
I have
if he
line
type
valid,
Darwin,
this,
provided that _ is 1 _ :_, sufficiently
but
it seems
very
sufficiently
small.
Is
small?
likely.
likely that M. Darwin would have ohtained stable orbits his study of the planets A for values of C smaller than
CHAPTER XXXII PERIODICSOLUTIONS OF THESECOND TYPE 385. Let us again consider the equations of No. 13 ,IF
dxi
dyl
dt - dye' with p degrees these equations by
the
period
The
x's
of freedom. will have T,
integer
the
numbers
However, this is is not zero. The
hessian x.
is
This to recall
zero,
YI'
k I, k_,
...,
_
statements such that, yp
may
F 0 does
what occurs Y3, Y%; Y5,
increase
be
in Y6
not
given in No. 42, when t increases respectively
by
arbitrary. of is
F 0 with invalid
depend
on
respect to when this
all
the
the three-body problem. represent, respectively,
the
variables
I would like the mean
the perihelions and of the nodes, and that variables x I and x 2 which are proportional axes.
us consider a periodic one solution will be
differences one period.
...,
when
longitudes of the planets, of depends only on the two first the square roots of the major Let ventions,
Y2'
(i)
F = F0+ ,_F,+...
only valid if the hessian proof presented in No. 42
particularly
is precisely that Yl, Y2;
dF
-i;F = -- 7/_'
According to the periodic solutions
variables
1362
F0 to
solution. According to the stipulated conassumed to be periodic, provided that the
of the y's increase by multiples of 2_, when t increases In actuality, F only depends on these differences.
by
Let
be
the
quantities
by
which
the
y,--yc, when
t In
increases Chapter
by
following
y=--y_,
/363
increase
y3--)'6,
)'_--y_,
y,--y,,
a period.
llI we
solutions corresponding k3, ky and k 5 are zero.
could to
only
establish
arbitrary
values
the
fact
of k I and
that k2,
there
are
assuming
periodic that
357
It may be inquired whether in this case, lust as in the general case, there are periodic solutions corresponding to arbitrary values of the five integer of the
numbers k, solutions second type.
which
I would
like
to
386. Do these solutions of the second type to answer in the affirmative, based upon reasons the fact that the form of the function F needs to order to obtain canonical equations to which the in
No.
when to
as
solutions
exist? It is a temptation of continuity and considering be modified very little in line of reasoning pursued
42 applies.
However, one difficulty is entailed. we cancel the term we have designated
the
disturbing
If
the
What happens to these solutions as _ and which is proportional
masses?
disturbing
masses
are
zero,
the
two
planets
Kepler. The perihelions and the nodes are fixed, the numbers k 3, k4 and k 5 can have no other value This small, itself
designate
difficulty
the two becomes Let
us
may
be
resolved
as
that
the
two
follows.
planets,
which
the
and it would than zero. If
planets will obey the laws of Kepler, infinitely small at certain times.
assume
obey
are
the
very
are
their
far
of
appear
masses
unless
laws
away
that
infinitely
distance
from
each
other, both describe a Keplerian ellipse. It could happen that these two ellipses will meet, or will pass very close to each other, in such a way that the distance between the two planets becomes very small at a certain time. At this time, their mutual perturbing action could become significant, and the two orbits could undergo large perturbations. The planets, moving /36____4 away from each However, these perihelions I would
other again, new ellipses
and
the
like
nodes
would then describe will differ greatly will
to employ
the
although it is not a collision planets do not come in contact only be rather small in order the smallness of the masses. However
that
may
be,
account, it is no longer are fixed for _ = 0, and
undergo word
Keplerian from the
considerable collision
ellipses again. old ellipses. The
variations.
to designate
this
phenomenon,
in the true sense of the word, since the two and since the difference between them need to have considerable attraction, in spite of
if we
take
these
orbits
with
collisions
into
valid to state that the perihelions and the nodes that consequently the numbers k 3, k4 and k6must be
zero. We must
thus
conclude
that
solutions
of
the
if w_ make _ strive to zero, they will tend to be series of collisions. However, this rough sketch more
358
detailed
examination
is necessary.
second reduced is not
type
exist
to orbits sufficient,
and
that,
with and
a a
387. Let us first consider the effect of the collision. Let E and E' be the ellipses described by the first planet before and after the collision; let Ei and Ef be the ellipses described by the second planet. It is apparent that these four ellipses must intersect at the samepoint, in such a way that the two planets describing these four orbits will pass through the encounter point at the time of the collision. As long as the distance between them is considerable, the two planets describe curves which differ very little from an ellipse. During the very short period of time when the distance between them is very small, they describe orbits which are very different from an ellipse. These orbits may be reduced to small arcs of curves C having a radius of curvature which is very small; these arcs differ very little from arcs of a hyperbola. At the limit, the very short time of the collision may be reduced to an instant. The small arcs C may be reduced to a point, and the orbit, being reduced to two arcs of an ellipse, has a hooked point. In order to define the orbits E, E', El, _i' it is necessary to know the magnitude and direction of the velocities the two planets P and P1 before and after the collision. _at are the relationships between these velocities? I would first like to note that the velocity of the center of gravity of the two bodies P and P_ I must be the samebefore and after the J365 collision. The magnitude and direction must also be the samebefore and after the collision. Wemust now consider the relative
velocity
of P with respect to PI;
the
same
before
the
direction
of
axes whose origin is at PI' in magnitude and direction,
and the
the magnitude of this velocity must be sion, but it may differ in direction.
after
Following is the the collision.
rule
for
Let us consider moving a line AB which represents, city
of P with
through
the
respect
point
PI'
to P1 since
determining
before the
body
the
collision.
which
moves
and
This at
after
this
colli-
velocity
let us consider relative velo-
line
the
the
AB must
velocity
pass
which
it
represents must collide with the point PI, which is fixed with respect to our moving axes. However, this is only valid at the limit. This is only valid because we regard the masses, on the one hand, and the distance at which other
the mutual attraction of P and hand -- i.e., that which could
P1 begins to be manifested, be designated as the radius
as infinitely small quantities. It would therefore be more the distance d from P to the line AB is an infinitely small same order as the radius of action. Let respect and the
A'B'
be
the
line
which
represents
the
to P1 after the collision. In terms distance from P1 to A'B' equals 5.
relative
the action
exact to quantity
velocity
of magnitude,
on of
A'B'
of
--
say that of the
P with
equals
AB,
359
k_1
We
finally
have
the
rule
for
determining
the
direction
point P1 and the two lines AB and A'B' are in the ties which are infinitely small of higher order). A'B'
may
be
determined
proportional
to
It may
are
6 and
thus
be
The only two the following:
magnitude and alone. These The
Let
toward
follows.
to
the
seen
The
square
that
of
the
tangent the
length
direction
and us
try
which
area
to
constants
compile
the
the
solutions
of
like
to point
out
must
AB.
of A'B'
may
not
orbits the
of
of
with
this
be
angle
is
arbitrary.
velocity
be
changed
in magnitude
by
collisions
second
The
(up to quantiof AB and of
upon our four velocities the center of gravity in
permanence of the relative may be given as follows:
the
of A'B'.
plane angle
of half
conditions which must be imposed permanence of the velocity of
direction; conditions
energy
388. limits
as
same The
type
tend
the
which when
collision. are
the
_ strives
/366 to
zero. I would
first
assumed in order that consecutive collisions ellipses described tive collisions.
by the planets P These two ellipses
they have a common section points and Let
us
that
at
least
that
we
are
collisions Let us Let E
and P1 in the interval must intersect at two
focus, they are in the same the focus are on a straight
assume
two
such an orbit may be periodic. never occur at the same point.
dealing
with
plane, line.
of two points
unless
an exception.
must
be
assume that two and E l be the
the
Let
Q
consecuand, since two
inter-
and
Q'
be
the two intersection points of the ellipses E and _i which I shall assume are not in the same plane. These two points are on a straight line with the focus F; let E and E'1 be the ellipses described by the two planets after the collision. They will pass through the point Q, where the collision has just been produced, and they will not be planes will intersect along the line FQ, point (which must exist if two consecutive same point) will be located on this line two ellipses E and El will have the same the points F, Q and Q' are on a straight '
in the same plane in general. Their so that their second intersection collisions never occur at the FQ. I would like to add that the parameter. Due to the fact that line, the inverse of the para1 1
meter
E l will
of
the
Under purposes
this
of
the points
ellipse
where
360
must
of
assumption,
clarity,
let
the
We may specify located on the same We
E or
the
we us
four
ellipse
shall
assume
collisions
employ
four
the
be _+
following
collisions;
two
ellipses
let
procedure. QI,
Q2,
Q3,
For Qh be
occur.
these four points arbitrarily, line passing through F.
construct
_,-
E and
E l which
provided
intersect
that
they
at QI
and
are
Q2,two
ellipses E' and E'I which intersect at Q2 and Q3, two others E" and E'I' which intersect at Q3 and Q_, and finally two others E'" and E'i' which intersect at Q4andQI. The orbit of P is composedof arcs pertaining to the four ellipses E, E', E", _" , and the orbit of P1 is composedof arcs pertaining to the 111
four
ellipses
El,
E'I, E_', E l
We shall specify constants must be the sions
(orbits
intervals. this is the
E
and
.
the energy and area constants same for the interval between
El)
for
the
following
interval,
arbitrarily. the first two and
According to the statements presented in only condition which must be fulfilled.
for
all
These colli-
the
the preceding
/367
other section,
In order to compile E and El, we shall proceed as follows. L_t us consider the motion of three bodies. Since we assume _ = 0, this motion is Keplerian, and the central body may be regarded as being fixed at F. We know the total energy of the system. The two planets P and P1 must leave the point Q1 simultaneously in order to arrive at the point Q2 simultaneously. When P and P1 go from Q1 to Q2, the true longitude of P increases by (2m + I)_, and that of PI increases by (2m I + i)_. We may still specify the two integer numbers m and m I arbitrarily. The problem has then been completely determined. It should be pointed out that the inclination of the orbits does not intervene. In order to resolve this, we may assume planar motion. The problem can always principle of Maupertuis, and Maupertuis always has a minimum. We must now area constants. the line FQiQ2.
be resolved. We need only apply the action, which is essentially positive_
determine the planes of the two We therefore know the invariable The areal velocity of the system
vector perpendicular we know. It is the planets, represented equal, respectively,
ellipses. We know the plane which passes through is represented by a
to the invariable plane, whose magnitude and direction geometric sum of the areal velocities of the two by two vectors whose magnitude we know, since they mp and mlp , where m and m I are the masses of the two
planets and p is the common parameter of the two ellipses E and E 1. therefore compile the directions of these two base vectors which are dicular to the plane of E and to the plane of El, respectively.
We may perpen-
v!
The
terms
E'
and
E_,
E"
Let
us
now
assume
and
El,
...,
may
be
determined
in
the
same
way. 389.
that
all
of
the
successive
collisions
occur
at the same point Q. The period will be divided into as many intervals as there will be collisions. Let us consider one of these intervals during which the two planets describe the two ellipses E and E l . As in the pre- /368 ceding section, we will specify the energy constant and the area constant which must be the same for all the intervals. We must construct E and E l .
361
kJ
Let
us
assume
that
during
the
interval
under
consideration
the
planet
P has performed m complete revolutions, and that the planet P1 has completed ml complete revolutions. We can arbitrarily specify the two whole numbers m and m I. Since we know these two whole numbers, we know the ratio of the major axes. Since we know, on the other hand, the energy constant, we also know the major axes themselves. On the other hand, we know the area constant. vector which represents the areal velocity of be decomposed an infinite number of ways into
the can
Consequently, we know the system. This vector two base vectors which
represent the areal velocities of P and PIWe shall arbitrarily this decomposition. If we know the two base vectors, we know the planes of the two ellipses and their parameters. The orientation of these ellipses in its plane remains to be determined. We will it
by
passing
the
Summarizing,
of
we
i.
The
point
2.
For
all
3.
For
each
the
areolar
In satisfy
can
Q and the
through
the
arbitrarily the
interval,
Q.
specify:
number
intervals,
point
of each determine
of
the
intervals;
area
the whole
constant
numbers
m
and
the
energy
and m I and
the
constant; decomposition
vector.
order
to make
certain
390.
ellipse
specify
Let
the
problem
inequalities us
tractable,
which
disregard
the
I will
exceptional
these not
arbitrary
numbers
must
describe.
case
where
all
the
collisions
take place along the same line or at the same point, and let us consider the case of motion in a plane. Let QI, Q2, ..., be the points where the successive collisions take place. We will arbitrarily specify the energy constant and the area constant which must be the same for all the intervals. Let
us
consider
one
two planets pass from of the radius vectors vectors,
nor
We know planets
has
the
of
duration
that
in
increased
the
intervals,
for
example,
Q1 to Q2We will arbitrarily FQ 1 and FQ 2, but not the angle of
this by
the
one
where
the
the magnitude these two radius
interval.
interval 2m_.
the
specify between
Let
the us
difference arbitrarily
in
longitude
specify
the
of whole
the
two J369
number
m.
Since we know this whole number, the two lengths FQI and FQ2, as well as the two energy constants and the area constants, we have everything needed to determine the orbits E and E]. This means that the principle of Maupertuis must be as was done in No.
362
applied. 339 and
However, the the Maupertuis
Hamiltonian action must
action must be defined be derived according
to the procedure of Nos. 336 and 337. Unfortunately, this Maupertuis action is not always positive and therefore one is not certain that it always has a minimum. Summarizing, we can arbitrarily
specify:
i.
The number of intervals
and the lengths FQI, FQ2, ...;
2.
The area constants and the energy constants;
3.
For each interval,
the whole numberm.
The collision orbits obtained in this way are all planar . Among the periodic orbits of the second kind which reduce to these collision orbits for _ = 0, there are certainly somewhich are planar. It is also possible that there are somewhich are not planar for _ > 0, and only becomeso at the limit. 391. Let us now see how one may demonstrate the existence of periodic solutions of the second kind which, in the limit, reduce to the collision orbits which we constructed above. Let us now consider one of the collision orbits and let t Obe a time before the first collision and t I a time between the first and the second collisions. In the sameway, let t 2 be a time between the second and the third collisions. For the discussion I will assumethat there are three collisions. I will call T the period in such a way that at the time t O + T the three bodies appear in the sameconfiguration as was the case at the time t O. As the variables, I will take the major axes, the inclinations and the eccentricities, and the differences of the mean longitudes, the longitudes of
the
perihelia
orbit is regarded configuration at Let
x_,
0 x2,
and
the
as the
periodic if the three end of the period.
tO +
T.
One
will
0 x H be
...,
t O fo_ the collision Let x_ be the values sion orbit, x$ their
nodes.
the
In
all, there
values
of
are
bodies
these
eleven have
variables. the
variables
same
at
the
The
relative
instant
orbit under discussion and consequently for _ = 0. 3_ of these variables at the time t I for this same collivalues at the time t2, and x$ their values at the time
have ,r_ = x, ° + _ i_,,._
where mi tricities
is a whole number which and the inclinations.
must
be
zero
for
the
major
axes,
the
eccen-
363
kj
Let
us
now
consider
an orbit
which
is slightly
different
from
the
col-
lision orbit. from zero. In
Let us assign a very small value to _, but different Q this new orbit, our variables will have the values x_ + _i 1 1 2 2 to, x i + 8i at the time tI, x i + _i at the time t 2 and finally
at the time 3 3 x i + 8i at the The
tO +
condition
Assuming the time
and
time
for
u = 0, t I, the
T +
_.
which
the
in order variables
solution
is periodic
that a collision _ must satisfy
with
period
occurs between two conditions.
T +
the
T is
time
tO
Let o 0
._I
be
these
two
Let
conditions.
us set 0_.0 _,. - r,,
/,(_0)::_?_,A(_7)=_; it
can
p.
By
be
holds
that
applying
the
for
In that
seen
the
order
that
_ = 0),
two
Let
I
Q's Yi
the
and
then
set
find
that
the
of
U.
The
Finally, t o + TT,
364
two
there
be
of
holomorphic
functions
of
Chapter
it
shown
a collision
conditions
the
us
and
principles
... ii);
II,
can
be
0v 7i s and
the
that
the
of same
8_'s.
Replacing of
0v Bi s are
the
(k=3,4,
B_'s
of _,
in
in
holds
between
necessary,
relationships
and
Iv B i s and same
are
then
setting
the
which
(I) by _ = 0,
their
2v B i s are holomorphic O's and true for the _i '
order
that are
there
be
necessary
a collision which
tl
and
t2
(assuming
write
as
follows
values
as
a function
I obtain
the
conditions
times
I may
I may
"Ivs functions of the Yi consequently for the 8_'s.
between write
the as
times
follows
t 2 and
f,< _i ) =f_(;_,' If and
of
I may
and
2_S 8i
the p,
and
are
if we
replaced then
set
by
their
p = 0,
) -- o.
values
they
as
a function
of
the
Iv Yi s
become
set
I then
find
tions
of
the
tions
of
the
The holomorphic
that
¥i2's and 2w Yi s, of
the of
_.
In
the
_,
and
of
T.
relationships with
Bg'Sl , the
B_I =
respect
to
8_ are the
8
's, same
and way,
therefore
y_'s,
_,
the
and
the
that this is necessary, the orbits traversed in
holomorphlc
B 13's are
whose
T.
equations
These The
func-
holomorphic
equations
discussed in the same manner as in Chapter III. of the second type could then be demonstrated. I do not believe deviate too much from
812, s are
existence
two
of
func-
terms
are
could
be
solutions
because these solutions actuality by celestial
bodies.
365
CHAPTER XXXIII DOUBLY ASYMPTOTIC SOLUTIONS
Different
392.
In
ourselves
to
order
Methods
to study
a very
special
of
Geometric
doubly case,
Representation
asymptotic that
of
solutions,
Section
No.
we
9:
shall
Zero
confine
mass
of
1372
the
perturbed planet; circular orbit of the perturbing planet ; zero inclinations. The three-body problem then has the well-known integral called the Jacobi integral
•
Returning tinguish
between
following
to No.
299
several
devoted
to
cases.
We
this saw
problem
from
on page
159
Eo.
that
9, we
must
dis-
must
have
the
we
inequality
_
__ .....
f*i
ft
,_ _ (_,+ ,,,)= v+ 2
-- (_' 4--_')> --h.
(1)
2
We then distinguished between the in which -h is sufficiently large
case in which m i is much (page 160). We saw that
smaller than the following
m 2, and
curve V ,-n' ($'+ ¢')= -- h
(2)
2
may be broken down into three closed branches which we have called CI, C2 and C3. Therefore, in view of the inequality (i), the point $, n must always remain inside of CI, or always inside of C 2, or always outside of C 3 (_, are the rectangular coordinates of the perturbed planet with respect to the moving
axes). We
for
shall
curve
assume
below
that
the
(2) to be
broken
down
into
point 5, _ always remains inside of the perturbed planet to the central for the distance r I between the two
lated
This hypothesis corresponds to on pages 199 and 200 -- i.e.,
Figure
9, We
and
shall
the
point
employ
the
x I, x2
value
the
closed
constant
-h
branches,
is and
large that
enough the
C2. In this way, the distance r 2 from body may vanish, but this is not true /373 planets. the the
remains
notation
of
three
given
following hypothesis, curve F = C has the on
the
in No.
utilizable 313,
and
which we form shown
arc we
formuin
AB.
shall
introduce
the Keplerian variables L, G, i, g. However, these Keplerian variables may be defined in two ways. Just as in No. 9, we could relate the perturbed body to the center of gravity of the perturbing body and of the central body,
366
and we could consider the oscillating ellipse described around this center of gravity. However, it is preferable to refer the perturbed body to the central body itself, and to consider the oscillating ellipse described around this central body. These two procedures are equally legitimate. Wesaw in No. ii that the body B may be related to the body A, and the body C may be related to the center of gravity of A and of B. It is apparent that we could also refer C to A,and B to the center of gravity of A and C. If A represents the central body, B the perturbing body, and C the perturbed body, it can be seen that the first solution is that which was adopted in No. 9. It may also be seen that in the second solution, which we shall adopt from this point on, the two bodies B and C are both related to the central body, since -- due to the fact that the mass of C is zero -- the center of gravity of A and C is at A. Wethen have [;'.=-Rq-G----
V/_--
:'t -+- G -4- _V/'7-'_t
L' where _ and 1 - U designate central body, r I designates nates the constant and r 2 designates Just
as
It
",
the masses of the perturbing the distance between the two
distance from the perturbing the distance of the perturbed
in No.
313,
we x_::
shall
v-F, =
I would like to stress the function F l always n cannot leave.
L--G,
to the to the
central central
body, body.
x,=L+G,
r Fo --= 7-L--_ +
_, F,,
¢' - ;" ' _-r,: ....
'2 G --
(xt x,)_ +
_ ¢'--:-,'] '......
t_
-.t-
("_-'-
.'_ -- .'r____._w. 2 '
"I)-
the following important point. It can remains finite in the region from which
_'._
of the 1 desig-
2y,=l+g--t;
We shall employ the method shall represent the configuration rectangular coordinates are X =
body body
body and planets,
set
_)'l=l--g+t, I"' = F_,-_-
(r_--I--,'])
2 V'i --
of
representation of the system
cosy,
y
¢.;;--7 __-,- .. ¢;, _o_y, Z=
=
by
_
¢____ _.,
be seen that /374 the point _,
given on page 200 , and we the point in space whose
siny,
_ = ¢., _o_y,
"_¢'_ _iny,
367
xl is constant, the point X, Y, Z It can be seen that, when the ratio x-_ describes a torus. This torus may be reduced to the Z axis when this ratio is infinite, and may be reduced to the circle Z-_-o_
when
this
ratio
The
just
small.
could
become
does
the
would
infinite
function
not for
be
dF 0
the
very
little
hypothesis
which
vanish because the be readily reduced (on page
2
201,
If x 2 is
dF 2 _-x2 can
because Y2
is
always Let
shall
are
We
saw
consi-
on page
201
that,
consequently
in
terms
n2
cannot
constant C (the constant C given in No. constant h given in No. 299) is larger 3 3 set _ instead of _ everywhere).
must
very
only
small,
become
a point
we
shall
infinite
except X,
Y,
therefore
for
Z, such
small
that
Y2
=
313 may than
have
x 2 = 0,
for very
of
from
which
it
follows
that
x 2. 0.
On
the
half-plane
we
have
When
Xl,
x 2, Yl,
to M I -- will The
368
under
except when x I or x 2 is ' dFl dFl which derivatives dy I, dy 2
dF0 d-_2and
Y2
vary
in
X > o. conformance
the point X, Y, Z will describe a certain creases constantly, reaches the value 2w,
given plane
region
a result,
dealing,
Y = o,
moved
the
dF' -- n_= d_,.
we
increasing,
M be
As
the
in
energy to the
we
not
for
dF0 and _2"
from_
with
true
dF' _,
finite
F I itself
r 2 = 0.
---n,=
differ
l,
dF 1 dFl _ and d-_2 remain
as
This
-Y_==
zero.
derivatives
deration, very
is
Xz-i
point
M 1 is
again then
be the
located
on
consequent
with
differential
equations,
trajectory. When Y2, which inthe point X, Y, Z -- which has
the half-plane of M,
according
in No. 305. Since Y2 is always increasing, has a consequent and an antecedent. There
Y = O, to
X > O.
the
definition
every point on the is only an exception
halffor
/375
k.J
very from
small x2 -- i.e., for points on the half-plane the origin, or very close to the Z axis.
ted
We shall have an integral to this word in No. 305.
invariant, Let us try
the
Due to the fact that the equations following integral invariant
which
in terms of to formulate are
canonical
are
very
far
the meaning attributhis invariant. equations,
they
have
_Z,r,_lx_dy, dy2.
Let The
us
set
invariant
x2 z =--Xl , and
will
We may (due
deduce to the
us
F' , z, Yl,
select
Y2
as new
variables.
become
- aF' ,w'-j-._i,,-i+_-n] " ._ (;v'(z: ,ly, 4r, _ ['_ aV'(t_.dr, at,
--
variant
let
the following existence
triple invariant the integral F'
of
/_i
from = C)
this
quadruple
dz +_:_" _'y' d.r, n, ¢_F'
In
this
triple
as
functions
replaced
integral, of
we
z, Yl,
assume Y2
xt
Let nate the become
us now take Jacobian of
us
--
which
,v t z,
of
the
are
d.r,
equations
F' ---- C.
X, Y, Z, and let us employ respect to z, Yl, Y2The
A to desig/376 invariant will
g
_n
it
follows
again
set
_ sinr,
2co,.r,'
Z= ¢t;+ _- 2cos.r_'
that X ::
us
means
dF'
n2--
set
_-
Let
that x,,x2,n,=--_zT_,
x_ dX dYdZ (xtnt+x,n,)_
R
from
by
the variables X, Y, Z, with
f
Let
in-
D -= [(R
R cosy',
-- ,)'
Y ::- R sinyt.
+ Z'][(R
+ ,),-t-
Z']-
369
k_J
A simple calculation
Our invariant
provides the following RD may therefore be written s.r_ v':.(:+ 45dx ,tY dz (.:r,,_ T-x, ,,,5 a o -"
The principles presented in No. invariant, in the sense of No. 305
305
.r,
D
n 2 and
small
R play
the
role
which
The tern under the sign x 2 -- i.e., for points
origin,
or
very
close
to
the
_ and
enable
nt
-_
x2
us
deduce
the
following
n,
O played
f is essentially of the half-plane
to
in
the
analysis
positive, except which are very
of No.
305.
for very far from the
Z axis.
393. This fact (that a point will no longer have a consequent if it is too far, or if it is too close, to the Z axis) could cause some difficulty, and it would be advantageous to avoid this difficulty by whatever method. We
could
employ
the
statements
presented
in No.
311,
and
we
place our half-plane by a simply connected curve on a surface. choose this curve on a surface in the following way.
turn
If x 2 is very small, the eccentricity is very small, in the opposite direction. The principles presented
applicable, first type ties
and we may affirm which will clearly
could We
re-
shall
and the two planets in No. 40 are
the existence of a periodic solution of the _377 satisfy the following conditions: The quanti-
¢'x_cosys,
¢x_siny_,
x,,
cosy,,
siny,
are periodic functions of the time t. These functions depend on _ and on the energy constant C. They may be developed in powers of _; the period T also depends on _ and on C. The angle Yl increases by 2_ when t increases by a period. that we have With have when
370
Finall_, _/_ cos x 2 = 0 for _ = 0.
our
called o, _ is very
method
of
Y2
and
representation,
is represented by small, this curve
V_
this
sin
y2
are
periodic
a closed curve K. is displaced very
divisible
solution,
Since little
by
_,
which
so
I
x 2 is very small from the Z axis.
It may be stated that it is displaced from it very little, in the that a circle having a very large radius is displaced very little straight line. Every point on the K curve is either very far from or very close to the Z axis. Under
this
assumption,
our
curve
on
a surface
S would
have
same way from a the origin
the
curve
K for the perimeter, and it would be displaced very little from the halfplane Y = O, X > O, except in the immediate vicinity of the curve K. It would be very easy to conclude this determination in such a way that every point on this surface would have a consequent on this surface itself. For this purpose, if I designate an arbitrary trajectory by (T) -- i.e., one of the curves defined in our method of representation by differential equations -it would be sufficient that the surface S was not tangent at any point to any of the trajectories (T). However, there is still another method, which does not basically from the first method. If we reflect on this a little, we will find this difficulty is similar to that in Chapter XII. We must therefore the change in variables similar to that performed in No. 145. Let
us
first
set
_,= ¢,_
where
then
S 1 is
_,,= d:,_-
have
a function
of
, t and
differ that perform
6'2,
dS
N2,
= _,
dS = "_"=dy-7
Yl.
X'I,
us
,
(IS
dS, .
dS
dS i
dS,.
,IS j z; + _, ,(_.,,
then
/37S
Let
set
(1)
finally ---7"
•
_,;.= t/2._, s,ny,.
I should not
be
first
changed
point
when
I pass
out
that
from
the
the
canonical
variables
Xl,
form Yl,
of the x2,
equations
Y2,
to Xl,
will Yll
62,
i
N2,
then
to X'l, Y'I,
I must
now
I know
that
_/_x2cos)'_,
_siny._,
6' 2,
choose F'
is in
D'2, and the
finally
function
a holomorphic the
region
to x'l, Y'I' x2,
yV2"
S I. function
under
of
%/_J:,cosy,,
consideration.
%/2"_x,siny,,
I would
like
it
to
371
remain a holomorphic function of the new variables _ For
this
phic
purpose,
functions
To
do
of
this,
cosj,, _sin J,
I would
like
the
variables
we
new
need
only
the old
2V_x_
assume
that
c°s sin
is
divisible
For
our
by
solution
o,
I would
r
F, = _,--- o,
to be
holomor-
_.
a holomorphic
function
of
_',, ,,,, =_
like
x', = x?=
to have
const.
let
be the equations of the periodic which are periodic of the period
Then
of
Yi
x'I.
periodic
Therefore,
COS sin
Yi, and
S I is
dT-x',_o_>,,, ¢_._i.r,, and
V_i
variables
C - _
will
also
be
solution. 2_ and may
a periodic
A, B, C are be developed
function
functions of Yl in powers of _.
of Yl.
Let
x 7 be
its
dyl mean
value.
We
may
obtain
another
periodic
function
d A
We shall duced
no
longer
assume
that,
for
a such
that
d:_
x' 1 = x_,
the
function
_S 1 may
be
re-
to _ -- B _'2+ AT,,.
This to be
will
reduced
It
is
clearly _I
of
same
may
time
Let
be with
in powers
372
/379
us
be
adopt
sufficient the
for
new
possible cos sin
the
new
equations
which
to
a function
may
be
expression
variables
of
the
periodic
solution
to
to obtain
Yl,
reduced
the
variables
(2)
_S I
divisible (2)
for
by
which
x'I, and
x'I = X°l•
X'l, Y'I, x'2,
Y'2-
may
be
which
developed at
the
%.J
The _2s
function
Os Y2, in
will
, o_/__v cos Vzx_i n yh.
In
equations
F'
which
'
also
was
holomorphic
be holomorphic
addition,
since
with
one
of
must
have
For @2.
degree reduced
the
small
following
values
In view
relationships
This
dF'
dF'
_---7
a_,
of
of
_'2
and
n'2,
relationships
constant
can
_'2 =
be
n'2
nothing = 0,
Thus, for F' = C, the terms in the expansion of F'.
the
The difficulty first degree in
and
that,
first even
arises
cos sin
2_xV 1 cOs sin
solutions
of
the
Yl,
Y'I,
differential
F' may for
else
for
xh
in -
than
developed the
terms _i-
the
be
of
energy
replaced
in
powers
terms zero
of
degree
constant by
the
of
the
C,
_'2
first
will
so
following
be
that con-
F'= C.
first
the
be
degree
fact
that
in
F'
_2
and
and
@2
F I include
will
vanish
terms
of
the
derivative
dF I _ includes
terms
l-l--- which
be-
x 2 = 0.
difficulty
degree
from
= O, x'I = x_
(3)
x'I = x_, the of
_=o,
the
_'2 = @2
-_-0.
_
x'I = x_ may
of
--
/38O
consequently,
infinite
for
_
dF' _
(3),
G=
This
to
for
in this expansion will vanish, and to a constant which is independent
the conditions ditions
come
respect
to
is
_,
and
respect
the
X'
we
with
_'2,
O,and
no
longer
_'2.
Therefore
dF' dx---_2 , which
the same sign. Therefore, the old variables by very stantly have
exists
with small
differs
now. the
longer have terms of dFl derivatlve d--x2 remains finite,
very
We
no
little
our new variables quantities on the
dF 0 frOm_x_2 , always which order
the
retains
only differ from of _, we shall con-
d--T> o.
373
With to that tion of
our
new
variables,
let
us
formulate
a convention
which
given in the preceding section, and let us represent the the system by the point in space whose coordinates are "-7
Y _
Z
•
v/.c, ,nyi
is
similar
configura-
!
--
d.x' 2 Everything can
never
which
vanish,
we
have
every
stated
point
on
have
a consequent.
There
It may now be stated that can only be some question
same
variables,
was
xlnl
may
--
be
assuming
that
is
.
without
since
exception,
dt will
the integral invariant is always positive. of doubt for the denominator which, with
+ x2n2
F'
holds
the half-plane,
and
- (x', which
still
However,
which
now
would
the
be
aF"_
a function
of
the
following
four
variables
written
dF' _; - _,(<, "' _-ffh
F'k. +_i _-_ +_; a_;/
_
In this form, it may be readily seen that with respect to the _' 's, the n''$ and _. reduced
the denominator is holomorphic However, for _ = 0, F' may be
to -L
_x_ + x',)t
and it may will still 394.
readily be be positive In
the
shown that for small
following
'
2
the denominator values of _.
statements,
we
shall
is
always
adopt
the
positive.
variables
It
defined
in the preceding section. We shall remove the accents which have become useless, and we shall write F, x i and Yi in place of F', x'i and Y'i" We then have the integral invariant (in the sense of No. 305)
374
dF (_T4
D
dF dX dZ
dl;
from which we have fJ ;: [( x -,):-_-
always plane.
I would first like positive, remains
If the
_(X
numerator
-
1) 2 +
x_
to note finite
Z 2 is
_
an
that when
z, ][(x -_-,)_+ z, ].
this integral it is extended
infinitely
is an
small
infinitely
invariant, over the
quantity
small
of
which entire
the
quantity
of
is half-
first the
order,
second
order, and the same holds true for D. If _/(X - 1) 2 + Z 2 is an infinitely large quantity of the first order, the numerator remains finite, while D is very large of the fourth order. All of the other quantities remain finite. I shall half-plane.
The
call
periodic
J0
the
value
solutions
of
and
the
the
invariant
trajectory
J extended
curves
over
which
are characterized by the fact that these curves intersect points whose successive consequents are finite in number. us refer to No. 312 and, in particular, to Figure 7 shown
the
entire
represent
them
the half-plane For example, in page 195.
at let
In this figure, the closed trajectory which represents a periodic /382 solution intersects the half-plane at five points M0, MI, M2, M3, M4 ' each of which is the consequent of the others. For purposes of brevity, I shall call such a system a system of periodic points or a periodic system. Two solution.
systems These
of as_jrmptotic solutions are
solutions correspond to each unstable, periodic represented by trajectories (in the sense of
No. 312), and the total group of these trajectories forms what I have designated as asymptotic surfaces. The intersection of an asymptotic surface with the half-plane will be called an asymptotic curve. Just as we saw in Figure 7, page 195, four branches of asymptotic curves (MA, MB, MP, each two of which are located in the extension of the other -- lead of the points M i of an unstable periodic system. There infinite infinite
an
number number
ourselves A
is
to
infinite
of
asymptotic
of unstable, periodic of systems of unstable
solutions
distinction
number
may
of
the
be
drawn
first
solutions periodic
type
between
curves,
which
because
there
and, consequently, points, even if we we
asymptotic
defined curves
in Nos. of
MQ) -to each
the
is
an
an confine 42
and
44.
first
375
kJ
family and of characteristic
the second family, depending on whether the corresponding exponent is positive or negative. Curves of the first
family are characterized by tbe following property: The tn-_-h antec edent of an arbitrary point is very close to a periodic point if n is very large. and not
For the
curves of the second family, it would be the tn-_hconsequent, tn-_-h antecedent, which would be very close to a periodic point.
first
On the figure shown on page 195 , the curves MA and MP belong to family, and the curves MB and MQ belong to the second family.
sense
These asymptotic curves may be regarded of Chapter XXVII, under the condition
the
as invariant curves in the that one of the two following
conventions is employed. Let us again consider 195, and we shall find the curve M0A 0 which has for successive consequents. If we consider the
the figure shown on page MIAI, M2A2, M3A3, M_A4, MoA5 five curves M0A 0 , MIAI, M2A2,
M3A3 , M_A4, this total _roup will clearly constitute an invariant curve. I_h we only consider the consequents in groups of 5, and if we designate the 5p--_ consequent, which it has been called up to the present, as the p_ consequent, it is apparent that only the curve MoAoA5 under consideration will be an in/383 variant
curve.
Two end at curves of the
curves
of
the
same
family
cannot
intersect.
These
two
curves
will
the same periodic point -- for example, the point M 0. These two will coincide (since M0A 0 with its extension MoP0 is the only curve first family which passes through M0), and we must determine whether
asymptotic curve can in the negative (No.
have 309,
a double point. page 186).
The
question
has
been
answered
Or, these two curves will lead to two periodic points of the same periodic system -- for example, to the two points M 0 and M 1 • If two curves, which would then be M0A 0 and MIA I, had a point in common Q, the antecedent of Q would have to be very close to M 0 for very large p, because Q would belong to M0A 0, and it would have to be very close to M 1 at the same time because Q would belong to MIAI. This is absurd. Or, different belong
finally the two curves would lead to two points belonging to two periodic systems. For example, let us assume that the two curves
to
the
For very to one of the the
second
first
Let
system
376
cannot S and
and
that
Q is
large n, the n_ antec endent points of the first periodic
Conversely, families
family,
at the there
same
is no
time. reason
This that
their
point
of intersection.
of Q would have to be system and one of the is two
also
very close points of
impossible.
asymptotic
curves
of different
intersect. S' be
two
unstable
periodic
solutions,
let
T
and
T' be
the
an
corresponding closed trajectories, periodic systems.
and let P and P' be the corresponding
Let _ andK ' be two asymptotic surfaces which pass through T and T', respectively, and which intersect the half-plane along two asymptotic curves C and C' -- one belonging to the first family, and the other belonging to the second family. What will happen if C and C' have a point in commonQ? The two surfaces E andK' will intersect along a trajectory T, which will correspond to a special solution o. The trajectory r will belong to two asympototic surfaces, so that for t = -_ it will closely approach T, and for t = +_ it will closely approach T'. For very large n, the _ antecedent of Q will be very close to one of the points of system P and its tn_-hconsequent will be /384 very
close The
to
one
solution
There
is
of
the
o is
nothing
points
of
therefore absurd
in
system
doubly any
of
P'. asymptotic.
these
results.
We must distinguish between two cases, however. The two solutions S and S' coincide, so that T first closely approaches T = T' then recedes farther away from it, and again closely approaches this same trajectory T = T'. I could then state that the solution S differs from S', and T differs from T'; I may then clinous.
The shortly. least in
existence of homoclinous solutions will be demonstrated very The existence of heteroclinous solutions remains doubtful, at the case of the three-body problem.
Homoclinous 395. intersect". asymptotic M3B 0 which The
o is homoclinous. Or, state that o is hetero-
Solutions
At
the end of No. 312, we found that "the arcs A0A 5 and B0B S However, the arc A0A 5 belongs to the curve MoAoA 5 which is an curve of the first family, and the arc BoB 5 is part of the curve belongs to the second family. line
of reasoning
is
general,
and
we
must
conclude
that
the
two
asymptotic surfaces which pass through the same closed trajectory must always intersect beyond this trajectory. The asymptotic curves of the family curves
which lead to of the second
the points of family, which
a periodic system always intersect lead to these same points.
first the
In other words, on each asymptotic surface there is at least one doubly asymptotic, homoclinous solution. We shall see very shortly that there is an infinite number of them, but we shall now show that there are at least two
377
of them. For this purpose, let us turn to the figure shown on page 195. Following the line of reasoning in Nos. 308 and 312, we find that the integral invariant J extended over the quadrilateral AoBoA5B5must be zero. It is for this reason that this curvilinear quadrilateral cannot be convex, and that the opposite sides AoA5and B0B5 must intersect. Let Q be one of the /385 intersection was chosen
points of arbitrarily
these two arcs. on the asymptotic
We should note that the point B 0 curve MA 0. If we place the point
A 0 at the point Q itself, this point A 0 will also be located on the curve M3B 0 and will coincide with the point B 0. If the two points A 0 and B 0 coincide, the same will hold true for their five consequents A 5 and B S. The quadrilateral AoBoAsB5 will therefore be reduced to the firgure formed by two arcs of a curve having the same end points. This figure cannot be convex, since the integral invariant extended over the quadrilateral must be zero. Therefore the two arcs A0A 5 and B0B 5 must have points in common,
other
than
their
There
will
(a point
end
points.
therefore
be
an
arbitrary
consequent
will
therefore
always
and
at least
two of
different
it
are
intersection
not
regarded
points
as being
differ-
ent). There Let
us
assume
that
the
the arcs A0A S and BoB S up will have thus determined point which
of view of are a part
same
be
at
points
to an
least
A 0 and
two
doubly
B 0 coincide,
the first point area which will
asymptotic and
let
Analysis situs) and which will be bounded of the two arcs A0A S and B0B 5, respectively,
end
points
--
i.e.,
Let
a 0 be
this
A 0 = B0
area,
and
and
let
us
at which they touch be convex this time by
solutions. extend C O • We (from the
two arcs having the
CO •
a n be
its
tn_h consequent.
like a 0 -- will obviously be convex and bounded one belonging to the first family, and the other
_he area
a n --
by two arcs of a curve -belonging to the second
family. ]he integral value.
Since
the
J will value
have
the
same
J0 of
the
integral
plane is finite, following will find that, if
the
line
of
value
for
s 0 and
invariant
reasoning
for
a n. the
presented
Let
j be
entire
in No.
291,
7' the
378
area
a 0 will
have
a part
in
common,
at
least
with
p
of
the
this
half-
areas
we
k.Y
Since n cannot be taken arbitrarily result:
part
Among the areas o_, in conmion with s 0.
How
may
it happen
there
that
large, I may stipulate
is
an
n 0 has
infinite
a part
in
The area s 0 cannot be entirely within has the same value for the two areas. For cannot be entirely within s 0. Neither can
number
common
c_, the the
of
the following
them
with
which
have
a
0_?
since the integral invariant same reason, the area a n two areas coincide. If one
part of an asymptotic curve (for example, belonging to the first family) coincided with its n-19-consequent, the same would hold true for its p-_ antecedent, no matter how large p may be. However, if p is large, this antecedent is very close to the periodic points, and the principles formulated in Chapter VII will demonstrate that this coincidence does not occur. We an. the
must
therefore
assume
that
the
However, the perimeter of s 0 is curve MoAoA 5 of the first family,
perimeter
composed and of
of
s 0 intersects
of an arc an arc
that
AoHoC 0 belonging
of to
l::j K_Co :-: _ o K_,Co belonging
to
the
curve
In the same way, the tn-_h consequent of as AoHoC 0 -- i.e., to posed of the same
M3BsB 0 of
the
second
family.
the perimeter of an will be composed of the arc AnMnCn, AoHoC0, which will belong to the same asymptotic curve a curve of the same family -- and it will also be com-
the arc AnKnCn, the asymptotic curve as
tn!khconsequent of AoKoC0, which will belong to AoKoC 0 -- i.e., to a curve of the second family.
Due to the fact that two curves of the same family cannot intersect, it is necessary that AoHoC0intersect AnKnCn, or that AoKoC 0 intersects AnHnC n. However, if the two arcs AoKoC 0 and AnHnC n intersect, their tn_h antecendents A-nK-nC_ n and AoHoC 0 will equally intersect. AoHoC 0 intersect the tr__hconsequent, or the However,
the
arc
AoKoC0,
all
of
its
It is therefore necessary tn__hantecedent, of AoKoC 0.
antecedents,
and
all
of
its
that
conse-
quents will belong to the same invariant curve of the second family, which was shown in the figure on page 195 by the total group of curves M3B0, MIB3, M_BI, M2B4, M0B 2.
this
]he arc AoHoC 0 is group _ curves.
the therefore
therefore
intersected
two surfaces _ and Z' which have an infinite number of
an
infinite
number
passed through the other intersection
closed curves.
of
times
trajectory
by
T
379
_hqrefore_ asymptotic,
on
the
surface
homoclinous
K there
is
an
infinite
number
of
double
solutions. q.e.d.
396. Let AoHoC0 be an arbitrary arc of our asymptotic curve of the first family, and let us assume that this arc intersects an asymptotic curve of the second family at two end points A 0 and C O • It may be stated that there will always be other points of intersection with the curve of the
second Let
these
family AoKoC0
two
be
points the
end
points,
Or,
these
the
two
these
arc
A 0 and
Either their
between
arcs
of
arcs
the
points
curve
A 0 and
of
the
CO•
second
family
which
unites
C O• AoHoC0
in which
two
two
and
AoKoC0
the
theorem
has
been
proven.
have
a point
in
common
other
case
do not
have
points
in
common
than
other
than
their
end
points A 0 and C O • lhe two arcs then bound an area s 0 which is similar to that which we considered at the end of the preceding section. _he same line of reasoning may then be applied, and we may conclude that the arc AoHoC0 intersects the curve of the second family an infinite number of times.
curve the
_herefore, there is an infinite number of other points on an asymptotic of the first family, between two arbitrary points of intersection with curve On
bitrary
where
of
an
the
second
arbitrary
solutions,
We may dense
family.
asymptotic there
is
surface,
an
not yet conclude on the asymptotic
infinity
between of
that the surface,
other
two
doubly
asymptotic
ar-
solutions.
doubly asymptotic solutions are but this seems very likely.
eve___-
the points of intersection of two asymptotic curves may be divided into two categories. The asymptotic curve may be traversed in two opposite directions. We assume that this direction is positive, if we proceed from a point to its consequent. Let A be a point of intersection of the two curves, and let BAB', CAC' be two asymptotic curve arcs intersecting at A. Let us assume that BAB' belongs to the first family, and CAC' belongs to the second family, and that -- when following the curves in the positive direction -- one proceeds from A to B', and from A to C'. Depending upon whether the direction AB' is to the right or the left of AC' the intersection point A will belong
to the Under
first
this
or to
assumption,
sected at A 0 and C O by an category A0 and C O belong closed curve. If the two end
380
points,
this
closed
the
second
let
category.
AoHoC0
be
an
arc
of
the
first
family,
inter-
arc AoKoC0 of the second family. No matter what to, the group of two arcs AoHoCoKoA0 will form a arcs have no other point in common except their
curve
does
not
have
a double
point
and
defines
an
area
s 0.
If
the
two
arcs
had
points
in
common
other
than
their
end
points, and if, for example, the two arcs AoHoDoH_C0, AoKoDoK_C 0 intersect at DO, we may replace the points A 0 and C O by the points A 0 and DQ located between A 0 and CO, and the arcs AoHoC0, AoKoC 0 by the two arcs AoHoD 0 and AoKoD 0. _his may be continued until we arrive at two arcs which
have
no
point
in
common
other
than
their
end
points.
Let us assume that the two arcs define an area s 0. According to the statements we have just presented, the arc AoHoC 0 must intersect the asymptotic curve of the second family an infinite number of times. _erefore, the curve of the second family must penetrate within s 0 an infinite number of times, and it must leave it an infinite number of times. It may penetrate it or leave it only by intersecting AoHoC0, because it cannot intersect AoKoC 0 which also forms a part of the curve of the second family. It is apparent that points through which it will penetrate into the area, and the points through which it will leave the area, will not belong to the same category. _herefore_ between two arbitrary there is an infinity of other points an infinity of other points be!onging
which order
intersection points of two curves_ belonging to the first category, and to the second category.
Let us employ (i), (2), (3), ..., to designate the successive points at the curve of the second family and the arc AoHoC 0 meet, taken in the in which they are encountered proceeding along the curve of the second
family in the positive direction. They will succession. Let us study the order in which along the arc AoHoC 0. This excluded
order --
for
cannot example,
be
completely the
(',m + i) (_.m), (2;_), as well as 2m + i and
belong to two categories in they are encountered proceeding
arbitrary,
and
certain
successions
are
following:
(2p),
/389
(',_m),
C2/* + i._, (_.t (_.,¢),
(_p i
'- ,),
(_,n--
the same inverse successions, and 2p + 1 are replaced by 2m - 1 and
,),
,)
(_.p) (2o
-,)
the similar 2 p - i.
successions
where
397. When we try to represent the figure formed by these two curves and their intersections in a finite number, each of which corresponds to a doubly asymptotic solution, these intersections form a type of trellis, tissue, or grid with infinitely serrated mesh. Neither of the two curves must ever cut across manner number
itself again, but it must bend back upon in order to cut across all of the meshes of times.
itself in the
in a very complex grid an infinite
381
k_J
The complexity of this figure will be striking, and I shall not even try to draw it. Nothing is more suitahle for providing us with an idea of the complex nature of the three-body problem, and of all the prohlems of dynamics in general, where there is no uniform integral and where the Bohlin series
are
divergent.
Different
hypotheses
I. We may or the group of of points occupies bility
assume points
belonging the entire
of 2.
the We
solar
may
are
possible.
that the group in the vicinity
of points of which
of two asymptotic curves E0, there is an infinite number
to E 0 -- i.e., the group E'0, the "derivative half-plane. We would then have to conclude system
assume
of E0" -that insta-
exists.
that
the
group
E'0
has
a finite
area
and
occupies
a
finite region of the half-plane, but does not occupy it completely. Either one part of this half-plane remains outside of the meshes of our grid, or a "gap" remains within one of these meshes. For example, let U 0 be one of these meshes bounded by two or more asymptotic curve arcs of the two families. Let us compile its successive consequents, and let us apply the procedure _390 presented
in No.
291.
Just
as
on page
u_, If
it is
finite,
the
ub
area
145,
U_, u_,
E will
let
us
formulate
u_ .....
the
following
_.
represent
one
of
the
gaps
which
we
just mentioned. It would appear that we may apply the line of reasoning employed in No. 294, and may conclude that this area must coincide with one of its consequents. However, this group E could be composed of a region of finite area and of a group located outside of this region, whose total area would be zero. According to page 151, we may only conclude that E I (the I th consequent of E) includes E, and that the group E I - E has area zero. In the same way, the groups E - E- I, E- I - E-21, ..', E-n% - E-(n+l)1 will have area zero (by area of a group, we mean the value of the integral J extended over this group). On the other hand E_(n+l)1 is a part of E_nl. When n increases indefinitely, E_nl tends toward a group c including of all the groups E_nl at the same time. The area and
equals 3.
that
of E.
Finally,
It would densed
in
curves
398. in
when
the
any
then
we
Finally, may
be
E coincides
assume
similar
that to
the
those
with
group
"perfect
its
I th
is part finite
consequent.
E' 0 has
area
zero.
groups
which
are
not
con-
interval".
We may represent the different intersection points of the two the following way. Let x be a variable which varies from -_ to +% asymptotic
curve
of
the
first
family
M0A 0 is
point M 0 up to infinity, and which increases by point to its fifth consequent -- from A 0 to AS, point, we shall assume that we are dealing with
382
every point which of this group c is
followed,
from
the
unity when we pass from one for example (to clarify this the conditions of the figure
shown on page 195). Let y be another variable which varies from +_ to -_ when the curve of the second family M3B 5 is followed from the point M3 up to infinity, and which increases by unity when we pass from a point to its fifth consequent. the different intersection points of the two curves are characterized by two values of x and y, and each of them may he represented by the point on a plane whose rectangular coordinates are x and y. Weshall thus have an infinite number of representative points of the /391 doubly asymptotic solutions in the plane. An infinite number of other points may be deduced from each of these points. If the point x, y corresponds to an intersection of the two curves, the same will hold true for the points x-_-I,y-_;
x+_,y+_]
...;
x+n,
y+n,
where n is a positive or negative whole number. In order to determine all the representative points, it is sufficient to know all those which are included in the region 0 < x < i, or in the region 0 < y < i.
these
We would also representative
ship This
with the order results in the
like to note that the order in which the projections of points will occur on the x axis will have no relationin which their following.
projections
will
occur
on
the
y
axis.
Let us consider several doubly asymptotic solutions. For t which is negative and very large, they will all be very close to the periodic solution, and they will appear in a certain order -- some of them will be closer to, and others will be farther from, the periodic solution. All of them will then recede appreciably from the periodic solution, and -- for t which is positive and very large -- they will all again be very close to it. However_ they will then appear in an entirely different order. Out of two solutions, if the first is closer than the second to the periodic solution for t = -% it may happen that for t = +_ the first is farther away than the second from the periodic solution, but the opposite could also occur.
the all
We have pointed this out in order to illustrate the great complexity three-body problem, and to show how many different transcendents out those which we know must be considered in order to solve it.
Heteroclinous 399. As number
Do heteroclinous
far as we of them.
can
solutions
determine,
if
of of
Solutions exis_
there
is
one
of
them,
there
is an
infinite
383
Let M0 be a point belonging to a periodic system. Let M0rA 0 and M0B0 1392 be two asymptotic curves bordering upon this point M0 -- one belonging to the first family, and the other belonging to the second family. Wehave just seen how these curves intersect, so that the doubly asymptotic, homoclinous solutions maybe determined. Nowlet M_ be a point belonging to M'0A_, M'uB'0 be two asymptotic and M'B' 0 belongs to,he second Let will
us
assume
correspond
However, at an infinite I shall periodic
that
to
curves, family.
M'0A _
a doubly
another M'A'Q
intersects
MoB0
asymptotic,
periodic
belongs
at Q0.
heteroclinous
if these two curves intersect at Q0, number of points Qn, the consequents state
system
this
precisely.
of which
M 0 is
For
a part
example,
is
composed
solution.
to
the
This
Let
first
family,
intersection
solution. they will of Q0"
I shall of
also
assume
five
intersect
that
points
the
M 0, MI,
M2,
M 3, M4. Then the fifth consequent of an arbitrary point of the curve M0B 0 will still be located on this curve, and in general -- if Q0 is on this curve -- the same will hold true for its tn_-h consequent Qn, provided that n is
a multiple In
the
of same
five. way,
let
us
assume
that
the
is a part is composed of seven points. Then, the same will hold true for its tn-_hconsequent multiple
have
of
periodic
system
of which
M' 0
if Q0 is on the curve M'0_ 0, Qn, provided that n is a
7.
Therefore, if an intersection
the two curves have an intersection at Q , they at Qn, provided that n is a multiple of 35.
will
still
Let QoHoQn be an arc of MoB0, and let QoKoQn be an arc of M'0A'0. Due to the fact that these two arcs have the same end points, together they will form a closed curve. We may pursue the same line of reasoning as in No. 396 for this closed curve. We shall find that, if the two arcs have no other point in common except their end points, double point, and defines an area which Nos. 395 and 396. If the two arcs have end points, we may obtain QoHoQn, QoKoQn which have an area similar to s 0.
used
The for
same this
points
may
points
of This
384
be
line area
obtained
of
two other arcs which are part of the two arcs only their end points in common and which define
of reasoning as was employed s 0, and we will find that an
intersection line
this closed curve does not have a is similar to the area s 0 given in points in common other than their
on
each
with
reasoning
the
of
the
other
shows
that
two
in Nos. infinite
curves,
395 and number
between
two
396 may be of other
arbitrary
curve. if
there
is
one
heteroclinous
/393
solution,
there is an infinite
number of them.
400. If there is a heteroclinous solution, the grid of which we spoke in No. 397 must be still more complicated. Instead of a single curve M0A0 bending back upon itself without ever cutting across itself, and intersecting the other curve M0B0 an infinite number of times, we shall have two curves MoA0, M'0_0 which must intersect M0B0 an infinite number of times without ever cutting across each other. In No. 397, we defined the group E'0 with respect to the point M0 and to the asymptotic curves MoA0, M0B0. Wemay also define a similar group with respect to the point M'0 and to two asymptotic curves _0_0, M'0_0. If there is no heteroclinous solution, these two groups must be outside of each other; therefore, they cannot occupy the half-plane. If, on the contrary, there is a heteroclinous solution, these two groups will coincide. It may be seen that the existence of such a solution -- if it could be established -- would provide an argument against stability. In Chapter XIII we studied the series of Newcomband Lindstedt, and we showedin No. 149 that these series cannot converge for every value of the constants which they contain. However, one question remains in doubt. Could these series converge for certain values of these constants and_ for example, could it happen that the convergence occurs when the ratio n__ 1 is n2 the square root of a commensurablenumberwhich is not a perfect square (see Volume II, page 104, in fine). However, if a heteroclinous must be in the negative. nl ratio-the series of Newcomb n2 our method of representation. tion
solution does exist, Let us assume that for and
Lindstedt
converge,
the answer to certain values and
let
us
this quesof the
return
to
solutions of the differential equations nl which would correspond to this value of--could be represented by certain n2 trajectory curves. The group of these curves would form a surface, having the same connections as the torus, and this surface would intersect our half-plane proceeding along a certain closed curve C.
side
The
The group E'0 which we just of this curve, or completely
mentioned would inside of it.
have
to be
completely
/394
out-
Let M 0 and M' 0 be two points belonging to two different systems. If MQ is within the curve C and M'0 is outside of this curve, the group E'0 with respect to M 0 would have to be entirely within it, while the group E'0 with respect to M'0 would have to be entirely outside of it. These
two
groups
could
not
have
any
point
in
common,
and
no
doubly
385
k_J
asymptotic,
heteroclinous
If we have
just
number
admit
the
presented
of values
commensurable
--
solution
hypothesis --
could
advanced
exist,
proceeding
in Volume
II,
from
page
MQ
to M_
104,which
.
I
i.e.,
if the convergence occurs for an infinite nl of the ratio-for example, for those whose square is n2' there would be an infinite number of curves C which would
separate the points belonging to different periodic systems. This hypothesis is incompatible with the existence of heteroclinous solutions (at least if the two points M 0 and M'0 which we are considering, or the corresponding periodic solutions correspond to two different values of the number nl.)
'
_F2
Comparison 401. Before trying to return to the example
shall totic,
homoclinous We
with
present of No.
solutions
may
No.
225
examples of heteroclinous 225, where the existence
be
of
solutions, we doubly asymp-
illustrated.
set --F --p + q'-- _s;n'Z2 --_z _(y)cosz
(p, x;
q, y)
We powers
are
then
the
two
formulated
pairs the
of
conjugate
function
S
variables.
of
Jacobi,
and
Let
us
consider
the
second
term,
S,_'+
....
neglecting
e 2,
and
obtain
So= Ao_ + v'_fV/;+ or,
assigning
the
value
zero
to
the
_'n"r__+,
constants
A 0 and
So = ± _. vtf,_ cosY; and
we
then
obtain St=: real
where
386
it
in
/395
S = So'+ St¢. then
developed
of S = So+ S,E+
We
we
_ is
a function
of y
defined
by
the
part +eix, equation
h,
let
us write
Weset
and assuming that h = o,
we to
obtained the two
?(y)
(pages 464 and 465, asymptotic curves of
:: siny,
the
other
e
+
it-2=
f"t
is
t',
_'
is +'= I/'_
_he equations
t
2
Volume II) two values of _ corresponding the two families. One of these values
+= and
_ __
of
the
two
I + t tz
asymptotic d p = __
it-ta £t
surfaces
real
q=_/_-_sia
It_+ dt tt
will
then
be
[+e,'_,];
part +¢_-.fi
real
part
[_e"_];
and
/396 .d P = s _xx real
part
[+'e"_];
d
the
In order to intersection
obtain the doubly asymptotic solutions, we of these two asymptotic surfaces. It will
for
us
the
Let
to equate us
two
values
of
p and
the
two
values
must determine he sufficient
of q.
set J _-=f**
t==dt
u _ _ Iogt. We
shall
obtain
_d real dx
part
[Jie-_,,+"_]
d d-_ real
part
[Jte-="+t¢]=o,
= o,
387
or,
setting
J = pe im, rl x
where
K is
a whole
This
is
the
In reality, one corresponding values of K.
2
7_
_f_
t-(o
KT,-!-
-
number. equation
of
doubly
asymptotic
this equation provides to even values of K,
totic
402. We may be surprised solutions, when we know
finite
The following approximations number of doubly asymptotic
us and
solutions.
with two different solutions, the other corresponding to odd
at not obtaining that there is an
more than two doubly asympinfinite number of them.
should provide us with no more than a solutions. How may this paradox be
explaine_ In the preceding solutions correspond certain
sections we saw that in an infinite number
arc
AoHoC0
with
the
different
us
assume
that
the
first
Let
of
consequents its
If we
order
The of
majority
develop
it is arc the of
as
in
powers
though
AoHoC0 no other arc the
doubly
we
Let
us
try
e and
doubly asymptotic intersections of a
another which
arc
AoKoC 0.
encounters
AoHoC0
N will clearly depend on the conthe larger it will be. It will
stop
at an
e as being
arbitrary infinitely
term
in
the
asymptotic
to
ex-
small.
longer encounters the consequents of infinitely AoKoC0, and for this reason we have not analyzed
Examples 403.
of
regarded
of
consequents
is the consequent of order N. The number stant E, and the smaller the constant is, become infinite when e is zero.
pansion,
the different to different
large the
solutions.
of Heteroclinous
generalize,
and
let
Solutions us
set
F = Fo+_Ft.
These
F 0 is a function of p, q and two functions are periodic, Let
us
consider
the
y, and F I is a function both in x and y.
p,
q, x
and
y.
curves Fo = const.
388
of
(1)
/397
in which we regard p as dinates of a point. Out
of
these
a parameter,
curves,
those
and
which
q
must
and
y are
draw
our
regarded
as
attention
the
are
the
having double points. These double points correspond to periodic of the canonical equations when we assume that _ is zero and that reduced to F 0. We
have
a double
infinity
of
curves
(I) whose
general
coor-
ones
solutions F may be
equation
is
F 0 : : _.
and
which
depend
on
two
parameters
p and
h.
I have Just stated that the most interesting ones are those which have a double point, especially in the case in which some of these curves have two or more heteroclinous
and
double points. solutions.
Just as in No. let us set
The
function
225,
S O may
It
let
is
us
be
try
We
then
of y
this
case
we
the
immediately.
dSo defined
that
to formulate
formulated
dSo where q is a function meters p and h.
in
shall
encounter
function
S of
We
have
shall
/398
Jacobi,
fq by
equation
(i)
and
depending
on
two
para-
obtain dFo dSt dFo dSi ,_t7 i&- + 7/,/- @ + F, ::o.
(2)
dFo dF o in--' -and FI, and we replace q by its dp dq value obtained from equation (i). Equation (2) is therefore a linear equation with respect to the derivatives of SI, whose coefficients are the given functions of x and y, which depend in addition on the parameters h and p. We
regard
Since
p as
a constant
F 1 is periodic
in x,
I shall F I : :
where
_m In
only the
depends
same
way,
on
y,
just
I shall
like
set
v,I_nl
the
eimz,
derivatives
of
F 0.
set
389
and
the
function
_m will
be
given
by
dl"o
whose
coefficients This
equation
are
the
may
clearly
of
example
q correspond
presented
These to
each
two
given
at
as
in No.
analytical The
period
47 such
This
as
uniform
If we
225,
q
had
of
y
periodic
may
of
each
seems
--
also
quadratures. in this way. We must (i) has a double point. such that two real
(this
is what
functions
of
occurs
in
the
sin
will
double
for
example,
assume
y, which
for
that
become
equal
!399
y = Y0.
these
two
values
of q are
other.
to us
function
function
several
by
asymptotic surfaces p so that the curve these constants are
point
we
then
the
integrated
value
q are
double
extension
function
of y.
225).
of
the
(3)
functions
be
each
in No.
values
other
Just the
to
equation
dFo d'b,n
Let us try to determine our first choose the constants h and In addition, I shall assume that values
the
to be
uniform
in y
and
periodic
of
y2"
take
the
points,
same
instead
value of
for
one,
y = Y0
we
could
and
y = Y0 +
still
regard
q as a uniform function of y of period 4_, if the number of double points were odd. On the other hand, if this number were even, we would have two values for q which would not be interchanged when y increased by 27, and which could consequently be regarded as two different uniform functions of y,
having In
have
two As
root,
tion
2_ order
a result,
since
the
period.
to formulate
double
the
points for two
our
ideas
more
corresponding y = Y0
values
and
of
q
for
to
similar
to
the
clearly,
the
values
y = Yl,
coincide,
Equation (3) is a linear equation is similar to the integration of
consequently
390
for
integration
we
shall
Y0
and
Yl
assume
(i) must have dF 0 consequently_--must
a double vanish.
with a second term, whose integraan equation without a second term, of
the
following
we
of y.
equation
and
that
equation
and
2_.
dFo 0-_- dFo dO
@
o
(4)
a_ dy
from which we have ,if*" u
,iq- .
0= e J
The function 8 thus defined is a holomorphic function of y for all real values of this variable, except for the values y = Y0, Y = Yl, which correspond to the double points. For these values, the function e -- which plays a role infinite. We
then
similar
to
that
of
t =
tan
_
in No.
226
-- becomes
zero
or
obtain
/4OO +,,,.-_o,.,, / _ --#Fo o ,,,,'b,,j__ -I- C,,, 0t,,, J ap
where
Cm
is
an
integration
constant,
s,-:
In
order
to obtain
from
..... I
......
,J
@
equations
which
values stants
as
values
to the
of asymptotic
surfaces,
YO,
qo;
Yl,
ql
are
In order to represent four-dimensional space,
the
the
we
may
write
dS
integration
constants.
Let us first neglect ¢. We shall set S = SO, and corresponding to the curve which has two double h and p = PO.
With this approximation, periodic solutions
where
in
suitable
have
-. '=c,. o,',,,
dS
assigning
we
differential
we shall assign the points to the con-
equations
have
the
following
,o - Pc,
q :'qo,
Y ::Yo,
(5)
P '= Po,
q = q,,
.r ----x,,
(6)
coordinates
of
the
two
our asymptotic surfaces, whose coordinates are
double
points.
we
take
may
a point
391
where
a and
need
only
b are
Equations dimensional Two longing
two
consider
positive
constants
positive
(5) space,
values
and (6) then corresponding
asymptotic to the first
surfaces family,
p +
represent to the
pass and
a
The
equations
of
the
two closed periodic each of belonging
surfaces
P =p0, As
we
have
seen,
the
Y = Y0 and for y = Yl, and which are periodic equations
of
our
equation
F0=
asymptotic
that
curves solutions. these to
employed pairwise. will
of
this
we
four-
curves -the second --
one befamily.
i.e.,
neglecting /401
be
two
roots
which
coincide
interchanged when y increases 2_. Let q' and q" be these
surfaces
I p_Po, :_po,
enough
h.
F 0 = h has
which are not in y of period
large
q + b.
two
through the other
asymptotic
are
and
However, with the degree of approximation these four asymptotic surfaces coincide
E --
The
which
of
thus
by two
for 2_, roots.
become
qq _= q" q'.
(7)
In order to determine the significance of these equations more precisely, let us distinguish between the different layers of our surfaces. We have four asymptotic surfaces. Each of them passes through one of the curves (5) or (6), and is divided into two layers by this curve, which I shall
designate The
divided
surface into
The divided
The divided
392
two
two
surface into
With following
two
surface into
the
the
of
surface into
The divided
by
two
the
layers of
the
first
notation:
family
N I and
N'I •
second
family
layers
N 2 and
N'2.
of
first
the
family
layers
N 3 and
N' 3.
of
second
family
N4
N'4.
the
layers
degree
equation
following
of
and
approximation
passing
through
passing
passing
through
through
passing
employed,
through
these
the
the
the
the
curve
(5) will
curve
curve
(5) will
(6) will
curve
layers
will
(6) will
have
be
be
be
be
the
\ V
f
It with N 3 +
can
this N' S .
Let
N,;
P----po_
q =q',
Y>Yo;
N',;
P----po,
q :
N_;
P=po,
q=q",
y.t--yo;
N'2;
P=:po,
q=q',
Y
Na;
P=Po,
q:-:q',
Y>Yt;
N_;
p = Po,
q ----q',
Y
N_;
P=Po,
q=q',
Y>Yt;
N';
p
:pc,
q --: q',
y _..yt.
be
seen
degree
us
that
of
the
two
surfaces
approximation,
proceed
to
the
NI
Just
following
+
like
N' I
the
q',
and
two
approximation,
y'
N_
+
N' 4
surfaces
and
coincide
N 2 +
let
us
N' 2
and
/4O2
set
S=So-÷-sSt. In
order
to
For
the
functions to
the
this
define layers
Sm analysis
call
a
is
the
and
on
For
be
the
regular regular
which
for
are The
thus
q q
=
y
equations
and
N'2,
y
= Yl.
of
by our
However,
result, after
our the
In
we for
order
Yl
to
provide
We
the
shall
note
the
call
its
Cm's
Sl. 2
the N_
designate
thus
dS,.,, "P°-_-s--d'_-"
Cm's
N'4,
the
two
I
that
the
_m'S
that the
SI
the
_m'S
_m'S
must
functions
S 1
become
q=
q,
+_
dS,., dy
." '
p ::po-i-_
q=q'-+'*
P=po+_
dSjA., dx '
q=q'+¢
dSt._. dy
be
refer
that
function
so
and
dS,.s -_y ;
the
the
only
constants.
so the
dS,._ . dx '
an
that
need
understand
dSt.l. dy "
has to
to
q = q'-+" _
that
and
so
We
determined.
layers
dSt._ dx ;
_'" N_,
Y0.
Sl. 4.
surfaces
_-N_;
=
order
choose
P::I)o-_-*
cease is
shall
Sl. 3 and
in
choose shall
N_-:-N_:
= Y0,
equations
value
we
constants
Y
determining
P
should y
shall
and
q',
thus
N'-I-N'z;
N_.-t-
behavior
we
C m.
these
=
II,
is
For
four
q
Volume
we
= Yl-
Nj
lar
Y0,
q",
determined
choose
completely
N'2, =
constants
for
S I which
N 2
y
must
466, for
and
q',
we
the
behavior
function
N 2
layers for
N'I,
sufficient
layers
choose
page
are regular for q = q", which is thus determined.
are
must
regular
given
Sl. 1 the
For
we
NI
have
condition
shall
SI,
function
irregular
valid,
even
SI.I,
for
behavior as
a
(8)
example,
for
first
y
=
has Yl.
As
a
regua
approximation,
exceeded.
a better
illustration
of
this,
I
shall
confine
393
myself
to Let
the y'
Let value large
and
M0 be
remarks.
y" be
the
two
point
values
of
The
value
which
must
indefinitely In
y such
y0Cy'
L
our
general,
serve
as
be
when the
that
asymptotic
curve
the
assigned
to
n
to
zero.
first
are
periodic In
to
the
order
of N I and
of
for
y,>y>Yo:
N'l
et
N_:
yo>y>y:--_.
yo+_>y>Yi;
N_
et
N'6;
yt>y:>Yo.
(6)
to
n is y".
on
which
Nt;
e,
our
the
chosen
and
/403
it
equations
for
NI
and N'_ coincide, doubly asymptotic
solution
(5)
for
the intersection will solution which will be
t = -%
and
very
close
to
the
t = +_.
to determine
this
intersection,
let
us
compare
the
equations
N'4
intersection
will
dSt.i,
¢ ---_-x
clearly
,
p=po'4-*-'--_,
be
given
d(S,.,-St.,) dx
Sl.l - St.4 is a function negative
y
N:;
dSH
the
depends
ct
p =/_o+
and
that than
Na et
periodic
solution
clearly
values
corresponds
assume larger
approximation:
For example, if the surfaces correspond to a heteroclinous, close
the
which
I shall of y is
E strives
following
Nt
very
of
y'. Let R n be its n IK consequent. enough that the corresponding value
increases
may
following
whole
powers
of x
and
y,
by
(9)
=o.
which
may
be
developed
in
positive
and
of _ielx.
The fact that it is a periodic function of x is important fore has at least a maximum and a minimum. Equation (9) least two solutions, which means that there are at least solutions In ponding to the
394
• the
same
to the surfaces
The
to us. It theretherefore has at two heteroclinous
way,
it
could
intersections N2 and N' 3,
preceding
analysis
be
of and does
shown
the two not
that
there
surfaces N4 corresponding yield
the
are
two
and N'2, to the homocllnous
solutions
corres-
two corresponding surfaces N3 and N'I. solutions.
404.
For
example,
let
us
Fo
set
--p--ql+2H.
sin'Y--Y°sin_Y--Y' 2
2
FI ----F"cosx sin(y --Yo) sin(y
The periodic solutions strive for t = -_ and
It will
be
noted
(5) and (6) toward t = +_ are then
that,
= 0, the function and does not depend F therefore has the
for
x=t, x =:t,
p=q=o, p=q:=o,
_ = 0,
F may
quantities
two dx _
solutions and
dy dt
which
were
these
called
the
heteroclinous
solutions
(5) --
Y:=Y0, y=y_. be
reduced
to -p
- q2.
Therefore,
for
p and q, function
and
(6) both
correspond
to
the
same
value
of
the
i.e.
quantities
n I and
/404
not be content with this example, which proves having the form considered in No. 13 can have
_Ix di ::"
However,
which
F depends only on variables of the first series on variables of the second series x and y. The form considered in Nos. 13, 125, etc.
Nevertheless, we shall that the canonical equations heteroclinous solutions. The
--Yl).
dx _,
dy dr.... dy dt
are
o. nothing
else
than
the
numbers
n 2 above.
Therefore, we find that doubly asymptotic solutions exist, which come infinitely close to two different periodic solutions for t = -_ and t = 4_. However, these two periodic solutions correspond to the same values of the numbers n I and n 2.
with
Therefore, equations
I shall formulate another example, in which we shall having the same form as those presented up to No. 13,
which have doubly asymptotic solutions coming arbitrarily ic solutions which are not only different, but correspond nl of the ratio--. n2 Unfortunately, I would like to show that values of U which are close to i, but I still fact that they also exist for small values of 405.
We
shall
take
two
pairs
of
conjugate
deal and
close to two to different
these solutions am not able to _.
periodvalues
exist for establish the
variables
395
v
or
J405 xl, y,;
by
x,, ,ys,
setting b = I/_-_l cos,y,;
This
change
equations.
We
in
variables
shall
does
_t
=
not
%/_zi
alter
pendent
shall
assume
of
and
Yl
that
Y2,
F 0 is
the
and
that
function i x2 = _, we
--0_
_
assume
that
for
x 2
=
shall
assume
It have
F
that
follows =
The
F0,
a
from
our
first
<
i holds
these
xl
solution,
for
=-_"
will which
I
tx I --_ --,
solution,
-,
The =
value
01 of
In
first n2
=
corresponds i.
the
ratio
order
to
These
nl
=
may
t,
call
1,
n 2
O,
FI,
I
shall
----- I--
cosvo_
and
from
which
we
written
i --
written
_,=
= sint.
not
r
be
t,
the
do
_I----
0,
solutions.
YI=
set
r
=
v_l =o.
_cost, =
be
o' 1 may
solutions
_
-ylt=O,
yl=o,
_=
set
n-JI. n2 define
inde-
have
periodic
_=I,
--_ 2
=
we
_1
.y|
shall
periodic
_i
396
I
if
call
_----o, to
two
.x2,
a.
special
1 --,
Zl-----
_s=1.
nl
two
_, : = sint,
which
Zl-----
x I and
have
quantity
shall
X2=
we
that,
have
_1= =cost; second
of
dFo dx---? --o.
the
hypotheses
equations
_i
The
the
a2
_,
dFo dx-'-"_= -- "
I
of
_-_I._
1 also
form
dF0
d_,
shall
canonical
_F,.
a holomorphic a2 for x I = --, 2
dFo
We
y/.
set F = Fo(1-- _)+
We
sin
sinto,
second correspond
corresponds to
the
to same
V
assigning
a value
which
is essentially
positive
I shall then assume that (due to the small quantity) we have the following
very
FI
,
=
,
_ql -_-_._
a function the period
for
for
_ =
0 and
Since _2 = i -values, as well
in as
the
variable
fact that P is for r > 0
r.
1406
a positive,
_
of _, which is 27, and finally
r
(1)
2
regular for every real value of _, which vanishes with its derivative
_ = _ 2"
function
I shall
the
(r--,)_ _+_¢ ___ 4,(,_)
'2
where _(w) is periodic with
to
(i) would
assume
that
such a way that its derivatives
for
it of
be
infinite
r <_p,
for
the
r =
0 --
function
nevertheless remains the two first orders.
i.e.,
F 1 takes
finite
and
for
on
_i =
arbitrary
continuous,
It may be readily verified that for _ = i -- i.e., for F = F I -- our equations still have two periodic solutions a and o'. For the first of these solutions, we have _ = 0, and for the second we have _ = 2" It may be immediately concluded that have these two periodic solutions.
will
at
406. We least that If we
central hardly
The
Jacobi
assumed
that
method
leads
7 \dr h
is
e =
0, we
and the integration the general case.
, (4s], where
every
shall now integrate our equations r constantly remains > p).
first
forces, true in
for
a constant.
Let
to
would
would
the
be
partial
be
us
the
dealing
differential
--.f-
--_
of p,
case
immediately
, (<_s]-,-_,_(,._,_
+ 2;.i \
in
value
of
with
our
_ =
equations
i
(assuming
the problem
possible,
This
of is
equation
¢(,,,_ ,.,
----
-=z
]/i
set
' (i-'_(<'_)----D, where
k is
a second
constant,
s=d7
The
general
solution
of
and
h
our
we
).____
equations
shall
have
......
is
therefore
1407
397
k.7
d,_
')_,+(h
-Oq'_
f
v/_3"_ -'_z"-"_("
r dr
-,)L
(2)
h'-4- t,
(B)
_--z k',
¢'_k -_ _q, +_
,., v/_.-z,-'--- _.k -- _'_,'-,)'
where h' and k' are two new constants. We shall obtain our two periodic following
particular
values
solutions
o and o', assigning
the
to the constants 5 2
k =o,
h= --, 2
,_.'_-,_ =o,
k==o,
h=
k'
--, '2
_-"
2
Let us assume that we would like to employ equation (2) to define r as a function of h' + t. If we assign values which are close to zero and _2 -to the constants 2 We shall set
k and h, r will then be a periodic
u --n[t
function
of t + h'
-+ It'),
where the number n is chosen in such a way that r is a periodic function of u with the period 2_. This number n, which is a type of mean motion, will naturally
depend on the constants
h and k.
dr
In the same way, _
will be a periodic
function
of u.
For k = 0, we simply have
r=, 407.
We therefore
have two periodic
solutions
o and o' which
will be
represented by two closed curves, if we may regard the _'s and the n's as the coordinates of a point in four-dimensional space. Two asymptotic surfaces pass through each of these curves -- one belonging to the first family, the other belonging to the second family. We shall see that the /40____8 four surfaces coincide pairwise, as occurred in No. 403 (equation 7), when E is neglected. In order to obtain
398
the equations
of these surfaces,
it is sufficient
to
_2 assign
the
values
zero
and
_-
.'(_!-
These seen sign
the
shall
= 7'
is
We
obtain two radical.
assume
that
positive
We shall values of
I)NI
equations
that we only of the second
We
for
are
to
for
now try _ which
the
k
and
q-(_,-
h. I)_,
asymptotic
of
these
other
thus
_---"
surfaces
e_
the i.
have
( r --I)',
surfaces,
value
to formulate are close to
r
=
of
function
every
We
which of
for
_ =
i.
corresponding
vanishes
It to
for
m
may
the
=
0
be
double
and
m.
equations
of
asymptotic
surfaces
have F = F, _-(j-- M)(F0 _. F,);
where
F 0 and
sequently
of
The
F 1 are r,
_,
equations
holomorphic
functions
dr _
and
d_ _.
of
our
surfaces
may
of
be
the
_'s
and
the
n's
and
con-
written dS dS
(_,--')_,--(_,-')n, = _, where tion
S
is
a
function
of
r
and
m,
satisfying
F
where
we
Let
have
us
dr dm ---- and--by--dt dt
replaced
develop
S
in
powers
we
shall
have,
as
the
first
--
l)!r_l-F-
i -
....
_)S,+(,--S),$,÷
/409
t,)_2--'--
F_
dSI -¢-(I-
dSo
for
the
equations
of
our
asymptotic
equa-
dS -r 2 d_"
approximation,
(_.2-
differential
i and--
dr
dSo (_t
partial
¢0n$1._
dS
of
S = So+(,-and
=:
the
_)r
"_F
'
dSi
surfaces.
399
We
have
already
obtained
dS_2= * ¢_ ,_
dr We
must
now
determine
doJ
S I.
For
this
purpose,
dS0 dS, I dS0 dS, -'-'7 r/I --7 dl + r' de _
In and
by
and
_.
the , fi
The
second
dSo ,_,_....
*= _/_ /7-
equation
dr _
term, This
•
dm and_must second
be term
--
'
we
have
equation
--Ft--Fo.
replaced
is
the
--
by
therefore
a
dSOdr =
known
_2
(r
-
function
1) 2
-
of
r
becomes
"' v'_' -- (" - ')' dS, _/ __+_/_-@ dS, _ = ,"(F,-- Fo). Let
us
set
f
It may and we
may
Our
be
seen
regard
equation
that
S as a then
r and
_2
function
dr
_
(r -
of v and
respect
No.
second
Just
to
400
i.
be
employed
corresponding
coincide,
of v,
is periodic
with
becomes
is
a known
function
,a(F,--Fo.
of v and
This equation thus has exactly the same form 403, where v plays the role of x, and m plays
be
functions
m, which
to v.
It will will
term
periodic
_.
dSt __ _-_+ d---f-_ dSt -3-_=
The
1) 2 are
as and
to
handled to
four
in No.
in
the
determine
same the
asymptotic
403,
it will
consequently
However, this has only I do not know whether
way.
four
The
as equation the role of
procedures
functions
SI.
1,
(2) given y.
presented SI.2,
Sl.3,
in
in No.
403
Sl.q
surfaces. be
found
heteroclinous
that
these
solutions
been established for this is still valid
asymptotic
surfaces
/410
exist. values of B which are close for small values of _.
The result is therefore incomplete. However, I hope that the reader will pardon the length of this digression, because the question which I have posed, rather than solved, seemsto be directly related to the question of stability, as I indicated in No. 400.
ENDOFTHETHIRDANDLASTVOLUME
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401
-,,...I
11f _f
7-
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_if
f
I
i,,_SJ
