JOSEPH EDMUND WRIGHT
OF
JOSEPH EDMUND WRIGHT
DOVER PUBLICATIONS, INC~ MINEOLA, NEW YORK
Bibliographical Note This Dover edition, first published in 2013, is an unabridged republication of the work originally published by Hafner Publishing Co., New York, in 1960. The 1960 edition was itself a republication of the work originally published as Volume 9 in the series, "Cambridge Tracts in Mathematics and Mathematical Physics" by Cambridge University Press, in 1908.
Library of Congress Cataloging-in-Publication Data Wright, Joseph Edmund. Invariants of quadratic differential forms I Joseph Edmund Wright. p. em. "An unabridged republication of the work originally published by Hafner Publishing Co., New York, in 1960. The 1960 edition was itself a republication of the work originally published as Volume 9 in the series, 'Cambridge Tracts in Mathematics and Mathematical Physics' by Cambridge University Press, in 1908." Summary: "This classic monograph by a mathematician affiliated with Trinity College, Cambridge, offers a brief account of the invariant theory connected with a single quadratic differential form. A historical overview is followed by considerations of the methods of Christoffel and Lie as well as Maschke's symbolic method and explorations of geometrical and dynamical methods. 1960 edition"Provided by publisher. ISBN-13: 978-0-486-49768-6 (pbk.) ISBN-10: 0-486-49768-2 (pbk.) 1. Differential forms. I. Title. QA381.W8 2013 515'.37-dc23 2012046797 Manufactured in the United States by Courier Corporation 49768201 2013 www.doverpublications.com
PREFACE
THE aim of this tract is to give, as
far as is possible in so short a.
book, an account of the invariant theory connected with a single quadratic differential form. "It is intended to give a bird's eye view of the field to those as yet unacquainted with the subject, and consequently I have endeavoured to keep it free from all analysis not absolutely necessary. It will be found that the rest of the tract is independent of Chapters III and IV. These chapters are included so as to give an account, as far as possible complete, of the various methods that have been applied to the subject. The most successful method is that outlined in the remainder of the book. This method, begun by Christoffel, owes its modern development mainly to Ricci and LeviCivita, and it is hoped that this tract may induce some of its readers to turn to their papers. J. EDMUND WRIGHT. Fellow of Trinity College, Cambridge Associate Professor of Mathematics, Bryn Mawr College, U.S.A. TRINITY COLLEGE.
J'Uly 1908.
CONTENTS PREFACE
•
•
•
•
INTRODUCTION
CHAPTER
•
•
I.
Historical
II.
The Method of Christoffel .
•
•
III. The Method of Lie
•
v.
Applications : Geometrical . Dynamical •
•
•
•
•
•
•
1
•
•
•
•
5
•
•
•
•
9
•
•
•
•
29
•
•
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44
•
•
•
•
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61
•
•
•
•
IV. Maschke's Symbolic Method
•
...
PAGE 111
80
INVARIANTS OF QUADRATIC DIFFERENTIAL FORMS INTRODUCTION
1. IN order to discuss in detail the geometry of a plane it is convenient to introduce coordinates. A point in the plane has two
degrees of freedom, and therefore to determine it two independent conditions must be satisfied. These conditions may be that two independent quantities• (e.g. the distances from two fixed points) take the values u, v at the point, and we then say that the coordinates of that point are u, v. If we suppose one coordinate given, the locus of the point will be a certain curvec, e.g. u = const., and, generally, a curve in the plane is given by a functional relation
f/l (u, v) = 0. For the metrical geometry of the plane we need an expression for the distance between any two points in terms of the coordinates of the points. Theoretically this can be calculated if the distance between an arbitrary point (u, v) and any neighbouring point (u+du, v+dv) is known. Suppose that (tD, g) and ($ + d$, y + dy) are rectangular cartesian coordinates of the two points, then :D, y are functions of u, v ; also if ds denote the distance between the two points, ds' =d:rr + dy2, or tJsl =Edu2 + 2Fdudv + Gdv 2, where
-~~ ~~ E -au + au '
F=~~ ay ay G=~~ ~~ a~u av +au av' av + &UI •
We have in fact a quadratic form in the variables du, d'D, for IY, and the coefficients of this form are functions of u, v. If E, F, and G are * These quantities have not necessarily any obvious geometrical significance. 1
2
INTRODUCTION
given it is possible to determine the equations of the straight lines of the plane in terms of u, v, to find the angle between any two of its curves, and, generally, to develope its metrical geometry. Now if the system of coordinates is given, E, F, G can be determined, and the, converse question at once arises, namely, if three functions E, F, G are given as three arbitrary functions of u, v, is it possible to take u, v as coordinates of the points in the plane so that the element of length shall be given by dr = Edu2 + 2Fdudv + Gdv' ? It appears that this is not possible unless a certain relation 1
K=. 2../EG-P
{ a[
.F
aE
1
ao]
i; E../EG-F2 a;·- ../EG-P" &;,
2 oF_ _ 1 BE_ F BE]} =o av ,JEG-F2 ~u JEG-F 2 &v EJEG-F1 ti;J is satisfied for all values of u, v. This condition is also sufficient. If, instead of limiting ourselves to a plane, we consider any surface in three dimensional space, we have again two coordinates for any point ; the element of length is given as before by tlr =Edu2 + 2Fdudv + Gd'IJ, and there is a surface corresponding to any arbitrary functions E, F, G of a, v. (The particular case EG =F 9 is excluded.) It appears however that for a given surface the expression K has the same value at any given point on it, whatever coordinates u, v are chosen. Let u, v and u', 'II denote any two sets of point coordinates on the surface, then u'=f(u, v), v'=~(u, v), where /and~ are arbitrary functions ofu, v, and we have the theorem : If by ang tranllformatim~ u' =I (u, v), v' = q, (u, v), Edu9 + 2Fdudv + Gdvs becomes E'du'1 + 2F'du' dv' + G'dv'1, tlum K =K', wluwe K' is Kin tke accented 'IXJiriables. + !_ [
!a. Deflnition of a differential invariant. Any function of E, F, G and their derivatives satisfying this condition is called a di.ffwential in'INWiant of the form Edu9 + 2Fdudv + Gd'V'. The idea of a differential invariant may be extended by taking account of any families of curves on the surface, say ~ (u, v) = oonst.,
1, 2]
DEFINITION OF A DIFFERENTIAL INVARIANT
3
t/1 (u, v) =const., etc. When we transform to new variables u', v' we have Edu'+ 2Fdudv+ Gdv=E'du'1 + 2F'du'd'D' + G'dv'2, • (u, tJ) =~, (u', v'), Y, (u, v) =t/1' (u', v'), etc., and a differential invariant is defined as a function of u, v, E, F, G, 4>, y,, and their derivatives (u, v being regarded as independent variables) that has the same value whether written in the original or in the transforined variables. Invariants which involve only u, v, E, F, G and their derivatives are called Gaussian invariants, while those which involve also deri.. vatives of q,, t/1, etc. are called differential parameters. Thus for example K is a Gaussian invariant and _
1
{
aij;i ~
of/l a.p
a;;; '}
,1q, = EG-FJ Bp;u -2Fa; iii+ Gav
is a differential parameter of the quadratic differential form Edu1 + 2Fdudv + Gdv. If the quadratic form is interpreted as the square of the element of length of a surface in space, K and A~ have also geometrical interpretations. K is the Gaussian or total curvature of the surface, and if 4~ = 1, the curves const. are the orthogonal trajectories of a fam~ly of geodesics on the surface. The extension of these ideas from two to m variables is immediate, and the quadratic differe11tial form in m variables may be regarded as the square of the element of length in the most general m dimensional manifold. The main point is that invariants are independent of the particular choice of coordinates, in other words they are intrinsically connected with the manifold itself. The course of ideas is as follows. We start with a given manifold, which possesHes certain properties. Some of these may be independent of each other, some may be consequences of certain others, and there are relations connecting these. We may develope the discussion on the lines of pure geometry, but we are compelled, sooner or later, to appeal to algebraic methods. These · methods involve the introduction of coordinates, and properties of the manifold are then expressed by means of algebraic equations. An algebraic expression has some interpretation in the manifold taken . together with the coordinate frame used, and a complication has been introduced, for the discussion will now involve those additional properties which are not intrinsic to the manifold, but arise out of the
"'=
4
INTRODUCTION
particular coordinate frame chosen. If however we work only with invariants, we avoid this latter class of properties and are able at the same time to use the powerful methods of analysis. The geometry of the manifold thus breaks up into two parts : (i) The determination of all invariants and all relations connecting them. (ii) The geometrical interpretation of all these invariants in the manifold. S. So far there have been considered only invariants arising through the quadratic form that is equal to M. These are all the invariants when we consider the manifold in itself, but if we suppose it existing in, say, Euclidean space of higher dimensions we introduce other invariants connected with the relation of that space to the manifold. For example, in the case of a surface in space, the totality of invariants is only given when two quadratic forms are taken account of, the additional one being that which determines the normal curvature at any point of the surface. The surface, in fact, is not intriusically determinate by means of the single form, but may be bent provided there is no tearing or stretching, or as we say, it may be deformed, without alteration to diP. (For instance any developable may be defo1·med into a plane.) The discussion when there are two or more forms is similar to that when there is only one. The invariants arising from the single form are called deformation invariants. Thus far it is suggested that the invariants are essentially connected with differential geometry. This is by no means the case. They are connected with a certain form, and any interpretation of this form leads to a corresponding interpretation for the invariants. Consider in fact any dynamical configuration with Lagrange coordinates ~, ~, ... , u,.,. The kinetic energy of this system is
l" f",l=l
ar,uru,, where ara is a function of the variables u,
and dots denote
derivatives with regard to the time. By a new choice of coordinates we effect a transformation of exactly the same type as that already considered, and again we have a series of invariants of a quadratic form, and these are those quantities which are dependent on the configuration itself as distinct from the particular system of coordinates.
CHAPTER I HISTORICAL
4. Group. Invariance necessarily carries with it the idea of a transformation. Suppose we have a set of transformations in any variables whatever, and suppose that each of the set leaves a 'certain function of these variables invariant, then any transformation compounded of two or more of the set will also lee,ve that function invariant. If any such transformation as this is. not one of the original set we add it to that set, and we may thus continue adding new transformations until we reach a closed set, that is one such that if you apply in turn any two of its transformations the result is another of its transformations. Such a set is called a GROUP, and it is clear that any invariant whatever is invariant under a group of transformations.
6. In the case considered in the preceding pages there are a certain number of quadratic differential forms l"' a,.,d:crd:c,, '1',
•=1
together with a certain number of functions ~(a?., ... , :c.), and the group of transformations :c~ =:c~, ('!Jl, ••• , y.), (i =1, ... , n), and we suppose that under a member of this group l"' ar,dterdm, becomes
"· •=1
ft.
~
..,.=1
a'r,d!lrdg,, and that
~
becomes ~'.
Then there are deducible
rela.tions for a'r, ~', and their various derivatives with respect to the y's, and for dm1 , ••• , d:c,. in terms of the original magnitudes ar,, ~, etc. In other words there exists a set of transformations for all the variables mentioned. It may be proved that this set is a gToup, and this group is said to be emteruled from the original group. o~r problem is the determination of all the invariants of this extended
grot:p.
6
HISTORICAL
6.
(CH. I
Christoffel.
There have been three main methods of attack. The first, his· torically, is by comparison of the original and transformed forms, and in this way invariants are obtained by direct processes. The fundamental work in this direction is due to Christoffel* (1869), though the first example of an invariant, the quantity K, was given by Gausst in 1827. Invariants which involve the derivatives of the functions are called differential parameters. Lame t, using the linear element in space given by d~ =dar+ dy2 + dz2, gave this name to the two invariants
{Al~)s =(~Y + (~Y + ~)' -
a2~
at~
As~ =a:r + iii
a2q, + a.z2 ,
and Beltrami§ adopted it for the invariants that he discovered, those involving first and second derivatives of a function ., taken with a form in two variables. lr1 the course of Christoffel's work there arise certain functions (i/crs); these were originally found by Riemann in 1861 in his investigations on the curvature of hypersurfaces. For a surface in space they reduce to the one quantity K. 7. Ricci and Levi-Oivita. To Christoffel is due a method whereby from invariants involving derivatives of the fundamental form and of the functions ~ may be derived invariants involving higher derivatives. This process has been called by Ricci and Levi-Civita C()'IXJlriant derivation, and they have made it the base of their researches in this subject. These researches have been collected and given by them in complete form in the Matlt6matiscluJ Annalen II, and on their work they have based & calculus which they call Absolute differential calculus. They give & complete solution of the problem, and show that in order to determine all differential invariants of order p, it is sufficient to determine the algebraic invariants of the system : (1) The fundamental differential quantic, * Crtlle, Vol. 70 (1869) p. 48. t Duqui1itione1 gerurale• cif"ca 1uperjlcie• cu"'a1. ~ Ll~m
I of
1ur le1 coordonf&le1 cunJiligr&el (1859). Darboux, TMorie glnerale de1 I'Urjace•, Vol. m. pp. 198 Iff· gives an
Beltrami~&
work together with a bibliography. II Math. Ann. Vol, 54 (1901) pp. 12G •qq.
account
6-9]
HISTORICAL
7
(2) The covariant derivatives of the arbitrary functions ~ up to the order p., (8) A certain quadrilinear form G4 and its covariant derivatives up to the order p, - 2 •.
8. Lie. 'rhe second method is founded on the theory of groups of Lie, and is a direct application of the theory given in his paper Ueber Di.fferentialinvarianten t. This theory involves the use of infinitesimal transformations, and the invariants are obtained as solutions of a complete system of linear partial differential equations. Our problem is discussed shortly by Lie t himself, for the case n = 2. ~orawski § considered this case in detail, and gave the invariants of orders one and two. He also treated the question of the number of functionally independent invariants of any order. C. N. Haskins II has determined the number of functionally independent invariants of any order. Forsyth~ has obtained the invariants of orders one two and three for a quadratic form in three variables, and of genus zero, that is to say, for ordinary Euclidean space. He has also obtained the invariants of the first three orders for any surface in space ••, that is, for two quadratic forms in two variables, one perfectly general, and the other connected with it by certain differential relations. The problem for the differential parameters has been solved by this method by J. E. Wright tt.
9. :Maschke. The third method is due to Maschket:, who has introduced a symbolism similar to that for algebraic invariants. He developes processes similar to that of transvection, whereby an endless series of invariants may be constructed. * Zoe. cit. p. 162. t Math. Ann. Vol. 24: (1884:) pp. 1;87 sqq. :1: Zoe. cit. § Ueber Biegungsinvarianten in Acta Math. Vol. :&VI (1892-1898) pp. 1-64:. 11 Tram. Amer. J'lath. Soc. Vol. 111 (1902) pp. 71, 91; also ib. Vol. v. (1904) pp. 167' 192. 1T Phil. Trans. Series A, Vol. 202 (1908) pp. 277-883. ** Phil. Trans. Series A, Vol. 201 (1908) pp. 829-402. tt Amer. J'oum. of Math. Vol. uv11 (1905) pp. 828-84:2. ~ Trans • ..4.mer. Math. Soc. Vol. I (1900) pp. 197, 204:; and Vol. IV (1908) pp. 4:45-469.
8
(cH. I
HISTORICAL
The geometrical interpretation of the invariants has been discussed at length by Forsyth*, and a considerable part of the work of Ricci and Levi-Civita deals with geometrical applications. A general account of the whole subject was given by Maschke t at the StLouis Exposition, 1904. An account will now be given of these three methods. * See his two papers already quoted, and Rendiconti del Oircolo Matematico di Palermo, Vol. 21 (1906) pp. 115-125.
t
Bt Louis Congress of Art• and Sciences, Vol.
1.
pp. 519, 580.
CHAPTER II THE METHOD OF CHRISTOFFEL
10. The quadratic form in two variables. Let there be two quadratic forms F 5 adal' + 2bd/l)dy + cdy?. and F'sAdXS+2BdXdY+OdY2, and suppose that :c, '!I may be ex .. pressed as functions of X, Y so that when these values are substituted in Fit becomes F'. We have then ada:' + 2bdf.Cd'!J + cdy2 = AdX2 + 2BdXd Y + Od Y 1•
In this we write d:r =
[i.ax + :;. d' Y, with a similar expression for d'y,
and the equation takes the form Pd.XS + 2QdXdY + Rd¥2= 0 where P, Q, R are certain functions of :c. y, X, Y and the derivatives of te, '!/ with regard to X and Y. Now X, Y are independent variables and therefore there exists no relation among the differentials dX, d Y, and hence P, Q, R are all zero. Thus the necessary and sufficient conditions in order that F shall be transformable into .F' are P =0, Q = 0, R = 0, or written at length.
a(:~y +2b :~a~+c (a~Y =A, aa- a:c
( o:c ay
0.11
oy )
a11 a9
aa:;ray +b axay+ayax: +c a;tay=B, ote \2 o:c ey ( ay )51 a ( BYJ + 2b ay ay+ c ay = o. These three are differential eq nations of the first order for m, y as functions of X, Y. If they can be solved their solution gives the transformation whereby F is changed into F'. Now there are three equations, and they involve only two dependent variables tc, g; hence they cannot in general co-exist unless there be relations between
10
THE METHOD OF CHRISTOFFEL
(CH. II
a, b, c, A, B, 0 and their derivatives. Our first problem is to find the conditions in order that they may co-exist. By differentiation we obtain six equations in the six second derivatives of cc, g, and these may be solved for the second derivatives in question. If the original three are differentiated twice there are obtained nine equations involving third derivatives, and by means of the equations for second derivatives these may be reduced to a form in which they involve first and third derivatives only. There are only eight third derivatives of two functions z, y, each of two variables X, Y, and therefore by eliminating them from this last set of equations we get a new equation, which, since it involves first derivatives only, must be added to the original three equations. It happens that from these four equations the first derivatives can be eliminated and tl1us there is given a relation between a, b, c, A, B, 0 and their first and second derivatives. This relation is precisely K = K'. We can now proceed step by step to find the equations involving higher derivatives of :e, y, and then by elimination to find other relations among the coefficients a, b, c, A, B, 0 and their derivatives. In the case considered, that of two independent variables, these relations all follow from the equivalence of K and K'.
11. The quadratic form ln n variables. The general quadratic in n variables may be treated in exactly the same manner ; the statement of the work is much simplified by the use of cert&in abbreviations which we proceed to define. The form F itself is written I o,.,dm,.d~E, and the form F' is ~ a',.,dy,.dy.,
"· ,
'"·,
the summation being always from 1 to n for each of the
letters under the sign of summation. The y's are taken as the independent variables, and the :e's are assumed to be functions of these. The determinant of n rows and columns, whose elements are the quantities ar,, is called a. The cofactor of the t-th row and sth column in a is written 4,.,. The quantity
! [-aa a,t+a.!. a~-a-~ agAJ is written [gk, k], '/IJA 'IIJg 'IIJt and l [il, k] ~rt/a is wri~n {il, r}. k
AlKlfa is a'") (seep. 20).
[gk, k] and {gk, k} are called Christoffel's three-index symbols of the first and second kinds respectively. The expression [gk, k]- a_!_ [gi, k] + l a_!_ ~, ~1& JJ
({gi,p} [kk, .\]- {gk,p} [ik,p])
10-13]
11
THE CHRISTOFFEL SYMBOLS
is written (glclti) and is called sometimes Christoffel's four-index symbol and sometimes a Riemann symboL In the case of a quadratic form in two variables, a2K is the only Riemann symbol. These symbols were discovered by Riemann, independently of Christoffel, in his researches on the generalisation of curvature for manifolds of '' dimensions. It is clear from the definition that [gh, k] =[ltg, k], and therefore that also {gk, k} = {kg, k}. Also, the three-index symbols of the first kind are linear functions of the first derivatives of' the coefficients of F: conversely these first derivatives may be expressed linearly in terms of the symbols. We have in fact aaaik = ~~
[il, k] + [kl, i].
It is further to be noticed that just as the symbols of the second kind are expressed as linear functions of those of the first kind, so also those of- the first kind may be expressed linearly in tkrms of those of the seQond kind. The typical equation is [ik, l] =~ az., {ik, v}. I'
19. The Riemann symbols are not all linearly independent.
We
deduce from the definition that there are atnong them the relations (gkik) =- (gkki), (kgki) =- (gkki), (iltlcg) = (glcki), (kigk) =(glcki), (gkki) + (gkik) + (gikk) = 0, and it readily follows that there are only /yn2 (n2-l) independent
Riemann symbols. 18. We now pass on to consider the case of two general quadratic forms F and F'. Symbols derived from the form F' are distinguished by means of accents. If F can be transformed into F' we have the relation F = F'. This is equivalent to jn (n + 1) differential equations of the first order for the :c's as f11nctions of the g's, of which a typical • one 1s "~ a... a.v.. a~. , () ~~ a = a a.fJ • • • • • • • • • • • • • • . • • • • • • • • • 1 . r, B v9a. '!J• From these by differentiation are obtained ln51 (n + 1) equations for
second derivatives. Suppose these to be solved for the second derivatives : we get a set of equations of the type tP 3:r " { •L } am, om, ~ { R \.}' Oler ~~ ~~ + ol/lif tAl, ,. ; - :i:" = a~'~, A a ............(2). .i(f
"I! a. v9-
i, k
cJ!}a. u8fJ
A
'!/A
12
THE METHOD OF CHRISTOFFEL
(CH. II
As in the particular case first considered, the number of equations of this type is exactly equal to the nun1ber of second derivatives. When, however, the set (1) is differentiated twice we obtain in1 (n + 1)1 equations involving third derivatives. There are only tn2 (n + 1) (n + 2) third derivatives of n functions of n variables, and thus by elimination are obtained !n2 (n + 1)2 --ln2 (n + 1) (n + 2)=T\n2 (n2 -l) new equations not involving third derivatives. These may be reduced, by means of the equations (2), to a set involving first derivatives only. Similarly from the equations for fourth derivatives may be deduced a new set of equations in first derivatives only, and so on for all higher derivatives. We call the set obtained from the equations involving 1·th derivatives the (r -l)th set. The exception to this is the set (1), which is the first set. This is correct because there are no equations of the first order arising from the equations involving second derivatives. The number of equations in the (t· -1) th set is
(n+r-1)! 1 !n (r- ) (n- 2) ! (r + 1) ! · The second set may be got directly from the equations of type (2). If we call the one given (o.{j), then
a
ag; (a,B) =0,
a Uufl (ay) =0,
a a and hence a,i,. (a,B)- ay;(ay) = 0.
This last equation does not involve third derivatives and may be reduced by means of the equations (2) so as to be of the first order. The totality of new first order equations thus obtained are, after some algebraic modifications, reduced to the forn1
(ao~/3y )' =
~ ~
g,h, i,k
7~ 1 .) ozg (u~lt't ;kl "':fa.
oQJA
0.1:-t oa:k ( ) a ~~ . . . . . . . . . . . . 3' "'9~ '!Jy "'liB ~
where a, {J, y, 8 take all values from 1 to n. These constitute the second set. As the number of linearly independent Riemann symbols is f?J:n2 (n2 - 1), the number of equations in this set is also -/-,;n2 (n' -1), as it should be. A simplification may now be introduced into the calculation of the remaining sets, for it may be seen without much difficulty that the third set may be obtained by differentiating the second set and eliminating the second derivatives involved by means of the equations of type (2), and in general the rth set may be obtained from the (r-l)th in exactly the same manner.
13-15]
THE QUADRILINEAR FORM
13
G4
14. The qua.drllinear form G4• We notice that the equations (1) and (8) are similar in form, and just as the former are the conditions for the equivalence of two quadratic differential forms, so the latter may be regarded as the conditions for the equivalence of two differential forms of the fourth order. Let there be four sets of differentials d<1>y, d<2>y, d<8>g, d<4)'!J of the variables y, and let the corresponding differentials of the variables :c be d'1>$, d<2>x, d'8>:c, d<4>x. Then if G4 l (glclti) d(1)tcgd(2):ekd(s):ehd(4)m,,
=
g,k, h, i
the relations (3) are equivalent to the single equation G4' = G4 • In this case we are compelled to take four different sets of differentials, for all the equations of (8) could not be obtained from a form in which any pair of the four were made equal to each other. In particular, for example, the form l (g/clti) d:c0 dtckdmhd:xi vanishes identically. Thus the second set of equations are the conditions for the equivalence of two quadrilinear forms. It will appear that this result is general, in other words that the rth set arises from the equivalence of two (r + 2)ply linear differential forms. 15. In fact let Gp. be any J.t-ply linear differential form, and Gp.' its transfonned. If the general coefficient of GP. is (i1 i2 .... ilL) then the equivalence of Gp. and Gp.' leads to a set of relations of type
(a.1 a.s • • • a.p. ), =
~ ~
ib ..., ip.
(. .
• ) ax, ami oya.l oytJ.a
a:x,IJ. ( ) ••• ••• ••• 4 oyD.p.
Zt Z2 • • • ~~~- --L - . ! • • • • · • -
which are of the same form as (3). Differentiate (4) with respect to '!/«and substitute from (2) for the second derivatives of the :c's. Mter a little reduction the equation becomes of exactly the same form as (4) except that 1-' is changed into 1-' + 1. The new equations obtained may thus be regarded as conditions for the equivalence of two (p. + l)ply linear forms G~J-+1 and G'~J.+l The relations which connect the coefficients of Gp.+l with those of Gp. are of type (i1 is ... i,.,. i)
=~ (i1is ... i,.)- ;[ti~, A}(~ ... i,.) + {iig, A}(i1Ai8... i,.)+ ..
J. (5)
where there are 1-' terms in the square bracket on the right, A replacing each of the letters ih is, ... , ip. in turn. A fonn such as G4 is called a covariant form of the original form
14
THE METHOD OF CHRISTOFFEL
(CH. II
F, and we now see that from any covariant form of order p. there may be derived another covariant form of order p. + 1. Further, the equivalence of the two quadratic forms F and F' leads to the equivalence of a series of covariant forms G., G,, ... derived from F, with the corresponding sequence derived from F'.
16. Sufficiency of the conditions obtained for the equi· valence of two forms. Now suppose that for given initial values of the variables y we can find initial values of the variables :c and of the first derivatives :
which make these two sequences equivalent and also make F=F'. Then from the equations (2) we can find uniquely the initial values of the second derivatives, and similarly we can find the values of the third and higher derivatives. There are no contradictions, since in addition to the relations given by the equivalence of the two sequences F, G4 , G5 , ••• and F', G4', G5', ••• there are only enough equations exactly to determine all the higher derivatives. Hence we can find for each of the x's its initial value and the initial values of all its derivatives for a given set of initial values of the y's, and it follows that the differential equations (1) can be formally satisfied, and there-
fore that the transformation of F into F' is possible. We thus have the important result: The 'necessary and sufficient conditions i1~ O'rder tkat it sltall be possible to traruiform a quadratic jor1n F i'flto arwtlter quadratic form F are tkat tke equations in tke variables :c, g,
~,
derived from tM
equivakl~ce
of t/i,e two sequences F, G., G,, ... and F', G.', G,', ·~· sltall be algebraically compatible. We shall now prove this result in another way, and incidentally
show how the finite equations of the transformation may be obtained. Let the quantities :;: be denoted by u..', and consider a linear trans-
formation between two sets of variables X and Y given by the scheme
"
~=~uCLiY". •=1
(i=l, ... ,n) •................. (6).
Now suppose that different sets of variables Y(l), Y<2), etc. are taken and let the variables X obtained from them by the above transfor· mation be denoted by X(t), X<2), etc. Let G4 denote 2 (gkki) xg<1, xk<2) xh<s, xi<•,,
15-17]
FINI'J.'ENESS OF THE CONDITIONS FOB EQUIVALENCE
15
then G4' is the corresponding expression in the variables Y. It is now 1... .... · · d .In t he quant1t1es ·• :c, y, am c1ear tu~t t he equations obtatne BY, are precisely the conditions that the sequence F, G4 , G,, ... may transform algebraically into the sequence F', G4', G6', • • • by means of the scheme (6). These are necessary conditions for the equivalence of the differential forms l!"' and F', but they may not be sufficient, for it may happen that the coefficients u of a possible transformation obtailled cannot be regarded as first derivatives of n functions m with respect to n functions '!/· For example consider d:e12 +dx22 =dy12 +dy22• In this case 0 4 , Or, etc., and the corresponding forms, vanish identically. Hence the two sequences can be transformed into each other by any linear transformation Xt == Ut Yi +u2 Y2, X2 = 111 Y,. +v21'2 which transforms X12+X21 into Y12+ Y22. Such a transformation is given by u1 ==cos 8, 'tt2 =sin 8, v1 =sin 8, v2 == -cos 8, where 8 is any angle whatever. But if the differential forms are to be tranaformed into each other we must have OXt
Ut = Olft J Hence we must have
03Jt Ug=
au.
OXg
03Jg
= OYt'
01/2'
Vt
~
5-~
tJ2=
01/2.
out '
~ = ay; ' oy2 and these two conditions are 110t given by the algebraic equivalence of the two sets of forms. They may easily be seen to impose on 8 the additional condition that it must be an absolute constant.
17. FiniteDJ;tBB of the number of oonditlons for equivalence of two forms. There are thus certain integrability conditions of the type Oua.- OufJ O!J~-
ajj;_
to be satisfied by the coefficients u. We shall now prove that tltere 6:cists a finite number q sucll tkat, if tlte forms F, G4' ••• , Gq are equivalent algebraically to F', G,', ... , G(/, tlwn tke integrability cxmditions can be satisfied and tke tra'llllformation is possible. Assume a linear transformation (6) and let it transform Gp. into Gp.'· We must have These may be regarded as algebraic equations for the variables
16
THE METHOD OF CHRISTOFFEL
(CH. II
u, :e, n2 + n in number, as functions of the y's. (F may be taken to be G~ and it may easily be proved that G3 derived from it is identically zero.) If these eqnations cannot co-exist, this will appear after a finite number of forms G have been considered. There are two other possibilities : (1) An order q - 1 can be determined such that
F= F', G4 = G4', ... , G9 -1 =G'q-1 determine the u's and x's as functions of the y's, and these values make Gq =Gq' an identity. (2) The relations up to order q -1 give only p independent equations for the u's and x's, where p is less than n2 + n, and if any values whatever of these variables be taken which satisfy the relations up to order q- 1, they also satisfy those of order q. In the first case the relations of order up to q- 1 give n2 + n independent equations for the u's and x's. Let
!" (u, x, y) = 0, (lc= 1, 2, ... , n2 + n), be these relations. Since they determine the u' s and x' s, their Jacobian with respect to these variables does not vanish. Write
o:e,- ui s Oya.
a.
(
i)a. ' (Jya. ~ {rs, ·} ua.r u s -au; + .-, .,., ' ~
13 -
~{ a,..,a ,p}'uPi = - (
..rif
P
iIJ) • • • ( 7),
a,..,
and suppose that f~c = 0 is obtained from the relation G~.. = GA'. Then after differentiating Jk with respect to '!Ja., substituting the values of the derivatives from the equations just given, and using a relation from the set given by GA+l = G'A+1 we obtain an equation
l
~ (i) + ~
t c.i:Ct
a
i~
(
ia) au; Ofk. = 0
a,..,
..................(S).
By giving k all its possible values and keeping a fixed we thus have a set of homogeneous linear equations for the n2 + n quantities
(!). (!p) obtained by giving i and P all possible values from 1 to n. The determinant of the coefficients of these equations is the Jacobian just mentioned, and therefore it does not vanish. Hence all the quantities
(!). (a~) vanish and therefore the integrability conditions
follow from the algebraic equivalence of the two sequences. In this case, therefore, the transformation of F into F' is possible in only one way, the equations of transformation are obtained from the algebraic
17]
FINITENESS OF THE CONDITIONS FOR EQUIVALENCE
17
equivalence of the two sequences, and the integrability conditions are satisfied of themselves. In the second case n" + n - p =k, say, of the functions u, :e may be chosen arbitrarily, and the others are determinate functions of these. Call the arbitrary ones Zh Zs, ... , z.,, and the remainder of the u's and x's ZA+lt ••• 'z.,+Pt where of course k + p =n2 + n. The equations (7) are all of the form
~- 1/t~.a. (Z, '!J) = (.\a) ........................ (9),
where A takes all .values from 1 to n2 + n, the .;'s are known functions of their arguments, and (.\a) is a quantity
(!) or (a~) .
Let the relations whereby z.,+t' etc. are determined in terms of zlJ ... , be Z.,+r = cl>r (Zit ... , Z,, y) •.................... (10),
z,
then the ~'s are known functions of their arguments, and the equation (8) becomes in this case (It + 'I', a) =
l
t=J
~ (ta) .•......•.••.....•... (11). VL./t
The equations (7) or (9) may be regarded as differential equations of the first order for the Zs, the quantities (Aa) being supposed given. If we write down the conditions of coexistence these are seen to be of the form a(.\a)- a(>..{3) = ~ .A ~.a. (>..a), ayfJ oya. Aa. where the A's are certain functions of the Zs andy's. By means of (11) these may be turned into equations of the same type, involving only however, the quantities (Ao.) where A takes values from 1 to k. The equations (9) may be reduced to two sets by means of (10) ; one is a set
: - x~.a. (Z
1 , ••• ,
z,., y) =(Aa), (~ = 1, 2, ... , It) ......(12),
and the other may be seen to vanish identically in virtue of (10) and (11). Now the conditions of coexistence of (12) may easily be seen to be the same as those for the set (9), and these al'e seen to be satisfied if all the quantities (Aa) are zero, (A= 1, ... , k). It follows from (11) that in this case all the remaining quantities (Aa.) vanish, and the Zs
18
THE METHOD OF CHRISTOFFEL
(CH. II
left arbitrary by the algebraic conditions must be determined to satisfy the equations obtained from (12) by making all the right-hand members zero. These equations possess solutions involving It arbitrary constants, and we see that in this case the transformation ofF into F' is possible in a.: " different ways. 18. Connection of differential with algebraic invariants. In consequence of the theorems just proved, the problem of the equivalence of two quadratic differential forms is reduced to that of the equivalence of two sets of algebraic forms, where one set is obtained from the other by a linear transformation. The necessary and sufficient condition that it may be possible thus to transform one set of forms into another is that the algebraic invariants of the one set shall be equal to those of the other. It is convenient to extend our definition so as to include relative invariants ; a relative invariant is an expression which, under a transformation, repeats itself multiplied by some factor which depends only ou the transformation. Let I, 11, etc. denote a complete system of relative algebraic invariants for the first set, and 1', /1', etc. the corresponding complete system for the second set; we have /' = kl, /1' = k1lt, etc. and it is a known theorem that the quantities k are all powers of the determinant of the linear transformation. But this determinant is the Jacobian of the transformation performed on the variables :v; it therefore follows that tlte invariants /, l1, ... of the algebraic furms F, G4, ••• are a complete system of relative differential invariants fur tke quadratic differential form F, and if 'U/Tuler any trarulformation suck an invariant I becomes lei, then k is BO'me power of tlte Jacobian of the transformation. If we t.ake account of differential invariants which involve the magnitudes dm themselves, covariauts we may call them, it is clear that they correspond exactly to the covariants of the algebraic forms
F, G., .... 19. In the case where the equations do not determine the transformation of F into F' uniquely, it is easy to see that F must be transformable into itself, for since two different transformations give F' from F, the first of these followed by the inverse of the second gives a transformation of F into itself. Such a case arises, for example, when F is dt for a surface of revolution in space of three dimensions. It is clear that the conditions for this to be possible are expressed by the identical vanishing of certain of the invariants already obtained.
17-20]
CONNECTION WITH ALGEBRAIC INVARIANTS
19
20. Differential parameters. Now suppose in the general case that we· wish to obtain all the invariants when account is taken of systems of functions f (xb ... , tttn) associated with the quadratic form. We have
~=:Sj[_~;
O'!Jr
t O:Vt Oyr
hence any invariant which involves only first derivates of I is taken account of by adding to the set of algebraic forms F, G, the linear form
l/i~, i
of'
where A=~ . tiJr,
For second derivatives the case is not so simple. We have, in fact, ilf = ~ ilf Omp CliVq + ~ Clj if'IVL •
oy,.ay,
:p, fl
OXp03Jq O'!Jr
ay,
p
ay,.oyB
03Jp
and the second term on the right shows that the second derivatives cannot by themselves be taken account of by means of a linear transformation. If the form F is used, however, there are the relations (2) for second derivatives ·of the x's with respect to the y's, and by means of these equations we have immediately f,'' _ 1 J; am,~ ,., - .r>, q pq ay,. ay, ,
where
.J'
-
Jpq-
02 /
03JpOaJq
-lA {pn2' A} O:CA a.f
I
Now fpq differs from the corresponding second derivative of /by terms involving only first derivatives of/, and it follows that any function, and in particular any invariant, which involves only first and second derivatives of /, may be expressed as a function of the quantities f.,/IJfl. But it is clear from the equations of transformation of the quantities/_pq, that any invariant involving them may be taken account of by adding the quadratic l /pq~Xq' to our set of algebraic forms. JJ, fl
The extension is immediate, and just as the coefficients of G"+1 were obtained from those of Gp. (see equation (5)), so may the coefficients f 11qr of a cubic form be derived from those of the quadratic; any invariant involving only first, second, and third derivatives of I may be expressed in terms of f., /pq, f.qr, and is then seen to be an algebraic invariant of the forms F, G, and the three forms ul =.s 17P, u~ = ~ /pqXpXq', u. = l 1pqrx_pX'q' x,.". ~
p,q
p,q,r
20
[ca.
THE ABSOLUTE DIFFERENTIAL CALCULUS
11
Generally, any invaria.nt which involves derivatives up to the rth off is an algebraic invariant of the forms It~ G,, •.. , U't, U,, ... , Ur, where the coefficients of the successive forms U are calculated in exactly the same way as are the coefficients of the successive forms G. 21. The Absolute Differential Calculus. These ideas are at the base of the "Absolute Differential Calculus" of Ricci and Levi-Civita. .A. system of functions Xr r, ... r.. (r1 , ••• , rm = 1, ... , n) is said to be covariant if the transformed system Yis given by y; l r ftls•••llll ax,l a.-r,,, a:v,,. rlrll •••r,, = ;t,l . 0- ••• t a ,si, ...,Bm ":Jr1 y,., '!Jr,. 1
a
wtt
1
and the notation Xr r. is always used to denote an element of a covariant system. A contravariant system, an element of which is denoted by x(r•... r"'), is defined as one which has the transformation scheme 1
y
•••
l
(r1 ••• rm) =
, , ••• , '"' 1
X(•l ··· '•)
01!
O'l!
.-i!.!.l. ...i!.!!
OIJI
~
o:c.,_ aa:,. ··· a:c,.,. ·
If X is any function of the variables m, and Y the same function expressed in the variables y, the equation Y =X shows that X may be regarded as belonging either to a covariant or to a contravariant system of order zero. Since the differentials dy satisfy the equations dy,. = ~ dx, ~ , the '
cJ:C,
differentials d.c form a contravariant system of unit order. The coefficients a,., of the fundamental quadratic form Fare an example of a covariant system of order two, and if the magnitudes a<JHJ.) are the coefficients of the form reciprocal to F (seep. 10), they are a contravariant system of order two. The laws of composition of systems are very simple : (1) Addition. If %,. Sr.... ,..,., are two covariant systems of the same order m, the system X,.•... r.,. + Srp.. r,. is covariant of the same order. This result holds also for contravariants. (2) Multiplicatioo. If x ...... r,., s......,, are two covariant systems of orders m and p, the system x,.1 ... r"' 'E,l ...•, is covariant of order m + p. This theorem also is true of contravariants. (3) Composition. If X ...... r ..p .. a, is a covariant system of order ?11 + p, and s<• is contravariant of order p, the system l s<•t···•,,..x:r, ... r"''' ... '" 1
1
.,..... ,
···'')
., ... ,, ,
20-22]
COVARIANT AND CONTRAVARIANT SYSTEMS
21
is covariant of order m. Similarly the system ~
...... lb ..., Bp
s
x
lt•••I.P
is contravariant of order m. In particular, if m = 0, we have an invariant in either case, so that from two systems of opposite nature and of the same order an invariant is derived by composition. An example of composition is given by the relation a<"')= l a
between the coefficients of the fundamental form F and its reciprocal. Again, by means of the fundamental form, we can de1ive from any covariant system X,. .... rm a contravariant system x(r, ... r.,.) defined by the equations s, ... ,lim
and similarly from a contravariant system s
It is easily seen that if X<,·a ... ,.., is the system derived from X,., ... ,.,. , then Xr 1 ...... is the system derived from x
1 ...
r,, ..., ,.,.
1 •••
or every invariant composed qf a covariant system and a contravariant S!JStem qf tke same order is tM same as tlte invariant composed of tMir reciprocals. 99. The system c. There is a certain system of order n of much importance in the theory. Let a denote the discriminant of the form F, and give Ja a definite sign for some given set of variables m; make the convention that this sign does not change when the transformation is made to a new set of variables y, provided the Jacobian of the :c's with respect to the y's is positive. If however the Jacobian is negative let the sign of this square root be changed. The system •r, ... rn is now defined to be zero unless all the r's are different, and, the r's being all different, is equal to + Ja or to - Ja according as the class of the permutation (r1t·1 ••• rm) is even or odd with respect to (1, 2, ... , n). This system is
THE ABSOLUTE DIFFERENTIAL CALCULUS
22
[CH. II
covariant, and the elements of the reciprocal system are equal to zero 1
orto±-Ja·
If 4 (z1 ••• Zn) is the Jacobian divided by Ja of n functions z1, ... , z,. with respect to the :.v's we have the identity
which shows that ~ is invariant, and at the same time renders applicable to it the processes of the Absolute Differential Calculus.
28. Covariant and contravariant differentiation. From a covariant system of order m may be derived by the method of Christoffel a system of order m + 1. With the present notation equation (5) becomes
a
Xr, ...rmrm+l =ax
m+l
("'f"r
m 1••• , . , . ) -
l l {rzrm+l' q} X,., ...r,_1qr,+1...,..,. ... (5'),
1=1 q
and this gives the elements of the derived system. 'rhe elements of the derived system are called the covariant derivatives of the original system. If we substitute in (5') for the X's their reciprocals, w~ have a set of equations connecting the elements of a contravariant system of order m + 1 with those of a similar system of order m. If these equations are solved for the elements of order m + 1 they give
X("~···r,.tm+l, = l
t
a
{_!_ (X'"l"'""'>) + l ~ {tq,1·z} XtrJ ...,.,_tqr,+t'".""'>} a.x, l=l q
...... (5"), and we have the equations of what may be called the contravariant
derivatives of a given system. It is to be noticed that the derivative of a particular element is indicated by writing an additional suffix or index at the right end of those denoting the original element ; for example the covariant derivative of Xpq with respect to ::c,. is Xpqr, and this is in general not the same as XFq· The laws for differentiation of sums and products of systems of the same kind are exactly the same as those for ordinary differentiation. For example, from the eqnations
Zpq= Xpq + Ypq, Zpqrst = Xpq
Yrsh
22-24]
SECOND DERIVED SYSTEMS
23
it is easy to deduce the relations Zpqk =
X pqk + Y pqk,
Zpqrstk
= X pqk Y,.,t + X pq Yrstt,
and similarly for any covaria.nt or contravariant systems. If Z r1···rm- "~ y(s, ...,,) ..t1.r "'V' rm8, ... s,, 1 •••
s., ..., Sp
the elements of the first derived system given by the equation Z,.l···rmk =
l
z,...
Y('a···•p)Xrp ..r.S:···Bpk + l
Bb ••• , Bp
s 1, ••• , Bp,
rmk
t
are readily seen to be
y<s•... Bpt) atk Xr(•••rm8l···'P'
and there is a corresponding formula for a composite contravariant system.
24. Now let X,.1...r. be any covariant system whatever, and form its second derived system. We have the identity
where ahkr,p is written for the Riemann symbol (kkrlp). ('rhis notation is justified since the Riemann symbols have been shown to be elemeuts of a covariant system of tl•e fourth order.) It thus appears that the element Xr,···rmhk is not in general equal to the element Xr1...rmkh· In fact, if all covariant differential operators are interchangeable, all the
Riemann symbcls must vanish identically. If the fundamental form is ~d~t2, all the Riemann symbols do vanish, and it readily appears from
'
the result of Christoffel already given that if these symbols vanish for a quadratic form, that form must be reducible to the sum of the squares of n perfect differentials. (There are some very particular cases of exception to this, of no importance in our theory.) In this case covariant differentiation reduces to ordinary differentiation. The number of linearly independent Riemann symbols has been proved to be ft n2 ( n2 - 1). In particular for n = 2 there is only one such symbol, G say, where G = (1212), and for n =3 there are six
symbols. The six equations obtained by Lame (see his Lefons sut• les coordonnees curvilignes) in connection with triply orthogonal systems of surfaces, are got by equating to zero the particular values the Riemann symbols take for the form Pd.'f!J+Qdy2+Rdz2. These six equations for a general quadratic form in th1·ee variables are given by Cayley ( Coll. Math. Papers, vol. XII. p. 13).
24
THE ABSOLUTE DIFFERENTIAl, CALCULUS
(CH. II
A certain amount of symmetry may be introduced into the case of n =8 if it is postulated that we may replace one index by another when their difference is 8. The linearly independent symbols may then all be expressed in the form ar+l, r+2, s+l, •+2t and if a,(rs) be written for this symbol, the system a
m
l
E(rB)
x,.l .. .,·,.ra = G l
l=l
'r,B
for
n = a,
l
s,t
l
E(rat)
Xr •... rmBt = l
q,s,t
a(rB) E,,.l
Xr1 .. r,_18J"t+t··· r.,.,
f',B
m
a(q•) a(t•t)
2
C,·,at ~·, •• r,_l Ql"t+1 •• r,..
l=l
25. In the general case we have a quadrat.ic differential for1n F. Its coefficients form a covariant system of the second order. The system derived from this is identically zero. We have, however, the quadrilinear form G4 , and its coefficients are a covariant system of the fourth order. The covariant derivatives of this system are the coefficients of G,, and similarly each form Gp. has coefficients which are the covariant derivatives of the coefficients of G~£-t· Again suppose we have any number of functions ll, V, etc. associated with F, and suppose that the transformation that changes F into F' changes U into U'. For the equivalence of U with U' it is clear from what has been said, that the algebraic fonns U1 , U1 , etc., where U~~>+t is the form obtained by covariant differentiation from U~~-, must be linearly transformed into U 1', U2', etc. by the transformation which changes the sequence F, G into the sequence F', G'. It may easily be shown that all invariant relations arising frotn the differential form F and the functions U, V, etc. must be invariant relations under a linear transformation for the system of algebraic forms
F, G4, GG, .•. ,
ult u2, ... , vh
17''2,
••• ,
etc.,
which we call the set S. In particular, all differential invariants of F, U, V, etc. must be algebraic invariants of the set S. We notice that the coefficients of any form in 8 are a covariant system, and that all the forms except G4 are obtained by covariant differentiation. Further, if 0 be any algebraic covariant of S, it is clear that the coefficients of 0 are a covariant system. From a covariant system of order r and a contravariant system of order s there can be constructed by composition a covariant system of order r- s or a contravariant system of order s - r according as r is
24-26]
INVARIANTS OF BINARY •.,ORMS
25
greater than or less than s. If in particular ,. = s we have an invariant. Initially we have the covariant systems given by the coefficients of the set S, ann the contravariant system c. From these we obtain uew covariant and contravariant systems by composition, and, in particular, invariants are thus obtained. By repeated application of the principle of composition new systems may be obtained, and it may be proved that all covariant and contravariant systems, and all invariants, may be thus obtained. 'fhe theorem is one on invariants and covariants of algebraic forms and is not difficult to prove. ~6.
Application to the theory of invariants of binary
forms. The method of proof will be illustrated for the case of a single binary form, though the process is perfectly general. We recall the ordinary symbolic notation for algel>raic invariants and covariants. Let a0xn + na1 x"'-1!1 + •.. + an'!ln be written symbolically ( «tiV
+ arf!j)n,
where the symbols a1a2 have no meaning by themAelves, but a 1n =au, «t"'- 1 a 2 = a1 , etc. Then any expression involving the a's may be expressed in terms of the a's. If, however, we have a term, a0~ for example, it becomes in terms of the a's a 1tn- 2 a22, and this might equally be the term a 1t. To avoid ambiguities of this kind we must arrange to have no expression of other than the 1~th degree in at and ~ combined, and we must therefore introduce equivalent symbols. Let there be introduced equivalent sets of symbols atCls!, f3tP2, etc. such that the form is symbolically
(a1x + Ur1!J )n :: (13tx + /32!1)"'
=etc.
Then a0a2 is a1"'Ptn-2/Js2, and similarly, if enough equivalent symbols are used, all products of the coefficients of the form may be expressed without arnbiguity. If the determinant a1/32 - a2{31 is denoted by (af3) it is easily seeu that (a/3), though without actual meaning by itself, satisfies the conditions for an algebraic in variant of the binary form. Hence products of such determinants satisfy these conditio11s, and therefore, if the product ha.s an actual interpretation in terms of the a's, it is an invariant. For example, (af3)n is such an invariant, and if n = 4, ({3-y )2 ( 'Ya)2 ( a/3)2 is an invariant. It is also obvious that a1:c + a 2y is a symbolic covariant, aud hence if this expression be written 4a:, any product of factors such as (a{J), aol) is a covariant, symbolic usually, but
26
'fHE ABSOLUTE DIFFERENTIAL CALCULUS
(CB. II
with an actual interpretation if it is of the nth degree in each of the sets of symbols a., {J, -y, etc. Further, it is capable of proof that any invariant or covariant whatever of the binary form may be expressed as a sum of products of factors of the types (o.[J), aCI:. By a slight extension of the symbolic notation any form such as aoO:ta'2 + llrta't'!/2 + ~X2!/1 + aa'!/t'!/2,
linear in sets of cogredient variables x1y1 , :r2y2, etc. may be symbolically expressed as (at.t't + o.i!/t) (a1' a-2 + a2' 'U2), etc., where a, o.', are symbols to be interpreted as before. (In our particular case, for example, a1o.1' = ao, «tG.t' = ah etc.) It may be proved that any invariant is a. sum of products of factors such as (aa'), (ap), aCI:, etc. Now let a base system of variables ~h '1/t be introduced, and let Jrc denote the determinant of the coefficients whereby the variables x, '!It are derived from this set. 'rhen if xh !/t are transformed into x/, yt' by any transformation, J: =J: J(zt'
yt').
~
(1/
:Ct!ft
Next introduce a system 'r• such that Era= 0 if t• =s, and is equal to ± 1/Jz if r+s, the sign being determined as before. Similarly introduce a system ,<,.•) = 0 or ± J~. 'l'hen it is easily seen that the former system is covariant, the latter contravariant. NOW
( a{3)
:= (a 1/32 -
/3 =: ~ "l E(r•)ar/Ja,
a 2 1)
..,~
rs
and The system .x, y satisfies the definition of a contravariant system of the first order. (The terms covariant, contravariant, applied to systems in the absolute calculus, are retained for the algebraic theory, though no special meaning is now attachecl to them.) Hence, if we change our notation, we may call them a:<1), x<2), and 0.~
= lo.,.:c(r). r
Thus any invariant or covariant consisting of a product of factors of
the types (a{J), a~, is clearly obtah1ed by composition from the systems ,(r•>, o.,., g;A.r>, and when the products of a's are replaced by their actual values in terms of the a's the theorem is evident. For example, consider the invariant of a single quadratic Jar, a.~rt-c{•) = (Ja,.x(r))2. rs
r
26-28]
INVARIANTS OF BINARY FORMS
27
The invariant in symbolic notation is 1 (a/3)2 aJ. 2 Jt(rt)a,./31 XJE(JUI)ap/3q z
1"8
1
=J. 2 2
z rspq
pq
E(ra),(J>q)a1.ap/31 /3q
1 ~ ,(ra),(pq)a a =-- J.z2 rspq ~~&~ rp ~q•
The factor Jz2 Jlnd the base system of variables have been introduced because symbolic algebraic invariants and covariants are merely relative, whilst those of the absolute calculus are absolute.
It is now clear, generally, that all the covariant and contravariant systems arising from given systems may be obtained by composition, and further, we have obtained a useful result in connection with algebraic invariants. This result is that if tlte notation for tke coefficients of given algeb'raic furms is p'roperly clwsen, any invariant or covariant may be expressed 'IW'n-symbolically in suck a form t"hat its in/variance is at once in evidence. 27. To return to the general theory, the result obtained is that all the differential invariants of order ~ ~t may be .expressed in the form of invariants of the absolute differential calculus, by means of the coefficients of the following forms : (1) the fundamental quadratic form; (2) the associated functions I and their covariant derivatives of orders up to ~t ; (8) a quadrilinear form G4, and its covariant derivatives of orders up to ~t- 2. 28. It is now clear that if any problem is given in which a
quadratic differential form is fundamental, it suffices to replace ordinary differentiation by covariant differentiation with respect to the quadratic form to obtain the equations of the problem in an invariant shape. It may happen that we have the equations expressed in a simple form due to the choice of a particular set of variables, and by means of this theory we may express them in terms of a general set of variables without going through the process of calculation. As an example consider potential functions in space of three dimensions. They satisfy the equation
28
THE ABSOLUTE DIFFERENTIAL CALCULUS
[CH. II
when the coordinates are rectangular cartesian. The quadratic form is 11ow dx2 +dy2 +dz2, and \vhen the equation is written %a(re) V,.,=O, its invariance becomes intuitive and \Ve have its expression in a general set of variables 3l1 , x2 , x3 • For other examples see a paper by the author, Bull. .Amer. .Math. Soc. (1906), p. 379.
One of the most important properties of a covariant or contravariant system is that if all its elements vanish they do so independently of the particular variables chosen. In other words, tqe system of equations, e.g. Xrat = 0, is an invariant system.
CHAPTER III THE METHOD OF LIE
29. We now consider the second of th~ methods used to determine differential invariauts. This method depends on the use of infinitesimal transformations, and is originally due to LIE. It will be of use to recall the main points of Lie's theory. This we shall do for the case in which there are only two independent variables, though the ideas are perfectly general. Suppose that there are two sets of variables :c, g, and :c', g', connected by relations :c' =I (:x, y), g' = cfl (:c, g). These relations will define a transformation scheme, provided the variables :x, y can be determined from them in terms :c', y', i.e. provided the Jacobian of the functions j, ~' does not vanish. The operation of replacing 31, g by x', g', in any function V of 31, g, may be denoted by SV, that is to say, SV (31, g)= V (:c', y'). Now let T denote a similar transformation scheme, then by applying first T and then S we obtain a scheme represented by ST. For example, let 31' = aOJ + b, y' = ay + b, be the scheme S, and :c' = mt, y' ='!! be the scheme T. Also let V be OJ+ y. Then or+ '!I= T (:c +g), and (aOJ + b)2 + (ay + b)2 = S (:# + '!/) =ST (:x + y). Also QOJ + by+ 2b =s (OJ+!/), and therefore aafl + ay + 2b = TS (:x + y). It is easily seen from this example that TS and STare not in general the same, so that it is necessary to take account of the order in which the operations are performed. Starting therefore with the two schemes S and T, we obtain two more schemes given by ST and TS. We have also the scheme SS,
30
THE METHOD OF LlE
[CH. III
which may be written shortly 8 2, and in general we have schemes given by the operations SA TP-8" .... In exactly the same way, if any number of transformations are given, we can obtain others by repetition of these transformations. If it happens that all the transformations that can be obtained in this way are included in the original set, this original set is said to form a Group of transformations. For example, the set of six transformations :c' = :v, OJ'= 1- x, x' = 1/:r, x' = (:e -1)/x, x' =x/(ft -1), OJ'= 1/(1-:e), • ts a group. Another group is given by all the transformations of the form :x' =ax + by, y' =ex+ dy, where a, b, c, d, are arbitrary. A third example is the set of transformations :c' -I (:e), y' = yt./1 (PJ) where f and ~ are arbitrary fitnctions of :r. These three examples illustrate different types of group. The first contains only six operations, and in fact these may be obtained by repetition of the two x' =1- :r, x' = 1/x. The second contains a quadruple infinity of transfonnations, since it depends on four arbitrary constants. The third depends on two arbitrary functions, and in comparison with the second may be said to depend on an infinite number of arbitrary constants. A transformation of a group is said to be infinitesimal when the variables x', y' obtained from it differ infinitesimally f:rom the variables :c, '!I· The first of the three examples contains no infinitesimal transformations. The second contains, for example, the one given by o! = :c + ''!/' g' = y, where • is infinitesimal, and similarly the third contains infinitesimal transfonnations.
30. Continuous groups. The group is said to be continuous if all its transformations may be built up from infinitesimal transformations. The first of the examples given is not contiuuous, the second and third are continuous groups. We shall be concerned only with continuous groups in future work. (It is to be noticed that a group may be discontinuous and yet contain an infinite number of transformations. For example, the modular group :c' = ao; + by, y' =c:e + dy, where a, b, c, d are any integers such that ad- be= 1, is discontinuous.) A continuous group is said to be finite if all its transformations are determined by a finite number of parameters, whereas, if it involves arbitrary functions, it is said to be infinite. The second example given is a finite continuous four parameter group ; the third is an infinite
29-32]
31
INVARIANT OF A GROUP
continuous group. It tnay be proved that a finite continuous group which depends on r parameters may be generated from r infinitesimal transformations. The second of our three examples may be generated from the four infinitesimal transformatious = :c + a.y = :x =x + ym 4 =m 1 3 2 y' = '!I y' ='!I + Px y' = y y' =y 4- 8y where a, {J, y, 8 are infinitesimal. The third may be generated from infiniteaimal transformations of the type :c' =X+ (aJ), y' ='!I+ P'!PJ (:c), where~ and fJ are arbitrary functions of :c, and a., /3, are infinitesimal.
{:c'
{x'
{a/
{o/
ae
1 31. Invariant of a group.
Now suppose that V(:c,y) is a function of the variables x, y which is unchanged when any transformation of a certain continuous group is performed upon it. V is said to be an invariant of the group. Since the transformations of the group can all be built up from infinitesimal transformations, it follows that the necessary and sufficient conditions for invariance are that fr shall be invariant under all the infinitesimal transformations of the group. Let a/ = tc + ~dt, y' = y + 'f/dt be any infinitesimal transformation of the group. If Vis an invariant we must have v (:c + ~dt, '!I+ 'f}dt) = v (:c, y). Hence
av av OV=~~+7Jay=O.
By making use of all the infinitesimal transformations of the group, we thus obtain a set of linear partial differential equations which all invariants must satisfy, and it may be proved that this set is complete and that all its solutions are invariants.
sg. Extension of· a group. Now suppose that we have a group of transformations on the variables x, y, u, v, and suppose further that te, y, are independent variarbles, and that u, v, are functions of these. A transformation of the group determines :c', y', u', v' as functions of te, y, u, v. By differentiating the equatious of transformation, we may obtain equations whereby the derivatives of the transformed variables are determined from the original variables and their derivatives. The totality of all the equations of transformation now obtained may be shown to form a group which is said to be etetended from the original group.
32
THE METHOD OF LIE
[CH. III
Suppose further that there are other variables, a, b, c for exatnple, dependent on :c, y, u, v, which are such that there are relations of the type /(a, b, c, :c, y, u, v) =/(a', b', c', :c', y', tt', v') connecting their original and their transformed values. Then if these can be sol\·ed for a', b', c' we may add their solved expressions to the equations of the group, and thus obtain a further extended group .. Consider for example a transformation on two variables :c, y, of the type x' =:c' (OJ, y), y' = g' (OJ, g). We may regard te and '!J as functions of a variahle a which is invariant under our group. Then a.'= a. and we have d:v' ox' d:e o:c' dy --=---+-do. o:c da oy do.
. .1ar expression . £or da dy' . w1•th a s1m1 We thus have the transformation equations for the derivatives ~,
~,
and similarly those for higher derivatives may be determined.
Again let a, b, c be three variables, functions of x, y such that adur + 2bd::vdy + cd'!f = a'd:c' 2 + 2b'dz'dy' + c'dy' 2• This relation leads to three equations for a', b', c' in terms of a, b, c~ :c, y, and we thus have a group in these five variables which may be readily extended so as to give the transformatiou equations for the derivatives of the variables a, b, c with respect to :c and '!/·
83. Application to determination of invariants of a quadratic form. The problem of the determination of differential invariants of a quadratic differential form is now seen to be that of the determination of all invariants of a certain extended group. We start with a general point transformation on the n variables :e1 , ••• , :e, given by the equations z/ = m/ (zu ..., ten), (i = 1, 2, ... , n). Certain dependent variables /, functions of the :c's, are introduced,. which satisfy the relations /' =f. In addition there are introduced other dependent variables ar,, functions of the :c's, of which the transformed expressions are determined from the identity lar,'dxr'dx,' = la,.,ik,d:c,.
32-34]
33
TWO INDEPENDENT VARIABLES
It is convenient also to introduce as variables the total differentials dflJ, tP:c, etc. (These last might be introduced by adding an equation a'= a. to the equations of transformation, and regarding the flJ's as functions of a. The total differentials would then ·be practically the same as the derivatives of the :c's with respect to a.) The group of all such point transformations is then extended so as to include the transformation equations for the new variables introduced and their derivatives, and our problem is to determine all the invariants of this extended group. To do this it is necessary to obtain the infinitesimal transformations of the group, and from these we obtain a complete system of linear differential equations the solutions of which are the invariants.
34. The case of two independent variables. As a first case we consider a quadratic form in two variables te, g. Let
m' = IJJ + ~ (m, y) 8t, y' ='!/ + 'tJ (rr, y) 8t
be an infinitesimal transformation of the original group, then ~~ arbitrary functio11s of IJJ, 11· Let the quadratic form be
'fJ
are
Edar + 2Fdmdy + Gdy2, then E'd:c' 9 + 2F'd:c'dy' + G'dy' 9 = Edar + 2Fdrrdy + GcJy. Hence if E' =E + 8E, etc., 8EdafJ + 28Fd:edy + 8Gdy + [2Ed:cd~ + 2F(d:xd'YJ + dyd~) + 2Gdyd'YJ] 8t = o, and therefore - 8E = (2EEz + 2FrJ:JJ) 8t, - 8F = (.E~, + F~f» + Fq11 + G'YJm) 8t, - 8G = (2Ft,+ 2Gq11) 8t. .A.lso &lx = ( ~r»tk + ~,Py) 8t, 8dy = (7Jalltc + "'.d!!) 8t, and if/is an associated function, 8/=0. Now if z be any one of the dependent variables we must have dz - zJ:Jm- zrfly = 0, and therefore d8~ - z:xiJ8$- z,P8y- 8zatl:e- 8z,Py = 0, from which the increments 8z0, 8z'll, may be obtained by equating to zero the coefficients of d:c and dy, and the process may be repeated for higher derivatives. For instance, the increments of the derivatives of I are given by the equations - Vm =f m€:1) + f'll"'aH - 8j~ = f :J'U +/v"'v·
34
(CH. III
THE METHOD OF LIE
The infinitesimal operator of the group in the variables :r, y, E, F, G, /, /rll, / 11 , is thus
a
a
o
a n a ~a$+., 3iJ- (2E~., + 2F.,.,) OE-(E~11 +FE.,+ F.,"+ Gq.,) OF a o a - (2F'E'I/ + 2Gq'l/) OG- (/~.. +I'll.,.,) v.;- (1~'11 + fr"'!!) Of'll' :f
Since ~ and "YJ are arbitrary, it is clear that any invariant n1ust satisfy the set of differential equations obtained from the coefficients of ~' "YJ and their derivatives, in the equation 0/ = 0, where I is an iuvariant, and hence we have aJ aJ , ol aJ ol al- = 0, ri!J = 0, 2.E OE + F aF+ f., 'if.,= 0,
U
U
U
2G aG + F oJ!' + / 11 ~ =o, aJ
~M 2Jj aE +
ol
U
U G a.F + / 11 ¥., = o,
ai
2F aG +E aF+/., BJ; =o. This is a complete system of six independent linear equations in eight variables, and therefore it possesses two functionally independent solutions. There are thus two invariants in these eight variables ; one is j, and the other is readily found to be Beltrami's first differential parameter There are several points to be noted in connection with this example. In the first place, the finite equations of the group are not wanted, and therefore it is only necessary to calculate the increments of the various variables for an infinitesimal transformation. Secondly, the .method given for finding the increments of various derivatives would be very long in any general caseJ and some more convenient method must be used. It is to be noticed also that if the group be only so far extended that no variables have increments containing derivatives higher than the pth of~ and "r/, the variables occurring will be (1) tt, '!J, a:t, fly, . , ., dPrc, dPy, (2) the functions f and their derivatives of orders up top, (8) the functions E, F, G and their derivatives of orders up to p-1.
In this case the number of linear equations is equal to the number of derivatives of ~ and "YJ involved, that is 2 ( 1 + 2 + 3 + ... + p + 1) p + 1) ( p + 2).
=(
34-36]
35
n INDEPESDENT VARIABLES
The number of variables in the linear system is 2 (p + 1) + lk (p + 1) (p + 2) + fp (p + 1), where k is the number of different functions f. Hence if the equations are linearly independent the number of invariants is 2 (p + 1) + jk (p + 1) (p + 2) + fp (p + 1)- (p + 1) (p + 2). Again, there is no need to take account of more than two functions j, for any other function may be expressed in terms of these two, say j<1), j(a). All the derivatives of another function I may therefore be expressed as· functions of1<1), j<2) and their derivatives. It follows that any invariant involving derivatives of I may be expressed as an invariant that involves only 1(1), /(a) and their derivatives. If, then, k = 2 and the linear equations are independent, the number of invariants is i {Sp + 4) (p + 1). If the total differentials (1) do not occur, and if lc = 0, the number is apparently fp (p- 8). In this particular case the equations are, however, not independent. They are connected by oue linear relation, and the number of invariants is i (p- 1) (p- 2). These invariants involve only E, F, G and their derivatives. They are known as Gaussian invariants, and the first of them is the wellknown quantity K.
85. The general case of n independent variables. We shall now consider the general case of a quadratic form in n variables. Let the variables be 3J1 , ••• , lCn and let the infinitesimal transfonnation be given by 8lCt = ~t (lCl, .•• , tc,.) 8t, (i = 1, ... , n) ~~ (lC1 , ••• , lCn) will be denoted by~~ (lC). Also let/ denote any function of the {I)'s; the increments of the various derivatives ofI are needed and to determine them we use the method of Forsyth. 86. Determination of the increments. Let tEi denote the original, lC/ the trausformed variables, and let z 1 + k1 become :c,' + k,' (i = 1, ... , n). Then lc/ =lC/ + k/ -xl = lC' +let+ ~t (rc + k) 8t- [lCt + ~t (lC) 8t]. Hence kt'=kt+[~,(:c+k)-~,(111)]8t, and therefore if I denote any function of the variables 3J, and'/' its transformed, I (111 + lc) =I' (al +k') =I' (al + k + ~ (m + k)- (x) 8t)
e
-
-1
I
I
fl
Of' (
fCI
+ k)
(:c +k)+.l ce,(:c+k)-e,(:c)) a( , k) 8t+ .•., \=1
tDi
+ '
36
THE METHOD OF LIE
[CH. III
or, if small quantities of order higher than the first are neglected,
i (~ ( k)- t' (z J') af1f(zt(z++k,)k)- 0 dt +i=l t z + ' where d/dt operates only on x and not on k. By equating to zero the coefficients of the various powers of the k's, we have all the increments of the derivatives of f. Exactly as in the particular case of two independent variables, the increments of the quantities ar, may be obtained by comparison of the quadratic form and its transformed, and then Forsyth's method will give the increments of the derivatives of the a's. The increments of the a's are given by darB 0~1& 0~1& - dt =lar,,a- +la,..,-a . d/ (:c + /c)
·
h
![£•
h
Xr
Also if small quantities of order higher than the first are neglected, ar,' (x' + lc') =ar,' (x' + k + ~ (.:c + k)- ~ (z) 8t)
= ar,' (oe' + k) + l [t, (.x + k)- Et (.x)] ~'( (.x ~ ~) &, :e, + '
i
and therefore d { of"(x+k) o~"(m+k) dt ar. (.x+ k) + ~a,.,. (.x + k) 0 (.x, +It,)+~ a,.. (.x + k) O(m,. + kr) +l
i
[t, ex+ k)- e, (x)] ~·z,C.x+~~)} =o, (
where d/dt does not operate on k, and, as before, the increments of the derivatives of ar, are given by equating to zero the coefficients of the various powers of the lc' s in this expression.
37. Definition of Rank. Now suppose that the quadratic form l
ar,d~rda-,
is of such a type
m
that it may be written l du1, where the u's are functions of the :ds.
It
A=l
is clear that, unless the discriminant of the form vanishes, a case that we definitely exclude, m must be equal to or greater than n ; if m is taken to be !n (n + 1), the equivalence of ~he two expressions for the form leads to jn (n + 1) differential equations of the first order fbr the u's as functions of the .x's, and the theory of such equations shows that these equations always possess a set of solutions (subject to certain conditions of regularity of the quantities a as functions of the z's). Hence m is less than or equal to !n (n + 1). We thus have, immediately, a classificatioB of quadratic forms. A form is said to be of rank r when r + n is the least value of m for which the form can be
36-38]
DEFINITION OF BANK
37
m
reduced to '2duA1• Such a form may be interpreted geometrically as A=l
the squa.re of the element of length of an n-fold manifold in ordinary · (Euclidean) space of n + r dimensions.
38. Differential parameters for forms of rank zero. Let the form be of rank zero, then we have n
n
r, •=1
A=l
l ar,dttrd:c, = l
du~..2,
and it is easy to prove that if one set of functions u is given, the most general possible set is lriven by performing a general orthogonal transformation and a translation on the set u. The problem in this particular case is therefore seen to separate into two parts: (A) The determination of all invariants under a general transformation on the :c's, subject to the condition that the u's are invariant. (B) 'l,he selection from these of those functions which are still invariant when a translation and an orthogonal transformation are performed on the u's. The first of these is equivalent to the determination of all the invariants of any number of functions of the set of variables rc. If these functions are j<1) (:c), j(2) (0), .. ,. , the variables occurring in the invariants are (a) derivatives of the/'s, (b) .1h ' ••• ' :c, ' (c) d:c1, ... , d:c,, d 1:c1 , ••• , d 2:c,, ..• , d~'z,. We slightly extend the definition of an invariant by assuming that an invariant is a function I, that becomes 0~'I under a general transformation of the group, where 0 is the Jacobian of the transformation, and ~ is a number. For an infinitesimal transformation 0 is easily seen to be 1 + (l~)at, ,. O:Cr
and therefore the equation for I is
where
~ 3e
~= (:z~)l dt ~ ,. O:Cr ' denotes the effect of the infinitesimal transformation on L
'rhe increments of the various variables in I may be obtained without . OJ'f difficulty. Let/.......... denote a~1•1 • •• a~...,. where a.1 + e~t + ... ~ =p.
!38
THE .METHOD OF LIE
(CH. Ill
'rhen from the equation given above,
- g_tJ.•...•• -.i a.)., ...... aJ!.. +terms involving lower derivatives of the (s. "'
r:ci
t=l
Also let a:(A) denote d":c, then it is at once clear that £m}"l = (~f•l'l ic,)"~J +terms involving lower derivatives of the (s.
The expression (
l:ci(l)
~)"'
~.Xi
is supposed expa11ded by the multinomial theorem and then applied as an operator to ~J· Hence, if none of the increments of the variables in I contain derivatives higher than the pth of the ~'s, i.e. if I is an invariant of order less than p + 1, the differential equations obtained by equating to zero the coefficients of these pth derivatives are n
l
v(k)
a
k=l ~,
o[
8jl"'
ca 1 ••• a...
+
I
p '
I,
a[
I (.lh(ll)•• . ,. (m.(ll)a• a
al • ~ · • • • «n •
(PI= alt,
0,
where i = 1, 2, ... , n, and the a's take all positive integral and zero values subject to the condition that their sum is p. It is further assumed that there are n functions f. A complete set of solutions of this system is readily seen to be given by tPj<1), dP/(2), ••• , d~'f<"), and it is clear that these expressions are invariants, and therefore they satisfy the equations obtained from lower derivatives of the ~'s. The remaining solutions of the complete set of equations arising from the coefficients of the ~'sand all their derivatives of orders up top must therefore be of orders less than p. It therefore follows that the complete system of invariants is given by dl\j
and there remains yet one other solution, which is manifestly J, the Jacobian of the J.,s. Hence a complete functionally independent system of invariants of n functions I of the variables aJ1, ••• , iVn, involving derivatives and differentials up to tke ptk Qrder, is given by (A:::: 1, •.• , p; k = 1, ••• , n) For allwC6pt J, ~is zero, and for J JL is -1. tJAjf"),
and J. The most general invariant is therefore a function of the/'s and the above absolute invariants, multiplied by some power of J.
38, 39]
PARAMETERS FOB FORMS OF RANK ZERO
39
Now let then functions/ be Utt tt,a, ••• , un.. The second part of our problem is to determine those invariants which still remain invariant when a general translation and a general orthogonal transformation are performed on the u's. The variables entering are (1) ttt, u..a, .•• , un, and the total differentials of these variables; (2) a number of arbitrary functions q, (u) and their derivatives, and this number we may as before take as not more than n ; {8) the Jacobian J. Tbe translation may be taken account of at once. It is equivalent to the condition that the variables u do not enter explicitly into the invariants. The orthogonal transformation is a linear one on the variables u. We give the proof for two variables, though the method is quite general. Let du2 + d# = d U" + d V 2 where U, V are functions of ·u, v. Then if suffixes denote differentiations, this relation is equivalent to the three "Ut2 + V12=l, U1U2+ VtVt=O, U.l+ V2"=l. Hence 39.
Jl (~~):: ( Ut2 + Vt1) ( U22 +
and therefore J
(UV) uv
v~~)-( U1Us+ Yt Vs)2 =1,
is not zero.
Also from the above three equations we deduce by differentiation with respect to u, v, ~ Uu + JTl "Vtt = 0, U2 Uu + V2 Vu +: U1 U11 + V1 Vts =0,
vl v~~=o. ul ~1 + vl Vu = 0, ul Un + VI Vu = o, u1~2+
Hence and the detenninant of the coefficients of these two linear equations in U'tt, Vu is the Jacobian just mentioned, and is not zero. Hence U'tt, "Vtt are both zero. Similarly all the other second derivatives of U and V are zero and hence U and V are linear functions of u and v. As a translation has already been taken account of, onr transformation is homogeneous and linear on the variables u, v. Now, returning to the case of n variables, let us consider a general homogeneous linear transformation on the u's; the variables dAtu are transformed by the same linear transformation. The first derivatives of a function c/l are transformed by the contragredient transformation. In general, let Am denote
c/l, {'=1i "' :-}.,. uu,
40
THE METHOD OF LIE
(cH. III
where the U's are a set of auxiliary variables. Then .Am is an algebraic form of order m whose coefficients are the mth derivatives of ~~ except for numerical multipliers. The transformations for these derivatives of ,; are precisely those on the coefficients of the algebraic form Am, when the U's are transformed by the contragredient transformation to the original linear transformation. Also J in the transformed variables is J in the original variables multi}>lied by the determinant of the transformation. If the t.ransformation is orthogonal, it and its contragredient are the same; also its determinant is unity, hence J is an absolute invariant. We therefore im1nediately obtain the result : The functionally independent set of invariants of orders up to and n
including the pth of the quadratic for.m l dul aud n associated i=l
functions ~(t), ••• , ..P'"'>, are J, and the orthogonal algebraic invariants of the system of 11-ary forn1s A A(k), ( k = 1, 2, ... , n ; A= 1, 2, ... , p) 7l
(r = 1, 2, ... ,p)
l d'"ui U;. i=l '1f,
The linear forms l
tl'"u~ U1
may be omitted if account is taken of
'i=l
n
the invariants d'"c/l'",, and if the quadratic l
Ul~
is included we may
i=l
state the result in terms of a general linear transformation, since an orthogonal linear transformation is a linear transformation which leaves n
the quadratic 2 U 2 invariant. The final result is : i=l
The most general invariant is a function of the quantities arq,<"), the general algebraic invariants of the forms A and of l us, multiplied by some power of J. It is worthy of note that the algebraic forms A are the polar forms of the functions ~.
40. We now transform these algebraic forms from the variables U to variables X given by the scheme
" au a___! .J.'j.
U, = l
j=l 'ZJ
(i = 1, 2, ... , n,)
The Jacobian of this transformation is J, and the discriminant of the quadratic form l U 2 changes from 1 to J 2• Hence the most general invariant is a function of the quantities d"~, and of the general alge.. bra.ic invariants of the A's and 2 Ul, expressed in the new variables.
39, 40]
41
PARA:&IETERS FOR FORMS OF RANK ZERO
u,s become i=l
Let l"'
l"' a'I.,XrX,, then this is the fundamental quad-
~s=l
ratic form, and it will be shown that the coefficients of the forms A may be expressed in terms of the quantities a and their derivatives. We proceed to calculate these forms :
li=1~ ~~,
At becomes
and As is readily seen to be l cfJ,.,XrX,, r, s
{llei,
where c/Jrs denotes the covariant derivative of 4> with respect to the fundamental quadratic form. In the general case _ "' "' au, Ou:1 oPcfJ v v Ap- .l l 0 a A.,\Xp.Av·••,
aa ...
i,J, ..., =1 A,p., ..., =1 'lEA '31,_
Ut Uj· ..
where there are p letters i, j, ... , and also p letters A, Let
..
A!~; plJ~o., ,., ...
expression for Av·
p., ••••
denote the general coefficient in the above
Then
n
(jPcJ> { o2u, auj auk } oxP pBA,p., .•. =»+tBp,A,p., ··· +i,J,· ...l_ auI au'I} ••• .. loxp OleA Blep. o:c11 . •• ' , -1 tl
where the last l denotes that there is a term corresponding to each of the letters A, p., ••• , and that these are added. Also (fP~
n
=
oui auj - - ....
pBq " ., 2 • , '··· i,j, ..., = 1 o·uiauJ· .• o:cq ole,_
We solve the n equations obtained by giving q the values 1, ... , n for the quantities i oPrfl ~uJ au" ...
o:r,. Otl'v
j, k, ..., =1 OU(OUJ· ••
and substitute in the above equation.
a
O:cp p!J~.,,.. ...
=P+lBp,A,p., •.. + ~
We thus obtain
{n
"'
p+tBp,A,p., .•. =
l..! J
CPu, } 1
i~lq~l pBq,p., ,, ••• M,' O:cpO:C~o,
where M"q' is the cofactor of a~~- in J.
J•
This equation may be written
Wq
a
•
~ ,BA,,, ... +
tltl'p
0
CP~
CPu.
' a:cp ale,\ ' ao:po:cA' au,. au2 aa-1 ' Out
ale2
•••
'
alet
'
• •• ••• •• • • • • •• • J
............... '
au. Olet
• • • • • • • • • • ................. , • • •
• ••
-- , aa-.
a:cpOOJA
I
e. e
I
a e a e a e e
I
e
I
e e e
I
e •
I
e
I
e e e J
•
"-•
42
THE METHOD OF LIE
(CB. III
We multiply the determinant in this equation by J, and note that
J 1 = t:., the discriminant of the quadratic form, and thus obtain the result p+1Bp,A,p., ...
l !A
a
= Oa:p ,BA,,., ... + 0
'
pBt, 11-, ....... ,
pB2. p., ..., ••• , • • •
,Bn, "' v, ... '
(p~,
............... ' ............... ' ' ............... '
1]' (pA, 2],
lJtt a11 • • •
ant
'
'
'
a12 ~
.• I
a,.2
'
'
•••.•.•••••.••. ,
[p~,
n]
.
~
a2n
• •• ann
'fhe coefficients of the forms Ap+t are thus expressed in terms of those of the forms Ap, the a's and their derivatives, and we have shown that the coefficients of A1 and A2 may be expressed in terms of the derivatives of c/J, the a's and their derivatives. It therefore follows by induction that the coefficients of Ap may be expressed in terms of these magnitudes. The process whereby p+tBp,A,p., ... is determined from pB>..,p., ... is in fact that which has already been defined as covariant derivation, and thus the result is identified with that given by the earlier method. The initial limitation that the quadratic form is of rank zero may
now be removed, for it may be verified without difficulty that the invariants obtained are also invariants for a general quadratic form. When the form is of -rank zero it has no other invariants than these, which are differential parameters. When the rank is greater than zero there are other invariants which do not involve any associated functions ~~ namely the Gaussian invariants. 41. In the general case the quadratic is expressed as the sum of the square of m perfect differentials; where m is greater than n. Then it is clear that all the invariants of our quadratic must be included in m
those of the form :S du>..2 ; our form is in fact determined as an n A=l
dimensional manifold in Euclidean space of m dimensions, and it is obvious that among the invariants of the space must be 'included all the invariants of the manifold. The limitation is made by taking certain of the associated functions ~' equated to zero, to define the manifold. If we call them a'n+t, ••• , a:m, we have the system of equations .Tn+t =0, ... , d.xn+t =0, ••• . By means of these equations the forms A for the parameters may be reduced so as to contain n instead of m variables X, and hence the covariant. derivatives of & function 4> are seen to lead to the parameters of a general quadratic
40, 41]
GAUSSIAN INVARIANTS
43
form. The equations a:n+l = 0, ... , ~»m =0 lead to a set of algebraic .. forms whose coefficients are the generalisation of the fundamental magnitudes D, D', D" • of a surface in space. These last are not invariants of the original quadratic form since they cease to be invariants if the n dimensional manifold is deformed in the m dimensional space. Among them, however, ntust be include(! the Gaussian invariants (those involving the a's and their derivatives only), but these last have not been determined by the method of Lie e:~cept in some of the simpler cases. An idea of the algebraic work necessary for a direct application of the Lie theory to the problem of the determination of Gaussian invariants may be gathered from a study of Forsyth's memt)irs on the subject. * See Bianchi-Lukat, Dijferentialgeometrie, p. 87, also Forsyth, Mess. Math. Vol. xxxu. (1908), pp. 68 sqq.
CHAPTER IV !\IASCHKE'S SY!\IBOLIC METHOD
42. Explanation of the symbolic notation. 'fhe remaining method for the determiuation of invariants, by means of a symbolic notation, is due to Maschke. Its leading principle will present no difficulty to those fatniliar with the ordinary symbolic notation of algebraic invariants. The assumption is made that the quadratic form 'la,k dtet d:ck can be expressed symbolically (d/)2, where f is a symbolic function of the variables x. The derivatives of/, written f~t A, ... , In, have no meaning when taken separately, but when their products are taken two at a time, aik =ftfk. Any expression involving the a's may thus be expressed in terms of t.he fs. If, however, the expression in the a's is of order higher than the first, we must avoid ambiguity by introducing equivalent symbols. For example / 1',h2 might be interpreted either as a122 or aua22, but if lautl:cid:c~c = (d/) 2 = ( dcp )2= ... , then a122~ftf2cptc/J2, and au~=.h ~ , and there is no ambiguity. In general, enough equivalent symbols must be introduced to make the symbolic expression of the second order in the derivatives of each of the symbols/, ~' etc. 2
1
43. Invariantive constituent. Now call the fundamental quadratic form A, and let Ft, J!'l, ... , F"' be any n invariant expressions of A, then we have (.'= 1' ... , n) F 1,=F', where Fi' is .F' for a new set of variables y. Hence " a.1!"' " aF' l - dyk = l -d:x" k=I
and therefore
&u~c
, aJ?i'
1
k=t
na-k
\all'
ay,.j = r O:~Ja.
,
where r is the Jacobian of the transformation, and
I~~ denotes
the
42, 43]
INVARIANTIVE CONS'flTUENT
45
Jacobian of the Jil's with reference to the .x's. Further, if Ia.ik I denotes . the discriminant of A' Iatk' f = r Ia'ilv I ' and therefore ~ Iask, I - i cF'' d'!Jlt
Hence
- .1. I atk 1 ~
=
Ia1k ,- i j -a oF' • r:J:k I
aFt IS . an Invariant . . of A .
-;-
ti.l't
This invariant may be denoted by (F). If it should be necessary to put in evidence the first one, two, three, etc. of the F's, they are written in their proper places and the last letter is assumed to run on. For example (b, c, a) denotes the invariant derived from b, c, at, ... , a"- 2, and similarly for other such expressions. (F) is called an invalriantive constituent of A. Now if A is written symbolically d/2 =df/l2 = ... , it is clear that {, c/l, etc. are (symbolic) invariants. Hence if we form invariantive constituents of/, c/>, etc. and any number of arbitrary functions of the independent variables, all products of these are formally invariant. These products will represent actual invariants whenever they are such that all the derivatives of the j's, cJ>'s, etc. have an actual meaning. Further, an invariantive constituent may itself involve other invariantive constituents as elements, and thus higher derivatives than the first of the functions/, cfl, etc. may occur. It is to be expected that some functions of the higher derivatives off, for example, may be interpreted by means of the derivatives of magnitudes aa;, and it is easy to see that any expression built up of invariantive constituents is an invariant if it can be interpreted in terms of actual quantities, for instance, the a's and their derivatives and the arbitrary functions introduced and their derivatives. Now by actual differentiation
.J'jj ~ ~ a.x, =J, kz + fk./,h ~ Baw Bau . . . an d t here are s1m11ar expressions 10r -a , -a • . ~, Wt These give at once oatk
.J' I!_ oa'"] _[kl' t.]• JI,JkZ = 1~ [oaik -00z + Baa = a.xk ax, He~We tkere is an interpretation for tke symbolic product of type ft!H_. Further, by differentiating again
and hence
Amhz +Ji/klm =[kl, i]m, fcrfu-l~erlu=[ir, k],-[is, k]r, frf.- fsltkr= [ik, r],- [ik, s]r•
46
MASCHKE'S SYMBOLIC METHOD
[CH. IV
44. By means of the relations for second derivatives we have, for example,
1
(n- 1) ! ((uf), f)
=~~u,
where 4 1u is Beltrami's second differential parameter, a result that may be verified by direct calculation. Some of the simpler invariants are• (/)2 = n !, (u/)2 = (n -1)! i.\1u, ('11/) (vf) =(n -1)! A (u, v). In these examples f is used to denote a symbol by means of which the quadratic form is expressed, and u, t' are actual functions oi the variables. This notation will be generally followed, that is to say, f's distinguished by means of indices are the symbols of the quadratic form, the letters u, v, w, etc. are used to denote actual functions, and a, b, c, etc. are used in general formulae to denote quantities which may be interpreted either as symbols or as actual functions.
45. Symbolic identities. Exactly as in the ordinary symbolism for algebraic invariants, there are many different symbolic expressions for a given invariant, due to the fact that equivalent symbols are used. For example, suppose an invariant to contain two equivalent symbols/ and~. This must have the same value as the expression got by interchanging f and ~- But regarded as an algebraic expression with f, cp as actual quantities, it may change sign when these two quantities are interchanged, and therefore, if it does so, it must vanish. For many symbolic identities Maschke's paper quoted above should be consulted. We content ourselves with giving one or two illustrations. We have first, since (/)2 = n !, a constant, (J~) (f)i = 0, where the suffix denotes differentiation with respect to :r~,. Now let [/] denote (/) or any ordinary or covariant derivative of (f), and consider the product / 11 (uf) [/], where / 1, / 2, ••• , / " are the equivalent symbols for fused in (/).
(tif)
=ja,tl-i
U1,
u~.,
... ,
Ura
.h~,
A~,
... ,
J,;~
= laQI-i{u/}, say,
••••••••••••••••••••••••••• • • • • • • •••• •• • • • •• • • • • • • • •• •
* See Maschke, Trana. Amer. Math. Soc. Vol. n·. p. 441 sqq. (1908).
44, 45]
.and
47
SYMBOLIC IDENTITIES
fl {tif} [/] = ft'Ut,
Ug,
• • .,
Un
J?lt f,l, ... , • • 9 ,
fn 9 •••
• •
••
[/].
J?ft"", 12", ... , f,"' Now if flit''[/] has an actual interpretation so far as / 1 and fk are concerned; also it changes sign if these two equivalent symbols are interchanged. Hence it vanishes, at~d therefore fl {uf} [/] =ft1Ut /2• 2 ••• , /n•2 [ / ] , •
•
•
•
J'2", ... , .~n"' and this expression is easily seen to be equal to U1
f't 1, f',l, ··., 0'
J··2 2 '
0'
.••
• • •
0,
.•
••• ,
fn 1
I
[/].
.•
.••
•
•
hn, ... , fnn
If in the above determinant we interchange the equivalent symbols f 1 8J:l.d / 11, and let p take all values from 2 to n, we have n different forms for J'l {tif} [}']. If these are added, the determinants combine into {j' }, and it follows finally that 1
fl (uf) [I] ='I~ Ut (f) [f]. Similarly
n1
fl (u/) [fJ = u, Cf) [JJ,
or changing the notation, we have 1
fi(ua) [fa]=- u,(fa) [fa], n where a denotes a symbol. For example, if [fa] denotes (fa) or (fa), we have the two results h (j'a) (ua) = (n- 1)! u~c, Jk (j'a)t (ua) =0. If 'IJ2, 11, •.• , v"' are arbitrary functions we may readily deduce from these two results (j'a) (j'v) (ua) = (n- 1)! (uv), (fa)~ (j'v) (ua) =0. By a method precisely similar to that given above we may prove thatjilfs' (uvf) [/]is equal to 1
n (n _ 1) ( u"'k- ukv,) (I) [/],
48
MASCHKE'S SYMBOLIC METHOD
[ca.
IV
and hence, for example, A~" (uva) (f~a) = (n- 2)! (u~e"- ukvi) I t
[fUc (.xa) +A (.xa)"] [fa]=! [.xik (fa)+ :e, (fa)k] [fa], n
and it easily follo\\·s that :cik is equal to 1
(n _ 1)! {jj (fa) (xa)k +lilt (/a) (.m)}.
Again, it is easy to show that.h (ua)i (fa) is equal to A (ua)~c (fa), and assuming f and a to be symbols and u a function of the independent variables, we have the covariant l (/a)Ji (ua)k d(lJ,d:x", ik
whose coefficients can be shown t.o be equal to the second covariant derivatives of the function u. This covariant may be written
(fa)fz(ua)z.
46. For further work we require the higher covariant derivatives of a given function of the variables. Following Maschke's notation (which differs from that used in Chapter II) we use :c to denote any function not involving derivatives, and z
where
'= (n~ 1)!, and I
and a are equivalent symbols for the funda-
mental form. It follows by means of the expression already calculated for zfk that :c(lt) (.xa)t (fa). The higher covariant derivatives can now be calculated as soon as we know the covariant derivative of an invariantive constituent, for we know that to differentiate a product covariantively we apply the same rule as in ordinary differentiation, e.g.
="'
It can be proved that
45-47]
SYMBOLIC EXPRESSIONS FOR THE FORMS
G
49
and the higher derivatives may be calculated at once; for example we deduce from the above expression for a:
t (fa )A (f/>a ), where 4> is a symbol, and therefore I J<'·\) =~ can be calculated from the formula for second derivatives.
47. Sym.bolic expressions for the forms G. The expression of the quadriliuear form G":: l (ikrs) d<1)x,d<2)x"d(3)xrd<")x, t,k,r, s
in symbols is next required. From the definition of (ihrs) it is not difficult to deduce that (ilc'rs) =ftrfka-fufkr + E {.fi,f/J~w-.ftr4-kf} (4-a) (fa) =Ar {flea- •f/J1ea (~a) (fa)} -fu {f~cr- •4-~cr (4-a) (fa)} = Ar/(b) - fuf(kr);
this may also be writteu j(tr)j(ks) _ f(fa) f(kr).
Many different symbolic expressions may now be deduced for (ikrs), for example
(ilcrs) = ifc ~k {(fa )r (4-a), - (fa), (t/la )r}, and it is not hard to prove that A4-k (teM) (jq..a) = (n- 2)! (u(IJ"- u"v,). Using this, we have (n -1)! (n- 2)! (ikrs) =((fa) (c/la) b) (f'f/J'b ).ftc/lkf,.'f/J,', and therefore (n -1)! (n- 2)! G.= ((fa) (cf>a) b) (/'f/J'b )/mtf/J~/r' f/Jr»l. Ga may now be calculated, and (n-1)!(n -1)! (n- 2)! Ga =(((fa) (c/la) b) c) (/'c/l'b) (/"c) .fo,t ~«!'/rc~' c/lmt' f al', and there are similar symbolic expressions for Ge, G'l, etc. These symbolic expressions for the G's put in evidence at once the fact that they are invariants.
50
MASCHKE'S SYMBOLIC METHOD
(CB. IV
48. For further work on this part of' the subject Maschke's papers should be consulted. We give, however, one interesting application of the theory, taken from one of his papers•.
Let /, t/;», tf!, etc. denote symbols, and let any two dimensional manifold be determined in the n-dimensional space (not necessarily Euclidean) for which dr is equal to the fundamental quadratic form. This manifold is given by n - 2 equations
U1, U2, •.. , Un-2 = 0. :aiaschke has proved that for this manifold _ 2 ((/.P U) ( c/J.P U) U) (f
(I' 4>' D )2 (/"4>" U )2 • If n =2 the U's disappear and this expression becomes 2 1 ( 2 oF 02E o2 K = 2 (EG -lf""J.) OuOv- avt - Ou2 + ... , K-
G)
where This is the well-known Gaussian expression for K. Next let n =3 and let the space be Euclidean. Then we may take dr =dar + d'!f + dz2, and there is one expression U. Take the equation of the surface to be then U is F. \Ye get
F(x, y, z) = 0, If we substitute in the above general expression for K 1
K=-
o2F
(¥1)' + (~i/ + (Wz'Y ~~,
a:cay' atp
CPF
a;;'
o:coz'
oy2 '
ayaz' 'iPF
o'JF
~:coz'
ayaz'
aF
oF
-~:c '
aF
o2F
-ay ,
rf~F
o.e1
o:c • oF ay oF
'
oF oz '
0
The two expressions forK given above have no apparent connection. They are, however, two interpretations of one and the same general formula.
* Maschke, Trans.
A mer. lllath. Soc. (1906), Vol.
VI.
p. 98.
CHAPTER V APPLIOATIONS
49. We now consider some applications of the theory of differential invariants. Suppose first that the quadratic form is interpreted as the square of the element of length in a certain n-dimensional manifold. 'l'he invariants, though expressed analytically, are intrinsically connected with the manifold, apart from any frame of reference. They are differential, and involve only one set of the independent variables. They thus give the intrinsic character of the manifold at, and infinitely near, a particular point of it. If we take account of differential parameters, they express quantities intrinsically connected with the section of the fundamentaltnanifold by another manifold. But we have a set of quantities, defined geometrically, of just this type, that is to say they depend only on the manifold at or near some particular point and are intrinsically connected with it. Let us consider more in detail the particular case of a surface in Euclidean space of three dimensions. This is not intrinsically determinate if only one quadratic form, that for dB", is given. ~,or example, all developable surfaces have the. same quadratic form for ds2 as that of a plane. The catenoid (the surface of revolution obtained by revolving a catenary about its directrix) and the regular helicoid (the surface swept out by a line which is parallel to a fixed plane, intersects a fixed line perpendicular to the plane, and rotates uniformly about the fixed line as its point of intersection moves uniformly along that line) are two surfaces with the same quadratic form for dsl. Two surfaces which have the same quadratic form for dr are said to be applicable to each other. If we regard a surface as made of some perfectly flexible inextensible material, then it is clear that the surfaces into which it may be deformed are the surfaces applicable to it. There is a second quadratic form which we may associate with a
52
APPLICATIONS
[CH. V
given surface, namely that for ds2jp, where 1/p is the curvature of the normal section by a plane which meets the surface in the direction dufdv. The coefficients of the second form cannot be chosen arbitrarily if the first is given, though they involve a certain amount of arbitrariness. If both t.hese forms are given, the surface is intrinsically determinate. It follows that the geometrical magnitudes at a particular point on the surface are the differential invariants of two quadratic forms. Similarly, the geometrical quantities connected \vith any curve U = 0 on the surface may be expressed in tenns of differential paratneters that involve two quadratic forms. 'fhus the principal radii of curvature of the surface, the normal, geodesic, aud principal curvatures and the torsion of a given curve on the surface are differential parameters of this type. Among all these geometrical quantities, there are some that are the same for all surfaces applicable to each other. Such are, for example, the total curvature I/R1R2 of the surface, the angle of intersection of two curves on the surface, the geodesic curvature of any curve on the surface, and many others. These, it is clear, do not depend on the second quadratic form, and they are therefore invariants and parameters of a single quadratic form, that for dil'. It follows, conversely, that any invariant or parameter of the form for ds2 represents some geometrical magnitude associated 'vith the surface, and the magnitude is the same for all surfaces applicable to that given. On this account, such iuvariants are called deformation in variants.
50. Geometrical interpretation of invariants. In order to apply the invariant theory, we have now to interpret our invariants in terms of geometrical n1agnitudes, and also to interpret geometrical magnitudes in terms of invariants. It is perhaps worth while to point out the advantages gained by the use of invariant theory. In the first place we are able to apply in the simplest possible way the methods of analysis to geometrical problems, for we express all our data analytically, and yet avoid extraneous properties which arise through the relations of our surface to a particular coordinate frame chosen. Again, when we express a given quantity as far as possible by means of invariants, it may happen that its expression only involves invariants, and thus its invariance becomes intuitive. Also, if we know that a given quantity is invariant, we can often determine its invariantive form for some simple choice of coordinates, and then we are able
49, 50]
GEOMETRICAL INTERPRETATION OF INVARIANTS
53
to write down its expression for any coo-rdinates whatever. An example of this has already been given in the case of Laplace's equation 02Jr c2V a2V a(Cs + (Jy2 + ?z2 = 0. In this case we are dealing with Euclidean space of three dimensions for which ds2 = dx2 + dy + dz2, and the equation, in the notation of the Absolute Differential Calculus, is l a('·s> Vr8 = 0, for this particular T, 8
quadratic form.
It follows at ouce that 02V
02JT
a:r
02V
+ cy2 + oz2 • is au invariant of a general quadratic for1n in three variables, and its general expression is la<,·s) Vrs for any such quadratic form. Another example is given by Darboux (Theory of Surfaces, Vol. 111. p. 201). Let the quadratic form be ds2 =A 2 du2 + 0 2 dv2, then the geodesic curvature of the curves v = const is 1 oA -AO ov.
If
.Y) =la(rs)cJ>r.Ys, A2c/J = la (rs) c/Jrs,
A~= ~a(rs) ~r~8,
A ( ~'
then these three quantities are obviously invariants, and 1 2 A1'= 02 , A(v,Av)=-ofJav'
ao
A2v = _!__
~
(A) = - _!_0 aoov ..!. Op.
AO ov 0
Hence
1
--P-
A2'V
JF:V
+
3
1 A (v, Av) 2 (Av)i .
It follows that 1/p is an invariant, since the A's are all invariants, and further, the geodesic curvature of any curve q, (u, 1') = const., on a surface for which the quadratic form ds' is perfect]y general, is
_ A2c/> +!A (cf>, Acf>) J Acp 2 (~cf>)i . This example is a good illustration of the advantages derived from the invariant theory. We start with 1
aA
- AO &v'
54
APPLICATIONS
(OH. V
and express it as far as possible in terms of invariants. It happens that it is entirely expressed by means of invariants, and hence it is itself an invariant. Further, when it is thus expressed in terms of invariants only, its general value may be at once written down. We also note that the differential equation for all families of geodesics on the given surface is 2~ 2 cp. A~=~ (9>, A~), and this may be turned into a differential equation for the geodesics themselves by writing dv a,p;a~ du =-au ov'
with a corresponding expression for the second derivative
-·
d2v du9
•
61. Another application of the differential invariant theory can be made by means of the theory of algebraic invariants. All our differential invariants and parameters are invariants of algebraic forms. By means of the algebraic theory we can determine syzygies or algebraic relations connecting these invariants, and such syzygies, when expressed in terms of the geometrical magnitudes of the surface, lead to algebraic relations among these apparently independent quantities. Also the coefficients of the various forms are algebraically independen~ and thus all such relations are given by syzygies. A surface is intrinsically determinate if two quadratic forms are given, and hence all its geometrical magnitudes are invariants of two quadratic forms. We are thus able, by means of the invariant theory, to determine which among these magnitudes are independent, and, by means of syzygies, to determine all the relations connecting those that are not independent. For illustrations of this part of the subject the reader should consult the latter part of Forsyth's memoir t.
52. The case of a surface in ordinary space-the quadratic form in two variables. Confining ourselves to a quadratic form in two variables, we suppose that there are associated with the form two functions ~ (u, v), t/1 (u, v). In this case there is only one Riemann symbol, the quantity G. We * For other examples the reader should consult Wright, Vol.
(1906), p. 879. t Phil. Trans. (1908), Ser. A, vol. 201, pp. 369 •qq. XII.
B1dl • .Amer.
Math. Soc.
50-52]
55
THE SURFACE IN ORDINARY SPACE
may therefore take instead of the series of forms G4 , Gr,, ••• ,the quantity K, which is an absolute invariant, and the covariant derived forms of K. The set of algebraic forms is now (i) the fundamental quadratic, (ii) cp, .p, K, and their successive covariant derived forms. All the deformation invariants and parameters of a surface are therefore given by the algebraic invariants of these forms. If the quadratic form is Edu2 + '2Fdudv + Gdv2, we have the invariants
~...1..~
=~a(rs)...t.~r~s =_!_{Ea§. H ov -2F~~+G~ Olt ov au I } ' ~ (q, ) =la cp ,,, =_!__ { E aq:, a.p - F (~cp a.p + 05!_ B4') + G ~ a_r_} H ov av au av av ote au o·u ' (...1.. ·'') = ~ = _!__ a.p - ~ 01} H {ocp OU O·V OU OV ' ...1..
' "'
ru\ \!:/
~'
't'
2
1
r'l''
-
~E
2
2
(rs) ,1.. ·'·
~r'YB-
where His written for JEG- F 2• These are obtained from the quadratic and two linear forms ; they are not independent, for ~ 2 ( rp, .p) + ®2 ( q,, "') = ~q, . ~"'' and we thus have an example of a syzygy between invariants. This has a simple geometrical interpretation, for if the curves 4> and .P cut at an angle a, it may be easily proved that cos a= 11 ( q,, .p) sin a= ® ( cp, t/1) Jb.cp.~~' J~cp.~~' and the intet'Pretation is therefore cos2 a + sin~a = 1. 'fhe invariants are all interpreted geometrically when the geometrical interpretation of lirp is obtained. Let dn denote the perpendicular distance between the curves cp and c/J + dcp at the point (u, v), then dcp a~ ilu acp dv dn = au dn + av dn . Also, if ds denote an infinitesimal arc of the curve c/l = const.,
01 ~ + ~ t!!=o ouds a~ds ' E
~ ·t!! - G ocp- F 0c/> • E ~- F ocp . dn · dn -
and since
E(~) dn
du
2
dv •
dv
du'
+ 2F
Hence
56
APPLICATIONS
(cH. v
we see that each of these ratios is equal to 1/H 2 J ~q,, and therefore dc/l/dn = J~. We thus have the geometrical interpretation of b.c/J, namely, it is the square of dcfl/dn. 58. An important paratneter of the second order is that known as Beltrami's second differential parameter ll. 2 c/J:: la
'
::.!. ~ {! (G ocfl - F~)} + _!_ ~ {_! (E~- F~)}. Hou H olt ov Hov H av au
Beltrami's method of obtaining it is by an application of Green's 'fheorem. In fact it is easy to prove that if cfl and .p are two functions regular inside a closed contour on the surface, and if du_ denotes an element of area of the contour, ds an arc of the boundary,
if 1Jr·rA ( cfJ, 1/J) du = - j t/t ocfl fin ds - }J 1/1 ~"' du, and it follows, since all the other ter1ns of the equation are invariants, ~nd since the contour is arbitrary, that 4 2 cfl is an invariant.
. 54. All the parameters may be expressed in terms of three of them, and their derivatives. The invariants ll., ®, l12 once obtained, we can calculate from them many others, for example Jl./1 ~' il. ( rp, b.~), 42 (b.~), t1 ( b.cfl, 4ep), etc. We shall prove that all invariants can be obtained by algebraic combinations of these, a result due to Beltrami. Suppose tbat the quadratic form has been transformed so that cfl, tf! are the parametric curves, then the coefficients may be calculated without trouble and we have iJs2 =@I ( ~' 1/1) {Al/;d"'~ - 2L\ ( "'' 1/1) d#l/1 + At/Kit/}}. Now let I be any invariant involving the two functions
aA
iij = 0 (>.., "')/0 ("'· "').
Hence all the derivatives may be expressed by means of the ®'s,
52-55]
57
THE COMPLETE SYSTEM OF PARAMETERS
and the result follo,vs. If the invariant contains only one function rp, we may take c/J, and either b.c/J or b.2cp, as the two independent variables of the form. It follows, in this case, that all the invariants may be obtained by repeated application of the operations d, b.2 to the single function q,. Although we thus have a method whereby all invariants can be obtained, the result is not complete, for we have no clue as to the inde· pendence of the invariants, and frequently they are not thus expressed in their simplest for1ns. We know from the algebraic theory that there are three independent invariants of the second order involving one function cfl. These must be b.b.cf>,
a (cf>, b.cfl),
b.2 cf>.
There are four of the third order, and these are included in b.b.b.cfl,
b. (.P, b.b.tfl), b. (b.cfl,
b.~cf>),
b. (b.2cp),
~ (c/l,
b.2c/J), etc.,
but we have no im1nediate method of showing what are the relations that connect the last set. 55. Another method for deriving invariants of order 1· + 1 fro1n those of order r is the following : Let I be any invariant, and let ds1 denote an arc of the curve cf> = coust. We have
di = oi du + ~I dv dsl
ou dsl av dsl
~ d·u + OJ!. dv _ 0 J
au dsl ov dsl -
'
E(du)2 + 2F(~) (dv_) + G(t!!)~ = 1. ds1 ds1 ds1 ds1 Hence di/dsh which is obviously an invariant, and is of order one higher than I, is ®(I, cfl)/ J~. Simila:rly dljds2 =®(I, t/1)/ JiS:¥, where ds2 is an arc of the curve .y = const., and any function of these two quantities is an invariant of order r + 1, where I is of order r. In particular, suppose that , tf! are such that a (cp, tf!) = 0, then the curves cp =const., .p =const. cut at right angles. An invariant of order r + 1 is + ds1 ds2
(d[)2 (d[)2,
and this is readily seen to be AI; it thus does not involve rp, .p explicitly. For example, the Gaussian invariant of the third order may be written either b.K or (dK/ds1) 2 + (dK/ds2)1, where dsh ds2 are any two infinitesimal arcs through u, v cutting at right angles.
58
[cH. v
APPLICATIONS
56. Geometrical properties expressed by the vanishing of invariants. We next consider the geometrical properties involved iu the vanishing of certain invariants. If ® ( cfJ, .p) =0, from what we have already seen, 4> and .; cut at an angle zero; therefore they are coincident, that is to say, ~ is a function of ~. If A ( c/l, .p) = 0 the curves cfl and .p cut at right angles. Now At/> is A ( c/J, f/J) and hence, if 4cJ> = 0, the curve cfl =const. is at right angles to itself. This curve (of course imaginary) must therefore be such that the tangent at any point meets the circle at infinity. Such curves are analogues of the straight lines '!I + i:c =lc in a plane. Along such a curve ar~>
a~ OV
:.-- du, + -.:- dv = 0 Ott,
'
and it therefore follows that Ed~t2 + 2Fdztdv + Gdv2 = 0, that is to say ds' = 0. The curves are hence such that the distance between any two points on one of them, measured along that curve, is zero. Consider the more general case of Acfl = f (cfJ). Let the curves .P be chosen at right angles to the curves cp, then A ( cp, 1/1) =0, and therefore ..7 2 df/>2 dt/12 = t:t. c/l + ~' ((18
or if du=dcpjJJ·(cf>), ds2 = dlt2 + 0 2dl/J, where 0 is a function of u and .y. The curves u =const. and the curves cfl = const. are obviously the same, since ~"is a fttnction of c/J. Now if 1/p is the geodesic curvature of the curves t/J = const., and
Hence, in our case, 1Ip is zero. 'rhe 1/J curves are therefore geodesics, and the u's cut them orthogonally. Hence, if b.cJ> =f (cfl), the curves ~ = const. are the orthogonal trajectories of a family of geodesics.
57. In order to develope the properties of 11 2~, we consider the following problem. It is proposed to express the fundamental quadratic in the form P (.x) dp2 + 2Q (.x) dpdg + R (OJ) dq2, · where P, Q, R are given functions of the single argument :c, and :c, p, q are functions of u, 'l' to be determined.
56, 57]
THE VANISHING 01•' PARTICULAR INVARIANTS
59
We have 11p = R!(PR- Q2) == S, say, then Sis a known function of :c. Also Ra:c_Q~
11 (p, t::. p ) = S and
'
papR - Q2oq '
~
®
, ( ) a2 , , p, 11p = 8 'J· PR-Q 2
{!_ R _!_ Q } P - J p R _ Q2 op J p R _ Q2 aq J p R _ Q2 •
112 _
1
From these equations, by elimination of :c, o:cfop, orcfoq, there is obtained a relation between the invariants of the first and second orders of p. This relation, being invarianti ve, may be written in terms of the original form, and we thus have an equation of the second order for JJ as a function of u, f'· Taking any solution of this equation, we obtain without difficulty an equation of the first order for q, and then te 1nay be obtained by elimination. Consider, for example, the case in which Q = 0, R =P =:c. We now have for the quadratic form, m (dy + dq2), and 1 Ap = ~·
1 ax A (p, Ap) =- {ij Op'
8 (p, J.\p) =
1 a:c
/? aq•
A1p= 0.
The second order differential equation is thus 112 p = 0. · Conversely, let p be auy solution of ll. 2 p = 0, and let a function q be chosen so that 4 (p, q) = 0.
We have
!{! ~ {! (EOJ!.-F~)}=o ouH (G~-FBJ!.)}+ au av ovH av au ' E?J!. av ~-F(?E ov au ~ ov +~~)+ av au G?£ oze ~ au =0. From the first of these equations it follows that
(o
(E
_!_ OJ!.F~) =Of"_ _!_ ap- F~) =-Jt. H au av ev' H ov au au' where f is a certain function of u and f', and the second. equation becomes
~~-~~=0 au av av au •
Hence q is a function of/, and without loss of generality we may take it to be exactly j. We can now calculate Ag in terms of the derivatives of p, and we have in fact dq = Ap. Thus the quadratic form
becomes
i.p (dp' + dq'), and hence a: is 1/J.\p,
60
[ca. v
APPLICATIONS
Nolv if the quadratic form can be \vritten a· (dJJ2 + dq2), the surface is said to be divided iso'lnet'rically, or ?·sothermally, by the systems of curves p and q, and "·e have the important result that the solutions of ~2P = 0 give the fan1ilies of isothern1al curves on the surface. We notice that, if one conjugate pair of solutions JJ, q of i12c/J =0 has been deter1nined, then taking these as independent variables, we have _ 1 (o2) =~ + -:::-:; , .. :r tJP.. oq.. and hence the 1nost general solution of i1 2 cp = 0 is given by cJ> =It (p + iq) +A (p- iq), ~.,cp
whereft and}; are arbitrary functions of their arguments. Other examples of this theory (given by Darboux) are dSJ = cos2 :rdp2 + sin2 xdq2, in which case ~ (JJ, ~p) = 2~2P (11JJ -1),
ds2 =:cdp2 +
and
~
for which Exantples.
(p,
~p) =
! dq_B,
i1p i12]J.
Find the relation among the in\'"ariants of p if ds2=dp2fsin 2 x+dq2fcos2 .7). 9 (ii) ProYe that if the equations A2'/J==O, t. ( ~~cJ>~cJ>), cJ>) = 0, have a (i)
common solution, the quadratic can be reduced to LiouT"ille's form, ds 2=( U + l') (du2+dv2), where U is a function of u only, T7 a function of v only, and show how to find the parametric curves u, v.
As another illustration, consider the curves cp for which 112c1>/b.cfl =I Cc/J). If we take these for the parametric curves u, and their orthogonal trajectories for the curves v, \ve have dil =A 2dtt2 + C 2d#, and
~.u = }oo~, (~)'
Au=
1~·
Hence our eq nation becomes
~(log~) =/(u), and therefore 0 = AeU+ v where U is a function of u only, V a function of v only. Hence dt =A 2e2U [e- 2u du 2 + e2Y dv2] =A (du' 2 + dv' 2 ),
57' 58]
61
.APPLICABILITY OF TWO SURFACES
where tl is a function of tt, v' a function of v. The curves tit' are the same as the curves¢, and it follows that if l12¢/~¢ =/(¢),the curves cp form an isothermal system.
58. Applicability of two surfaces. We no'v consider the general problem of the applicability of two surfaces. Let the quadratics for the two surfaces be Edzt2 + 2.Fdltdv + Gd'lf,
E 1dlt12 + 2F;du1dv1 + G1dVt2•
We first calculate the invariant K for the t\VO forms, and we must have K(lt, v) = K1 (ut, vt)· This gives one equation for tt1, v1 in terms of u, v unless K is a constant. As \Ve can have no relation between tt, v· alone, X. 1nust in this case be also the same constant. Let K =:: K 1 = a, a constant. Choose on the first surface any point P, and take as para1netric curves v the geodesics through this point, the curves 'lt being their orthogonal trajectories. We have then for this surface, ds2 = du2 + 0 2d#', and 0 vanishes with tt, whatever be the value of v. Similarly, the second quadratic may be reduced to 2 1 o2C o C 2 2 2 dzt1 + 0 1 dvt • No'v K =- ~ 2 , ai1d therefore ~ 2 + aO= 0. Hence 0 lllt tJlt 0 = V sin Jau. Similarly 01 = V1 sin J azt1. If, then, we take u =u1 , Vdv= V1dvt, the quadratics are transformed into each other, and we see that in this case the surfaces can be applied so that a given point on the first corresponds to any given point on the second, and so that a given geodesic through the point on the first corresponds to any given geodesic through the corresponding point on the second. Now suppose that K and Kt are not constants. We must have AK = ~'IG, where the index denotes that the invariant is formed with reference to the second of the two forms. There are thus obtained t\vo equations for 'lt1 , e1 as functions of u, v. If these equations are inconsistent, the surfaces are not applicable. 'fhere are two other possibilities ; either they are independent of one another, or one is a consequence of the other. In the second case it is easy to see that AK must be a function of K, and A'IG 1nust be the same function of K 1 • Suppose that ~K =.,f(K), and form the invariant A 2K. Again there are the three possibilities before mentioned. If
A2K = ¢ (K), A2' (Kt) = cJ> (K1), take the parametric curves u so that du = dK/f(K).
Then A (u) = 1,
62 ~ 2 (u)
APPLICATIONS
= F(u),
(cu. v
say, and ds' = du2 + (J2d'f1, where the v's are the
orthogonal trajectories of the u's. Also .:11u= ~~; hence 0=
VeJF(u)du,
where Vis a function ofv. Now take Vdv=dv', and it follows that ds'::: du'J + 6 JF(u) du dv'B. Similarly, for the second surface ds' = dull + e JF(ua)dul dvl'2, and the surfaces are deformable into each other by means of u = u1 , v =v1 + const. It thus follows that if K=K1, b.K=li'Kt, 112K=~2'K1, lead to only one equation between u, v, u1, v11 the surfaces are applicable to each other in a single infinity of ways. Now l~t the three equations just mentioned lead to two equations between u, v, u1, v1. Suppose first that K and 4K are independent of ea.ch other, and take them for the variables in the first form. Similarly, let ~ and ~, K1 be taken for the variables in the second form. Then, if the surfaces are applicable to each other, these two forms must nO\V be the same. It follows at once from the general expression for the form given on p. 56 that we must have ~11K = 11'~'K,., 11 (K, ~K) = 11' (K,_, fl..,~). If K and AK are dependent on one another, we take K and A 2K as the variables of the form, and the necessary and sufficient conditions become in this case
tl.tl.sK =11'~;lit, 11 (K, 11aK) = fl.' (I1t, ~~,K1) in addition to K = K1, tl.K =fl.' K1, A2K = Aa'K1. 59. The quadratic form in three variables. For this case we only note the significance of a few of the more important invariants of lowest order. The geometrical interpretation of those of orders one, two and three for a form of rank zero, the case of ordinary Euclidean space, has been considered in detail by Forsyth in his memoir, already quoted, on the differential invariants of space. We note that there are six Riemann symbols. The explicit expressions of these have been given by Cayley, and they have been discussed in detail by Lame for the case in which the parametric
58, 59]
'l'HE QUADRATIC FORM IN THREE VABIABLES
63
surfaces cut orthogonally, iu his book on Curvilinear Coordinates. If they all vanish, the space is Euclidean. 'fhe invariant Ac/> la('·')~rq,, is (df/>/dn) 2, where dn denotes an element of arc perpendicular to the surface cp = coust. The invariant ~ (f/>, t/1) E la(ra) c/Jr'/1s
=
(~) (~) cos a,
•
IS
where
a
is the angle between the surfaces ,P, ap, and d1l1 is normal to ®(c/J, af!, x) '1.c(rllt)q,,..;,xe
is equal to
=
deb) (d"') .0 (dii il1i"t (d)() dn~ Sin
.p.
'
where sin 0 is the sine of the solid angle at which the three surfaces
~)~(~) (~ (dn1 dn2 dna
2
')
and it is therefore equal to ®2(u1, u2, Us).
sin2 0,
64
APPLICATIONS
(CH. V
Now suppose that Af/l =f (f/l), then, if we take
d~/JJ(~) = du, the surfaces ~ =const. are the same as the surfaces u =con st., and 11tt = 1. If we choose v and w so that ~
we have
(u, v) = 0, A (u, w) =0,
ds2 = du2 + a?:Jf]'lf + 2adlvdw + as:tJ,w.
The square of the element of length for any surface w=f(v) is thus seen to be du'J. + 0 2d'lf, and hence the curves dv = 0, dw =0 are geodesics on this surface. It follows that they are geodesics in the general space. Thus, if ~cp =f (f/l), the surfaces f/l are the orthogonal trajectories of a normal congruence of geodesic lines.
60. Condition that a family of surfaces form part of a. triply orthogonal system. We next consider the questions: (1) Can the manifold for which dSJ is the fundamental quadratic contain a triply orthogonal system of surfaces1 (2) How is the most general such system to be determined? It appears that, if the surfaces
=
P, Q, R are obviously invariants, and P = la(rs)VrW8 • If we differentiate this covariantively, we have from the rules for covariant differentiation, and since a(r•t> = 0,
(rs) * The method here used for the determination of this equation has not previously been given. For the equation itself, for the general quadratic form in three variables, see a bote published by the author (Bull • .A.nter. Math. Soc. Vol. xu. p. 879). The equation for the particular case of the form da:?.+ dy 2 + dz2 was first given in explicit form by Cayley, and is obtained by Darboux (Orthogonal Surfaces, p. 14 sqq.). The reader will find it instructive to compare the method given here for the general case with that of Darboux for the particular case.
59, 60]
TRIPLY ORTHOGONAL SYSTEMS
65
where the bracket (rs) denotes that the summation extends to these letters. Also (p) and therefore ~P1ut = Ja
=
A (pq) =A (w).
If we form the covariant derivative of I we have I, = lA (N)(v,.,wq + e,wq.,) + lA (pqk)ahkv,wq, (pqk) and hence, as in the previous case, li,uP~) = lA (Pv,put + la(At)v,up,, (At) Qq = lafht)whqut + la(M)w,uqt, ( kt) and therefore we can at once eliminate the second derivatives of ~ and w from the above equation. It then becomes "JJAufA) = lA (pq)Wq (Rp- 2a(At)Upe'IJA) + lA (N)vp ( Qq -loJ'tuqewA) + l.A (fKJ.k)a(ht)ahJcutvpWq, (pqlc'At) or, if we bring over the terms in B and Q to the left, and change the
66
[ca. v
API,LICATIONS
letters of summation so as to havevpwq in the typical term in each sum on the right, we hs.ve ~IAu(A)- ~A (TJq)Rpwq- 2A (pq)vpQq = :JA (pqt)a(ht)ahk'UtVpWq - 2A (Aq)atpt>uAt'VpWq - 2A (PA)a(at)uAtvpWq ( pglckt) ='J:8{1Hl>v,wq, (pq) where the coefficients EW«> form a contravariant system of the second order, and IJ(Ptl.> = B(qp) = lA (pqk)a'"'>ah~cUt- 2uM (a("'> A (Aq) + a(qt)A (A.P)). (kkt) Now, since I= 0, R =0, Q =0, it follows as before that I~a, Rp, Q'l are all zero, and hence the left side of the equation just given vanishes. It therefore follows that 2B(Pq)VpWq = 0. (pg) This equation is thus another consequence of the first three. It involves only first derivatives, in the combinations v,w,, v,w4 + 'l.'qW11 , of the two functions v, w. Its coefficients are, however, functions involving the third derivatives of t~. We now have three equations laCr•)v,.w,, 2..4 (r•>v,.w,, 2Bv,.w, linear an~ homogeneous in the six quantities '· V1W1 1 VtWI, 11aWs, 'V~s + 'VaW1, VsW1 + 'lhWs, V1W1 + VaW1, and, in addition, A ( u, v) laurv, s lu(•)v, =0, l1 (u, w) lu(•)w, = 0. Now from u,(l)vt + u<•lvs + u<3Jvs = 0, u(t)wl + u<2>w,. + u(s)wa = 0, we deduce by multiplying the former by w1, the latter by Vt, and adding, 2u(1)V1W1 + u(s) (V1U'a + VsW1) + u
= =
J
'
60, 61]
CONGRUENCES OF CURVES
67
which is the differential equation of the third order for u. This equation may also be written l «abc f.tJtff.gh'k A (ad) B(bu) a(•h) u(c) u(/) u(l:) =0, (abcdefgllk) and this shows that the above determinant, which is 2a-t times the left side of this equation, is an invariant. It is in fact the algebraic invariant of four ternary forms, three quadratic and one linear, which is linear in the coefficients of each of the quadratic forms, and cubic in the coefficients of the linear form. This result might have been obtained by contravariant differentiation, starting with the expression l ara u
would have been of the same character and the final result of tbe same form, the only difference being that each contravariant derivative involved would be replaced by its reciprocal covariant derivative with reference to the' fundamental form. 61. The quadratic form in n variables. A general method for dealiug with a manifold for which ds' is given is due to Ricci and Levi Civita. The variables being :c1 , ••• , a:n, they write d:crfds = "A,(r). (r = 1, 2, ... , n) Then A_(l), ••• , A(") is a contravariant system of the first order, and in fact l ara~Jr)~.(•) = 1. The reciprocal system ~1 , ••• ,A,. is given by r,s
A.,= lar,A(r>, and therefore ~A(')A, = 1, and also ~ a(r•) A,.AR = 1. r ' ~·
If we
suppose the quantities 'A(r) arbitrarily given functions of the independent variables OJ, the values of the ratios dtDr/ds are given for any set of values of these variables. There is thus associated a direction with each point in the manifold, and, if we start from a given point and proceed always in the direction associated with the point reached, we finally obtain a curve in the manifold. Hence the equations given above define a congruence of curves in the manifold, such that there is one and only one curve through each point of the manifold. Let p, define another such congruence, then the cosine of the angle between two intersecting curves, one belonging to the congruence A, and the ()ther to the congruence p, may be proved to be lA(r)l'r = lA,.p.(r) = l a,.,A
r
~·
~·
If in particular the curves of the one congruence are everywhere orthogonal to the curves of the other, lA.(t1p,. =0. r
68
[CH. V
APPLICATIONS
62.
Orthogonal ennuple.
Now take n such congruences, denoted by A.,"' or 'A,(r), (k = 1, 2, ••. , n), where the letter k denotes the congruence. (A., is one symbol and is not a covariant derivative.) Let it be assumed that all the curves of these congruences cut each other orthogonally, then lA.,trJAt;r = fJhk, (k, 1c = 1, 2, ••• , n) r
where '7Ak =0 if k =f lc, and fJ'AI, = 1. Such a set of congruences is called an ortkogonal ennuple; the notation [1], [2], ... , [tt] is used for the congruences of the ennuple, 1, 2, ••• , n are the curves of the congruence that pass through a given point, and s1, s~, ... , s, are arcs of these curves. Any covariant or contravariant system can be expressed in terms of invariants and the coefllcients of an orthogonal ennuple. For let Xr1••• r,_ be a member of such a system, then elL.... .. l ~,., . . (rl) ••• .\,.,. ... (r,.) ..X:r1"" ••...."' ' ""1'.. •• ,.. 68.
=
~·
r 1••• r"'
is an invariant, and on solving all the equations of this type for the X's we have · Xr1 • ,."' = ~ Ckr··h.,. A,,1tr1 • • • AA,.tr•' hl•••h,.
which proves the theorem. It follows that if the members of the system X are all zero, the invariants c are all zero. Thus any absolute system of equations may be expressed by the vanishing of a set of
invariants. A particular case of the theorem just proved is that the covariant derivatives of the A's may be expressed in terms of the A's themselves. Thus let 1~&~:~ = l Alt.(r) A,(•)AA/rs, then Ah.trs = l "Yw'A~etr'AI/B• ~·
~~
Now there exist relations among the 'A's, and therefore there are relations among their covariant derivatives, which are obtained by differentiating covariantively the equations of type ~Ah.(r) At~r r
= fJI&k•
We thus obtain SA_A(r)Aktrs + l'A,(")ae)t.k/r = r
or
0,
r
lA.,(r))..lcJrs + l'A,,.)..k(r) r r
=0.
62-66]
COEFFICIENTS OF ROTATION
69
On substituting in this the values of the quantities A.,.1r,, Ak1r,, we obtain l 'YApqApJ,Aq/,Ak(r) + 2 "Ykpq'Ap/,).gja'Ah(r) =0, 1',
r, p, tl
p, q
'2y'Al«J."Aq1, + lykhqAQJ, =0,
or
q
q
for all values of s. It therefore follows that y.,.kq + 'Ykhq = 0, and in particular 'YAhfJ. = 0. Thus the number of independent invariants y is in"(n-1).
64. Coefllcients of rotation of an ennuple. It is clear, from what has been said, that the geometrical properties of the ennuple are all contained in the invariants y. These invariants are called the coefficients of rotation of the ennuple. 65. Let I be any function of the variables, then Of jasA is an invariant, and it is equal to lAA
infinitesimal increments, and not a derivative, for I cannot be regarded as a function of n quantities s1 , 82, ••• , Bn·) Since this is an invariant, we obtain by covariant differentiation
~ j[_ = '$,A.,.Crljrp + 1J'Cr1A11frp, .,. .,.
oa:.v as,
and therefore Normal congruence. By means of this identity we can determine the condition that a congruence may be normal. A congruence is said to be normal if its curves can be considered as the orthogonal trajectories of a family of surfaces f (llh, ... , a:,.) = const. Suppose that [n] is a normal congruence, having the surfaces f as orthogonal trajectories. Then any curve of the congruences [1], [2], ... , (n-1] must lie in a surface /, and therefore 66.
Of -=0. a s,
(A= 1, 2, ..• , n -1)
70
[OH. V
APPLICATIONS
Also aaj cannot be zero, for the curves [n] are perpendicula.r to f. 8,.
Hence, if we substitute in the above relation, we deduce that YnkA
= Ynu'
where It, k take all values from 1 to n- 1. These conditions are also sufficient, for if they are satisfied a oJi' a aF
as" os" as, as"
is expressed linearly in terms of the quantities ~ap' where i, /l., I: take ~
.
all values .from 1 to n - 1.
This is the condition that the system of
~a'F r, =0 may have a common solution. If all the congruences are normal we have relations Yrkh =Yrhk, for all values of r from 1 to n, equations
where It, lc take all values from 1 to n excluding r, and also Yw + yw =0
for all values of k, lc, l. It therefore follows that all the y's with three different suffixes are zero; reciprocally, if all the y's with three different suffixes are zero, the ennuple is normal. Suppose that [n] is normal; then (k = 1, 2, ... , n - 1)
Also
(k = 1, 2, ... , n -1)
and thereforef,. is proportional to >.."1,.. Letf,. =,.,_>..,1,., then sincef,., =f.,. and
we deduce that
P-.Ant,. + plyniJ~Jr'>.;l• =P.~/1 + ~~}niJ'Att,~Jr • (i . ij If we multiply this equation by >..,.<•) and sum over all values of s from 1 ton, we get
ft.-1
lp,,'An(I))..Rfr + ~ l Yntn~/r = ~r' •
or, finally, if
i=l
66-68]
GEODESIC CONGRUENCES
71
67. Families of isothermal surfaces. A family of surfaces f is said to be isothermal if 2a(ra)j
=0.
,.,
Such families are the generalisation of equipotential surfaces in ordinary space. An isothermal family may be considered given if its normal congruence is given; let then [n] be its normal congruence. If we substitute the value off,., in terms of the A's in the equation ~a(r•)f,.,
~ p,a}r•) >..,,,.
we have
,.,,
or
=0,
+ JL l Yn(J a(rB) ~tr ~,. =0, i.Jr•
lJL,).,(•) + JL ly,"' = 0,
'
'
tt-l
v = - l 'Ynii.
that is to say
i=l
If, then, [n] is normal to an isothermal family, this value of v must make n
n-1
l
r=1
{v ~n1r + ~
i=1
'Y"''"' Ai/r} d:e,.,
a perfect differential. The conditions for this are Bv
a'YAnn
a-~A + ar,. + V')'IJ.nn + ~·u v
'a'8~:"'"'" +
n-1
~
i=l
,., -1
~ i'-tnn ( 'Ytkn - i'-tnA) = 0,
i=l
:k.
')'inn "YN = 2a".!!! +
s,.,
n-1
2
i=l
')'tnn "Ywu
(k,k=1,2, ... ,n-1). These reduce to a much simpler form if the ennuple is normal.
88. Geodesic congruences. Let the coordinates of a point on any curve in the manifold be expressed as functions of a parameter t, and let accents denote derivatives with regard to t, then for this curve
ss =:p,q ~ apqtcp'mq', and therefore the length of the arc joining two points whose parameters are t 0 and t1 is
s
=1to s'dt =J.t,J~a 1
to
m'(I)ll'de•
'PlJ.P
Let 8 denote a variation due to taking a neighbouring curve, then
72
[cH. v
APPLICATIONS
If the curve we are considering belongs to the congruence A, then :ep' = s'AfiJ), la'" AlP)= A4 ; J)
[pq, k] + [pk, q] =
also
aa~4•• ~JJ
When these values are substituted in the above expression for &l and the result divided through by 2s', there is obtained the equation 8s' =2 Apo:tp' + s'2 Bz~e 2 [kp, q] A(P)A,(q). k
J)
:p,q
Now from the definition for covariant derivatives we have
a Al&k =-a A16 ~k also
~ {Ak, q} Aq, q
{kk, q} = 2 a(qr) [kk, r ]. r
Hence, since it follows that
a
Au=~ A"-~
[kk, r] A(r),
and therefore, changing the letters of summation, we have lA(P)Aq, P
= ~A(P)-a
B~P
P
At
- l [kp, q] A(P)A(q). p,q
If we substitute from this the quantity l [kp, q] A{P)A,(q) p,q
in the expression for &', we obtain &' = lAp &rp' + s' l AU') B.x~e a~- s'l&c1c :SAIP>Aq,, ~J)
p,lo
J)
k
p
= g_ [~&c11] -s'~8x~o:SAIPl Aq,.
ut JJ Hence, on integration 8s =
k
J)
[l Ap8.xp]"to - ltof'ts'l&c~o'1.A1Pl Atpdt. k
J)
f'
If this variation is zero for all possible neighbouring curves, we must have, in addition to conditions at the limits, ~&c~c:SAlP)Atp = k
0,
J)
at every point on the curve, for all possible values of the 8Zs. Hence, lA")Aq, =0 fJ
68, 69]
73
CANONICAL CONGRUENCES
for all values of !c. A curve for which this first variation 8s is zero is called a geodesic. It is clear that if sis a minimum, 8s must be zero, and tlj may be shown that, subject to certain conditions of regularity, and provided the length of the arcs is sufficiently small, if 8s is zero • • • 8 IS a mtntmum. The congruence ~ is therefore composed of geodesic lines if lA(P)Akv=O. (k=1,2, ... ,n) f)
Suppose that the congruence [ n] is geodesic. An(kp
we have
Then since
= ~".1 Yn1J'AilkA;IP'
~ Yn?J'An(")Ai1k'AJ;p = 0,
pij
(k= 1, 2, ••• , n)
or
From this it follows that all the quantities Ynin ( i = 1, 2, ••• , n) are zero. Thus if [n] is geodesic all tke in'Dariants i'nin are zero, and conversely. In particular, if the manifold determined by the quadratic form is Euclidean, these are the conditions that all the curves of the congruence may be straight lines. In the general case, if the congruence is also normal to the surface/,
.fr = p)..n/r'
P.r/ P.
= 'An1r • Jl
Hence fr/ P.r =f,f ~. for all the values of r and s, and therefore p. is a function of f. Hence we may modify the function f so as to have exactly f,. ='Anfr for all values of r.
69. Congruences canonical with reference to a given congruence. In many problems associated with a quadratic form, it happens that one congruence is given. If this is (n], the 'Il-l other congruences that form with it an orthogonal ennuple may be chosen in an infinity of ways. Out of this infinite number of systems of n - 1 congruences, there is one (or more) for which many of the relations are simplified. We proceed to define a set of n - 1 congruences which are called canonical with reference to the congruence [n]. Write 2X,., =AnJr• + AnJsr, and consider the system of algebraic equations
,.
l
,.
An/r 'A(r) =
0,
f"=l
'An1qp. + l (Xqr + waqr) ). =0, f"=l
(q =1, 2, ... , n)
74
APPLICATIONS
(CH. V
for the unknown quantities ;\,(t), ••• , ;\.C">, w, p.. These equations are linear in p. and the ~'s, when these quantities are eliminated we have an equation ~ ((a)) = 0 of degree '" -1 for w. Suppose that the roots of ~are all simple. Then to any root, say w," there corresponds a set of values for the 'A's. We thus have n- 1 congruences which are obviously orthogonal to [n]; they are orthogonal to each other. For, multiplying the second of the above equations by 'At(q) and summing for all values of q, we have l (Xqr + w~~,aqr) A~~,
or
q, r
Similarly and therefore
(w,- w~c) 2aqr >..,
w,
and since + wk, [k Jis orthogonal to [ k]. The set of congruences thus determined is called canonical with refereuce to [tz.]. If the roots of ~ (,.,) are not all simple, suppose that one root is of multiplicity p. Then we choose p congruences associated with this root and orthogonal among themselves, and similarly, by taking account of all the roots, we can obtain a canonical set in this special case. The set is uot, however, definite ; in the p congruences associated with the root above mentioned, for example, there is the same amount of arbitrariness as there is in a linear orthogonal substitution on p variables. It is easy to prove that, for an orthogonal ennuple in which [1], [2], ... , [n-1] form the canonical system with reference to [n], 'YMk+')',.kll=O,
and also
(k, k=1, 2, ... , n-1; k+k)
w~c =- 'YMt·
If (n] is normal, it follows that, with the same restrictions on k, k, 'YMk = 0. In this case the n - 1 congruences are the n - 1 fatnilies of lines of curvature of the surfaces orthogonal to the curves [n] .
70. Geometrical interpretation of the invariants y. The invariants y of an orthogonal ennuple hav.e an important geometrical interpretation. This will be illustrated by a simple case. We shall suppose the quadratic form to be that for ordinary Euclidean space, that is to say, it involves three variables, and the six Riemann symbols are all zero. In this case there are three mutually orthogonal congruences [1], [2], [8]. Now consider a frame of axes given by the
69-71]
GEOMETRICAL INTERPRETATION OF THE INVARIANTS 'Y
75
lines drawn through a fixed point parallel to the tangents at any point to the curves 1, 2, 8 through that point. If we suppose the quadratic to be dtJJ'J. + dy + dz1, the direction cosines of the lines 1, 2, 8 are given by the scheme 1
2
8
~1/1
~211
As11
~1/2
~2/2
A312
A11s
~2/3
~S/3
:c
!I
z
If~+ d~, '!/ + dy,
z + dz is a point consecutive to~, y, z, then it is
easily seen that
·
dm =l A,11 dsp,
dy = l Ap12 dsp, dz = ~ Ap13 ds,. f) f) ... Also, in this ease, Aplrl = Aptr and A,.tre =if~>..,.,"', where a:1, a:., a:a e a:, y, z. p
Again (Darboux, T"Mory of Surfaces, Vol. I. p. 5) if p, g, r be the infinitesimal rotations whereby the axes are brought to their consecutive position,
=l
Aa(r)
'Ap(a) ~radSp
r, s,p
=lp Similarly
)'m, dsp.
q = l 'Ya1pdsp, p
r = l Ytspdsp, f)
and hence the y's are seen to be the coefficients of rotation of the frame of axes.
71. Relations among the derivatives of the y's. Of the y's associated with a general form there are in1 ( n- 1) algebraically independent. These are not however absolutely independent.
76
[CH. V
APPLICATIONS
In fact it may be proved by forming the second covariant derivatives of the 'A's that if "'/AI, kl:;
a
a
as; "'/Atk- as;; "'/hU +; {yll(l ("'/;lcl- "'/Jik) + "'/JM'YJtk- "'/,1111:"'/JR}'
then
'YM, kl
=l
'1, r,
A.h(q) 'Air) 'A"(s) A,
s, t
(qrst).
These are the generalisation of the relations given by Darboux (Tkeory of Surjaces, Vol. 1. Chap. v) bet,veen the rotations of a system of axes in ordinary Euclidean space. If n = 2, they reduce to the single relation a a 2 2 K. ; - i"121 = ;-- i"212 = i"121 + Y212 + . uS u81 2
This has an immediate geon1etrical interpretation, for y121 and y211 are the geodesic curvatures of the curves 1 and 2.
72. Condition that a quadratic form admits an infinitesimal transformation. As an application of the theory of congruences given above, we consider the following question: What are the invariant relations that must be satisfied in order that a manifold for which the quadratic form is given may be transformable into itself by an infinitesimal transformation 1 lit other words: What are the conditions that a given quadratic form may admit an infinitesimal transformation? An example of a quadratic form which admits an infinitesimal transformation is given by dr for a surface of revolution in ordinary space, for such a surface is transformed into itself by an infinitesimal rotation about its axis. In the general case, let the infinitesimal transformation be
X/ =l
e(r)
r
.o/ O.Vr
J
then, if (dF/dt) dt denotes the change in any quantity F due to the transformation, dXr = e(r) dt '
d (d ) = d~(r) =
dt
Xr
l p
()~(r) d O.Xp Xp'
darB= l~(P) oars dt f) O.Vp '
and therefore the change in the quadratic form is ~ l'(P) aa,., d d ~ a~(s) .3 d } .3 ~ 'i 0$ Xr z, + 2 ~ a,.B UJ:Cp :e,. ut. {
p,r,s
tt,s,p
p
a
/J:p
The necessary and sufficient condition that the quadratic form may admit the infinitesimal transformation is that this must be identically zero. Now ~r = l ar8 ~(s), 8
71, 72]
FORM ADMITTING INFINITESIMAL TRANSFORMATION
O~r _ "
O:Cp
and therefore
~a,., 8
0~(8) ':l
fJ3Jp
+
~ l:(B) ~~ 8
77
oa,. a . 8
Wp
Erp = aaer - ~ {tp, q} ell=~-~ [rp, t] fl'l,
Also
Wp
q
fJXp
t
and it therefore follows that
~a... :e<•l =E.,+~ [rp, t] fl'l- ~f<•l aaa,.8. s
t
vtrp
s
Wp
Again, [pk, q] + [qk, p] = ~pq, and hence the condition becomes fJ:Ck
~ ~(P) p, ,., 8
([rp, 8] + [8p, r]) dtc,.dx, + 2 l {Erp + l [ rp, t] t
t
- l e<•) ([t:p, 8] + [sp, r ])} dre,.dtcp = 0, 8
or finally Hence the necessary and sufficient conditions that the quadratic form may admit the infinitesimal transformation are fr8+ ~sr = 0,
for all values of r and s. (This result is due to Killing, Grelle, Vol. 109.) The quantities e,. determine a congruence of curves in the manifold ; these are the paths of points in the displacement. Let us write Er =p"A,.1,. ; then and it follows that
Ers = S PYniJAt/rAjJs + P8"-ntr•
Hence since Ers + Esr = 0,
ij
l p ('YniJ + 'Y~U1) "Aitr~/8 + p,A,,,. + p,."A,,8 = 0. iJ
(r, s= 1, 2, ... , n)
We multiply this equation by "At(8) and sum over the values of s, and thus obtain the equivalent set of equations (a) lp ('Yntk + ')'n.kt) "Ai!r + bk'Antr = 0, (lc = 1, 2, ... , n-1) i
(b) lp (')'ntn + 'Ynni) Attr + b,."A,,,. + Pr = 0,
'
where bk is written for l
p8 'A"(8 ).
8
Again, multiply (a) by "A"'(r) and the sum over the values of r; the results are i'nhk + ')'n.kk = 0, (k, lc = 1, 2, ... , n- 1) and p ('Ynnt + 'Y•Jm) + b~c =0. (k = 1, 2, ... , n- I)
78
[cH. v
APPLICATIONS
Similarly from (b) we obtain the additional equation bn =0. Also 'Ynu =0 for all values of lc, for any orthogoual ennuple whatever, and therefore the necessary and sufficient conditions that the quadratic form may admit the given infinitesimal transfonnation are, finally, (A) Ynhk + i'nkh = 0, (ll, k = 1, 2, ... , n -1) (B) Pi'nhn + b, = 0, (k = 1, 2, . I., n- 1) (0) bn =0, where b" = l p,A,.(•). 8
73. 'rhese eq nations are in invariative form, and we proceed to interpret them geometrically. In the first place the set of equations (A) include in them all the conditions satisfied by a canonical system of congruences with reference to [n] ; thus every systetn of congruences forming with [n] an orthogonal ennuple is canonical with reference to [n]. The remaining equations included in (A) are i'nM=O. (k=l, 2, ... , n-1) To interpret these and also (B) and (C) \Ve need the generalised definition of geodesic curvature. Assume our manifold to exist in Euclidean space of sufficiently high order, and in this space through a point P of the manifold .draw a vector of which the length 'Y is given by n
Y =i=l ~ ')'2intu .and the direction by that of the tangent to the line which passes through P and belongs to the congruence p. ·for which P.,. :=::
n % 'Yinn}ti/r• i=l
This vector possesses the properties (1) It is identically zero if [n] is geodesic. (2) Its projection on the plane tangent to the lines i and n is equal to the curvature of the projection of the line n on the same plane. (3) It is normal to the line n. For these reasons, is called the geodesic curvature of the linen, and the congruence p. is called that of the lines of geodesic curvature of (n].
Consider the lines p. of geodesic curvature of [k]. P.A/r =
l-yu.h~/r• i
We have
(r = 1, 2, •• ., n)
Multiply this by }..,.(r) and sum over r. There is obtained the equation ')'taM
= l,.
).,.(r) P.hlr•
72, 73]
FORM ADMITTING INFINITESIMAL 'l'RANSFOBMATION
79
Therefore the remaining equations (A), 'YnM =0, show that the lines of geodesic curvature of every congruence orthogonal to [n] are themselves ort.hogonal to [n]. Similarly the equations (B) and (0) show that the eosin~ of the angle between a line of the congruence [k], and a line of the congruence of geodesic curvature of [n], through the same point, is h~&/p. Also Hence, from (b), which is deducible from (B) and (0), P.'n/r
= Pr/P•
Thus the congruence of lines of geodesic curvature of [n] is normal, and its orthogonal trajectories are the surfaces p =const. In particular, if the congruence [n] is normal, "Ynhk
= ')'nM,
(It, lc = 1, 2, ... , n- 1)
and therefore all the 'Y's except (k = 1, 2, ... , n- 1)
are zero. If the system of orthogonal trajectories of [n] is I= const., we have fr =U A,,r,
and hence ~ = U sl\n/r -
UPr -
p
~
1\nl•,
f,., =I.,. and A,1,. = /,./u
and therefore since
a
~ (pa)
we have
aJ
a
am; = az..
Of (pu) ~ •
It follows that pa is a function off, say F' (f). If we now write F(J) = g we have prr1 =1, where Ur= a 11\,1r; hence finally 1 1 Ur = - ~tr and g,., = - 1 {p).ntr + p,.An1,}. p
Again
p
1 {2p,A,,(a) + .!prA,(r)}
2 a(r•)gr, =
~ 'A,.(r))._A(•)gra =- I t", B, h p
t'_, 8
8
t'
2b,. 0 ..__------
pI
-
'
and therefore, if the congruence [n] is normal, its family of orthogonal trajectories is isothermal.
80
APPLICATIONS
(CH. V
DYNAMICAL APPLICATIONS.
74. 1'he position of any material system is given with reference to a fixed frame of axes when the cartesian coordinates of every point of it are given; frequently, however, the points of the system are not independent of each other, and it may happen that the coordinates of all the points may be expressed in terms of a finite number of quantities. Thus consider a rigid straight rod in a plane. Its position, and therefore the coordinates of all its points, are known when we know the coordinates of its middle point and the angle it makes with the OJ axis. The position of a sphere in space is determined by the coordinates of its centre, the angle that a line fixed in it makes with the z axis, the angle that a vertical plane through the line makes with a fixed plane, and the angle that a second line, fixed in the sphere at right angles to the first, makes with the line that is in the vertical plane above mentioned and is also at right angles to the first line. Whenever a, material system is thus determined in position by definite values given to certain quantities, these quantities are called the generalised coordin~tes of the system. Any change in the coordinates corresponds to a change in the position of the system. Suppose the system to be in a given position. It may happen that each change in the coordinates gives a possible change in the system ; on the other hand, it may happen that, owing to certain dynamical restrictions in a particular problem, some changes in the coordinates do not give possible displacements. For example, if a sphere is moving on a fixed plane with pure rolling motion, all infinitesimal displacements are excluded which do not make the displacement of the point of contact of the sphere with the plane zero (to first order of small quantities). In the first case the system is said to be lwlonomic, in the second non-/wlonomic.
75. Consirler any holonomic system, and let its generalised coordinates be o:1 , o:2 , ••• , :e,.. Then, if dots denote differentiations with regard to the tiine, it may be proved that the kinetic energy, T, of the system is given by
(i)
f','
where the coefficients ara are functiona of tc1 , ••• , m,.. Again, suppose the system to be subject to exterual forces which depend only on
74-77]
81
LAGRANGE'S EQUATIONS
position and not on velocities, then in a small displacement d.x the work done by these forces is where t.he X's are functions of .xJJ ... , .xn. of motion of the system are
(ii)
!:._
(aT) _O.l'r oT =-X r,
dt OXr
In this case the eq nations (r = 1, 2, ... , n)
the equations of Lagrange•. 76. The dynarnical problem involved may be regarded as completely solved (if "·e neglect such problems as the determination of the internal reactions of the system) when we have determined the most general values of the .:es as functions of t, that satisfy the set of equations (ii), the a'R and the X's being given functions of the quantities :e. Now we may regard the .x's as the coordinateR of a point in a general n1anifold of n dimensions, and then any particular solution of the equations (ii) will give the :c's as fcnctions of one parameter t. The possible values of these variables will thus determine a curve in the manifold; if this curve is given, the complete particular solution of the problem will follow as soon as one coordinate is determined as the proper fuuction of t. Such a curve as that mentioned is called a trajectory of the configuration, and it is clear that we have gone far towat·ds the complete solution when all the trnjectories have been determined.
77. Any n dimensional manifold whatever n1ay be chosen for the geometrical interpretation of the dynamical configuration. The obvious one to choose is that for which ds2 =
~ aril:c,dx8, tt, B
and then It is clear that, since l.X,.d.xr represents the work done by the r
forces in a small displacement, it is independent of the coordinates chosen, and hence the magnitudes Xr are a covariant system of the first order with respect to the fundamental form 'lar,dOJ,.d:c,. (If the • For details of the above, the reader should see e.g. Whittaker, Analytical Dynamics, p. 82 sqq.
82
(OH. V
APPLICATIONS
system of forces is conservative, the X's are actual derivatives of a function U). We may therefore take
X,.=piJT, where P.t, ... , P.n are the coefficients of a congruence of curves. These curves are called the lines of force.
78. Expression of the equations in inva.riantive form. The equations (ii) may be expressed in invariantive form in terms of an orthogonal ennuple ; to express them thus we choose from the trajectories any cougJueuce, which we take for [n]. Then Xr = uAn(r), where u = B8 ,
~! =laraXa = luar,Xn(•) =uA.ntr.
and
ux,.
d
Heuce
Tt
a
(aT) ~-;t 1Xr
8
8
aT _ .\ " BAn1r • " oaPi • • -8 = a'l\.t&/r + CTtllli4 a mp- 1!" ..-. a tc11 mq, 3.'r
p
'IIJ'P
p, q 'IIJr
and, after some slight reductions, this becomes
,
Q-Antr + a' I .JAn(.P) A.,/rp •
Again and therefore
An/rp
= ~iJ 'Yn'U~I,.AJIP'
2A"(.P) Antrp = 2y,1n ~1r = p
v,.,
'
where [v] is the congruence of geodesic curvature of [n]. Hence, finally, the equations (ii) become (iii) d-An,r- u 9v,. = pp.,., (r =1, 2, ... , n) Now let t/1 be any congruence whatever, multiply the typical equation (iii) by .p(r) and sum over the values of r. We have a2Antr "'(") - ullv,..p(r) = plp.,.tfi{,.,, ,.
or since (iv)
l~,..pfr>
,.
r
is the cosine of the angle between f/l and.;,
ucos (n.P)- u~ cos (vt/1) = p cos (p,.p).
This shows that any line perpendicular to the line n and to the line p. is also perpendicular to the line v ; in other words : TluJ tangent to tlte line of geodesic curvature of any t'rajectory lies in tke plane determined by t'M tan.gent to tkat trajectory and tke tangent to til£ line offorce throogk tke poi'llt. If, in particular, ·the forces X are all zero, A,"' is proportional to v,. or else u and v,. vanish identically. Now [n] and (v] are at right
77-79]
FIRST INTEGRALS 01-~ THE SYSTEM
(ii)
angles, and hence the first case gives n1inimallines. Excluding these, we have the result: If the external forces are all zero, the trajectories are the geodesics of the manifold, and they are described with constant velocity. When the field of force is not zero, we deduce, if in (iv) 've take l/J to be (k], any one of the congruences of the ennuple, u=p cos (p.n), 1 U ')'nhn = p cos (p.k), (k = 1, 2, ... , n- 1) which are the equations in invariautive form.
79. First Integrals of the system (ii). The n equations (ii) are linear in the second de1·ivutives of the ala with respect to t. If they are solved for these derivatives they take the form (v) .x, =X(') - l {1·s, i} x,.tt,. (i = 1, 2, ... , n) '1',
s
Let J' =co'lst. be a first integral of these equations, then f is a function of the quantities m and :t, and dj ~OJ .. ~OJ. -t = ~a-::c, + ...,i fi(Ci ~- x, d , 'llJf
(vi)
=l
~ ..l"' (i) + li
i t 1Xi
{:£- :hi - ~ l flfCi
fiXt .,-, 1
{rs, i} Xr:t,} .
'!'his last expression must he zero identically. Suppose in particular that f is a polynomial in the first derivatives :t, and let the terms of highest degree in these derivatives be u. Then, since (vi) vanishes identically, the coefficients of the various quantities Xr in it must vanish se:parately, and hence (vii)
~
~
i
{au . aw,
flJ#,-
au ~~ {rs, "'} fJJr. x,. } = _ 0. a-;w, .,.,,
Therefore u =const. must be an integral of the equations (ii) when all the X's are made zero. That is to say, to any polynomial first integral of the general set (ii) there corresponds a homogeneous polynomial first integral of the differential equations for geodesics, in the manifold for which dSJ = l a,.,d:c.,.d(l),. '1',8 Assume that u = ~ c.,.J·· .r. tbr, •••:hr,., f't' ••• , t'm
then the system c is covariant of the rth order, and (vii) becomes l ,.h••
Cr 1••• rmrm+l Xr1• • • d:r•+l
'"'"'+1
EE 0.
84
APPLICATIONS
(CH. V
80. Homogeneous linear first integrals of the equations
for geodesics. For example, suppose that m is unity, th()n == lcr.Tr:
ft
r
and (vii) becomes r,s
'fherefore Cra + Csr =0. (r, s = 1, 2, ... , n) These conditions show that the quadratic form must admit the infinitesimal transformation lc(r)
aj,
r
O:Cr
and hence the necessary anrl sufficient condition that the equations for geodesics of a manifold have a linear first integral is that the quadratic form of the manifold admit an infinitesimal transformation. If the geueral system (ii), where the X's are not zero, also has the linear solution u = const., the retnaining terms from (vi) give ~crX(r)
=0,
r
and hence the additional condition is that the path curves for the infinitesimal transformation must be orthogonal to the lines of force.
81. Quadratic first integrals. Systems (ii) that possess a quadratic first integral are of particular interest, for a reason that will appear later. Suppose the first integral to be l c,.,thrtbs = const. r, s
Then c is a covariant system of the second order with reference to l ar8 dxrdx, for which Cra = Csr, aud our conditions are t",B l CrseX rXsXe := 0, ~ Crs X (r) Xa := 0. ~~t
~8
These give the equations (viii) (ix)
Crat
+ Cstr + Ctrs =0, ~cr,X(r)
= 0.
(r,s,t=l,2, ... ,n) (s= 1, 2, .•• , n)
T
Hence the necessary and sufficient conditions that the equations for geodesics possess a homogeneous quadratic integral are the equations (viii). If, in addition, a general dynamical system for which the X's are not zero possess that integral, the equations (ix) must also be
80-82]
85
QUADRATIC FIRST INTEGRALS
satisfied for non-zero values of the X's, and hence the discriminant of the quadratic l Cr8 XrX8 must vanish. If this discriminant vanish, the 'I",B
ratios of the X's can be determined, in general uniquely, from the equations (ix). If the quadratic integral is not homogeneous, let it be 2 CrsXrOJ8 + '2brtbr + U =const. r,s
r
In this case the system must possess the linear integral 2,. brtbr =const., and hence also the quadratic integral 2crsXrXs + u =const. The con.. ~· ditions for this last are (viii) and lcrsX(r) + te, =0. (8 = 1, 2, •.. , n) (x) ,. One solution of (viii) is obvious, for we know that the first derived system of the coefficients ars of the quadratic form is identically zero. Hence one quadratic first integral of geodesics is ~ a,.stb,.dJ, and since larsX(r) ,.
=Xr,
'r, B
the equations (x) give X,.+ Ur = 0, that is to say, the
forces X must be actual first derivatives of a function -u. The system of forces is therefore conservative, and the first integral is the energy integral. . In addition to the solution Crs =Or, the general form does not possess a quadratic first integral for geodesics, and all the forms which do possess such integrals have not yet been determined, though many classes of such forms have been obtained. The case of n =2 has been completely solved by Liouville.
82. Systems which have the same tr~eotories. Suppose that we have any dynamical system for which the equations are the set (ii). Any solution whatever is given by definite expressions for the n variables a:1 , ••• , rc. as functions of the single parameter t. Thus to each solution there corresponds a curve in any n dimensional space, the trajectory of the solution. Now suppose that we have any other dynamical system d
(xi)
(au) au
dt1 oa:,.' - aa:,. = Y,.,
(r =1, 2, ... , n)
where 2 U =l c,.,a:,.' a:,', and accents denote derivatives with regard to t1 • ,., 8
We enquire what are the conditions that the trajectories of this system are the same as those of the former system. If the trajectories are the same in the two systems, let a:, =if (t), (i = 1, 2, ... , n), be a solution of
86
[ca v
APPLICATIONS
the former; then if Xt = F, (tt) is the corresponding solution of the latter, that is to say the solution that gives the same trajectory, these two expressions for :e, must become the same when we choose for t an appropriate function of t 1 • We write t = 6 (tt) and take l ar8dterdXs r, 8 as the fundamental form, and we also make the assumption that the. discriminants of the two quadratic forms do not vanish. Confining ourselves to a particular trajectory, 've have ''6'2, :Xr = X,,fl , Xr = Xr 6" + Xr I
'LJI
II
•
Xr + l {pq, t'} :hpdJq ::: X
also
p, fl
d (~ ') - .1-z ..,. ~ dt .IIGi CratCll
. and (Xl') IS
1
~ a
Or
"'"'Crs:C8
11
OCpq
,
,_ v
~-- Xp .Xq -
p, q t 1Xr
8
~ oc,.ll +"'"' 8, P 0X p
:C8
t
1 .1 ~ ncpq .x,J - 2 .... ~p, q
titCr
.L
n
1
tCp Xq
I
= "lJ" .L r•
If we substitute in this for x', a!', ii, and replace the quantities : ; p
by the .covariant derivatives 'vith respect to the fundamental form "lar8 dterd:c,, this equation (xi) becon1es, after a few reductions, (xii)
lcr,th,()" + lCr8 X<')fJ' 2 + t 6' 2 l 8
(crpq
+ Crqp- Cpqr) XpXq = Yr.
~q
8
Now we haven equations of type (xii), and if any three of them were independent we could eliminate 8", 6'2, and thus remain with an equation involving only the first derivatives :t, 'vhich would be the same for every trajectory ; this is impossible, and it follows that two at most of the equations (xii) can be independent. One solution of these equations is immedi~tely obvious. If 6" is zero, it readily follows that Cr8 is Oa,.s where 0 is a pure constant, and then the Y's bear a constant ratio to the X's. We neglect this case, and consider that in which all the Y's are zero. There can be now only one equation (xii), and therefore lCr8 Xa lCr8 XCs) + ! l (c,.pq + Crqp- Cpqr) XpXq s - 8 p,q (xiii) l CkBXs - l ck8X(8 ) + ! ~ (Ckpq + Ckqp - Cpqk) Xp Xq 8
p, q
8
must be an identity, for all values of r and k. If we multiply up, and equate coefficients of the derivatives x on both sides, we have in the first place (lCrsXs) (lcksXCs>) (lc~,) (~CrsX<8 >). s
s
=
~
8
Hence, since the discriminant of U does not vanish, we must have lckBXC') = 0 for all values of /c, and therefore finally X<•l = 0.
'
82, 83]
87
CORRESPONDENCE OF GEODESICS
88. Representation of one manifold on another with correspondence of geodesics. Therefore, if two dynamical systems have the same trajectories, and if these trajectories are geodesics for one system, they are geode-
sics also for the other, and we now consider the question of representing one manifold on another so that the geodesics of the one correspond to geodesics of the other. The equation (xiii) gives (lc,.,tb,) (2 (Ckpq + Ckqp -
Cpqk) XpXq] :::
(2cksf.h,)
~q
B
B
This equation shows that
[~ (Crpq
+ Crqp -
Cpqr)
XptVq].
~q
2c,.rt, must be a factor of IJ
and the remaining factor must be symmetrical. Hence we have
l (Crpq + Crqp- Cpqr) XpXq:: (lc,.sct,) (2bpbp)
p,q
p
for all values of r, and the b's are quantities to be determined. From this equation, by equating coefficients of dJpfhq on both sides, and noting that Cpq =Cqp we have (xiv)
2 ( Crpq + Crqp -
Cpqr)
=Crpbq + c,.qbp,
for all values of p, q, r. Let a, c, be the discriminants of the two forms '2a,.,drcrtk,, lc,.,d$,.da:,, then c/a is an invariant. Also, if the cofactor of c,., inc is alc
variant. Multiply (xiv) by a/c(rq) and sum over the values of r; the result is 2al (crpq + Crqp- Cpqr) A;(qr) =cbp, r
for all values of q except q =p. If p = q the right side becomes 2oop. When these equations are summed over all values of q there is obtained the result
(n + 1) cbp =2a 2 (crpq + Crqp- Cpqr) k(F), q, r
and since /c(qr) =/c(rq) this becomes (n + 1) cbp = 2a 2 k(qr)cg,. q,,.
Now
88
[cH. v
APPLICATIONS
and therefore
= n~. {.,.2 s(n. -1)! k(rl•l)Cr 1
=2
,
181 p
+ 11- 1 similar terms}
1
k(rs) Crap.
f", 8
Hence
(n + 1) cbp = 2a (c/a)p,
and therefore
a ( log - . bp = 2 ~ n + 1 v:rp a
c)
Thus the b's are the derivatives of the function _!_ log (cfa), and n+ 1 from (xii) we have (J'' + l8' 2 lbp:bp = 0, p
1
and therefore
8' = 0 (afc) n+t;::; p., say,
where 0 is an arbitrary constant. If we write A,., =p.2c,., we deduce at once A rat+ Aatr + A,.,= 0,
and therefore the set (ii) admit the quadratic first Integral
.
2 A,.,:t,.x, = con st. ,
.. 84. Again, we write
2 (c,., - pa,.,) A(•) =0.
(r=l, 2, ... , n)
8
If this set of equations gives a system of non-zero values for the >:s, the determinant II Cn- pa,.,ll must be zero. To each root of this equation of the nth degree in p there corresponds a set of values for the A's, and, if the roots are distinct, these n sets determine an orthogonal ennuple. If the roots are not distinct, an orthogonal ennuple may still be determined in this way, the only difference being that k congruences appertain to a root of multiplicity k of the equation II c.., - pa,., II = 0. If we express our quantities in terms of an ennuple thus determined, we have
c,., ='Jp,.A~a 1,A161, A
and when these values are substituted in the equations of condition
83, 84]
89
CORRESPONDENCE OF GEODESICS
(xiv), we get a set of equations which reduces finally to the following
set: (a.)
(k:t:t+j~
({J)
(i+J)
(y)
(i+j)
(8)
= 1,
(k, i, j
2, ..• , n).
Suppose that the roots p are all different, then, from (a), all the y's with three distinct suffixes are zero, and therefore then congruences are normal. We take the normal surfaces of these congruences as the coordinates m, and then the fundamental quadratic form becomes lHl~dm.,9,
and ds, = /lsdm1• remainder are
" The set of equations (a) are
now satisfied, and the
2 (p,- PJ) .;._(log Hc) + aap, =o, tlleJ
'IIJj
a
&; (~LPt) = 0, (~)
(i +.i) (i+j)
;._. (p.p,) + p, aop. = o. u:c, w,
'rhese equations are integrated without difficulty. From (y1) we have f'p, = ~', where ~~ is a function of :c, alone, and then, from ( 81), P.~t is independent of m1 • Hence ifF: cp1 ~ ••• c/l,., p.Fis independent of :c,, and therefore, since it is symmetrical, it is a pure constant, 0. Hence p. = 0/F, and the equation (/31) becomes 2 (~- ~J)~(log H,) + ~ ~=0, ~J amJ
amJ
and therefore on integrating we see that t/lJHl/(q., - 4-J) is independent n
of
:CJ.
Let P, denote the product II' ( ci>l - ciJJ), where the value j J=l
=i is
excluded; then Fllc'/P, is a function of m, alone, and by properly choosing a function of m1 as a new variable instead of :e, we may make its value o;q,,.
90
APPLICATIONS
[cH. v
The quadratic form lar,tlm,d:c. can therefore be transformed so as to be ~ ~
Also
C·rs =
OP, _, 2 Ft/lc u·tec.
lp,,'A, ,,.'A,.'s = 0, h
aud cu =p,H,2 = P, ; therefore the corresponding form is lPid:ci2•
" We thus have the complete solution of the problem of representation
with correspondence of geodesics in the case when the p's are all distinct. The case when son1e of the p's are alike involves modifications but can also be fully solved. 'rhe general pi·oblem with which we started, in which the forces are not zero, has not yet been fully solved. The method of the Absolute Calculus removes all analytical difficulties not inherent in the problem, and it is possible that that method will ultimately give the general solution •. * On this problem see: Painleve, Liouville's JournaZ, Ser. v. Vol. x. (1894); Levi-Civita., .A.nnali di Jlath. Ser. u. Vol. xx1v. (1896).
(continued from front flap) THE ALGEBRAIC STRUCTURE OF GROUP RINGS, Donald S. Passman. (0-486-48206-5) A BOOK OF ABSTRACT ALGEBRA: SECOND EDITION, Charles C. Pinter. (0-486-4 7417-8) INTEGRATION, MEASURE AND PROBABILITY, H. R. Pitt. (0-486-48815-2) BASIC METHODS OF LINEAR FUNCTIONAL ANALYSIS, John D. Pryce. (0-486-48384-3) THE CALCULUS PRIMER, William L. Schaaf. (0-486-485 79-X) FRACTIONAL GRAPH THEORY: A RATIONAL APPROACH TO THE THEORY OF GRAPHS, Edward R Scheinerman and Daniell-{. Ullman. (0-486-48593-5) INVITATION TO DYNAMICAL SYSTEMS, Edward R. Scheinerman. (0-486-48594-3) INTRODUCTION TO MODERN ALGE.BRA AND MATRIX THEORY: SECOND EDITION, 0. Schreier and E. Sperner. Translated by Martin David and Melvin Hausner. (0-486-48220-0) FRACTALS, CHAOS, POWER LAWS: MINUTES FROM AN INFINITE PARADISE, Manfred Sc.hroeder.
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(0-486-48582-X) LINEAR ALGEBRA AND GROUP THEORY, V. I. Smirnqv. Translated and Edited by Richard A. Silverman. (0-486-48222-7) THE CHESS MYSTERIES OF SHERLOCK HOLMES: FIFTY TANTALIZING PROBLEMS OF CHESS DETECTION, Raymond M. Smullyan. (0-486-48201-4) KING ARTHUR IN SEARCH OF HIS DoG AND OTHER CuRious PJJZZLES, Raymond M. Smullyan.
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COMPLEX INTEGRATION AND CAUCHY'S THEOREM,
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: 1-;:a.r --~.
1':'1 1:.11 _:.11:1........ i
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MATHEMATICS
JOSEPH EDMUND WRIGHT
T
his classic monograph by a mathematician affiliated with Trinity College, Cambridge, offers a brief account of the invariant theory connected with a single quadratic differential form. Suitable for advanced undergraduates and graduate students of mathematics, it avoids unnecessary analysis and offers an accessible view of the field for readers unfamiliar with the subject.
A historical overview is followed by considerations of the methods of Christoffel and Lie as well as Maschke's symbolic method and explorations of geometrical and dynamical methods. The final chapter on applications, which draws upon developments by Ricci and Levi-Civita, presents the most successful method and can be read independently of the rest of the book. Dover (2013) republication of the edition originally published by the Hafner Publishing Co., New York, 1960. $12.95 USA
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