Geometry of differential equations

GEOMETRY OF DIFFERENTIAL EQUATIONS UDC 514.763.8

N. V. Stepanov

This paper contains a survey of papers on the geometry of differential equations, which appeared no earlier than 1972, continuing the general survey (RZhMat, 1974, IIA800), and considers in more detail a special cycle of investigations of the geometry of systems of partial differential equations, distinguished by the presence of practical applications. Then we continue the survey of new results on the geometry of an ordinary differential equation of arbitrary order, started in (RZhMat, 1978, IA645). There is constructed a general theory of invariants of equations, and classes of equations admitting a simplified coordinate representation are invariantly distinguished.

i. We shall start from the definition of the geometry of differential equations as the geometry of triples (M, G, E), where M is a differentiable manifold, on which there is defined the action of a pseudogroup G and there is given a system of differential equations E (in the general definition, no other preliminary restrictions are imposed on E, in particular, it can consist of a single differential equation also).

For a given E the manifold M, and

correspondingly, the pseudogroup G, can be suitably extended. group G are given in the traditional way,

The manifold M and pseudo-

while for giving the system of differential equa-

tions E, along with direct coordinate definition one also uses coordinate-free invariant methods; the introduction of linear relations between invariant forms of the extend pseudogroup G ([ii, 57], etc.), the use of constructions of jet-extensions (Ehresman jets)

[7, 16],

the choice of local sections in tangent bundles [20], but one should note that this variety of definitions is purely terminological and all these definitions, in any case are included in the scheme given by A. M. Vasil'ev [i0] in 1951.

The process of investigation in the form

free from technical details consists of seeking invariants of a triple and of studying the differential-geometric constructions, invariantly associated with the triple,

We note that

there occur (for example, in the group analysis of differential equations) problems of a somewhat different character also, where the triple (M, G, E) is given incompletely (most often E or G is subdefinite) and, starting from previously given conditions one is required to determine this element of the triple. As to direct connections between the geometry of differential equations and the proper theory of differential equations, despite the explicit presence of potential possibilities, for various reasons they are far from having been fully realized up to the present, although there is definite progress in this direction (cf., for example, V, I. Arnold's monograph [i]. We permit ourselves, borrowing to some degree an idea and terminology of A. M. Vasil'ev, to propose the following quite general classification of differential-geometric methods of the geometry of differential equations. Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 12, pp. 127-164, 1981.

200

0090-4104/83/2102-0200507.50

9 1983 Plenum Publishing Corporation

i. Classical method, included in the study of the geometry of differential equations by means of the direct application of the methods of classical differential geometry and characterized by the absence of particularly specialized apparatus. sibilities of this method are far from exhausted,

Although in our view the pos-

there are now noted only individual papers

realized by this method. 2. Contravariant method, consisting of associating with a system of differential equations, systems of operators and functions.

The method goes back to the works of Sophus Lie

and at the present time is very effectively used in the investigations of group properties of differential equations of the school of L. V. Ovsyannikov

[50].

3. Covariant method, in which one associates with a system of differential equations a system of differential forms and functions.

This method has its source in the works of E.

Cartan with later principal essential improvements Vasil'ev

introduced by G. F. Laptev [43] and A. M.

[i0] and also quite productively used both by individual geometers and by entire

scientific schools. The author realizes that the proposed classification is rather conditional and very probably

debatable,

available to himo

but one is forced to follow this scheme due to the absence of others One should indicate the current tendency to a fusion of the covariant and

contravariant methods, based on their duality. has been intensively developed by A. M. Vasil'ev

The theoretical foundation of such a duality [12].

2. In order to avoid superfluous repetitions, we shall base ourselves essentially on the existing surveys and bibliographies,

referring the reader to them when necessary.

Further,

let us agree to the natural and intelligible abbreviations of terms, which, by virtue of the theme of the survey, will be used many times on each page: ordinary differnetial equation differential equations

differential equation

(o.d.e.), partial differential equations

(s.d~e.), geometry of differential equations

(p.doe.), system of

(g.d.e.), etCo

Fundamental in this sense for us will be the directly preceding Bliznikas and Z. Yu. Lupeikis

(d.e.),

(1974) survey of V. I.

[7], differing in breadth of scope and the qualification of

the problems of the g.d.e, considered and equipped with an extensive bibliography.

In this

survey the history of the origins of the g.d.e, and problems leading to it is sufficiently well elucidate

(in this connection,

cf. also [57]), and almost all existing approaches to

the solution of problems of g.d.e, are touched on.

This makes it possible for us to restrict

ourselves only to a survey of results obtained since 1972, turning to earlier papers only in the case when, in our view, the corresponding facts have not been sufficiently fully elucidated in [7].

Keeping in mind that [7] is also composed taking account of earlier ones, we

shall in what follows refer only to surveys not occurring in [7]. The rapid development in its time of the projective-differential

geometry of differen-

tial equations is sufficiently completely reflected in [79] [(1943), but work done up to 1939 is considered],

and also in the detailed bibliographies

in Wilczynski

[i01] and Tzitzeika

[98, 99]o The methods and basic results of the group analysis of d,e. are recounted in the monograph of L. V. Ovsyannikov

[50], moreover there is a collective survey [51] of the work

of L, V. Ovsyannikov, N. Kh. Ibragimov, E. V. Mamontov, V. M. Men'shikov, V. V. Pukhnachev

201

(1977).

Results obtained in the geometry of o.d.e, are reflected in [57] (1978).

On the

connections of g.d.e, with the theory of nonlinear connections, developed by L. E. Evtushik and B. V. Tret'yakov,

there is detailed discussion in [20, 70].

To this same question is

devoted one of the sections of the collective survey of L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, and A. P. Shirokov [19], which is also recommended for getting acquainted with a sufficiently wide circle of ideas and methods of contemporary differential geometry, many of which find essential application in g.d.e. be obtained from [9].

Some information in this connection can also

We recall that almost every one of the sources indicated is accompanied

by a corresponding bibliography. The exclusion of the results and methods described in the above listed surveys, practically uniquely determines the content of the present survey. We purposely keep to a local point of view, not touching on questions of global differential geometry, since on the one hand these questions are considered in [7], and on the other hand there often arise situations when the global

(or local) point of view is not fixed

in advance, and many questions admit consideration in the local as well as the global aspect. And finally, it is not possible for the author to touch on g.d.e., developed by means of geometry "in the large," leaving this for a true specialist. For completely natural reasons, papers reviewed in RZhMat since 1979 and material of the Seventh All-Union Conference on Contemporary Problems of Geometry are only partially used in the survey. 3. In the present point we shall consider, basically, methods, results and publications,

independently of directions and

appearing no earlier than 1972 and not considered in other

papers. A clear description of classical methods of g.d.e, and a large collection of results concerning equations of Pfaff and Monge sov [56].

are contained in the collection of papers of D. M, Sint-

By these same methods are studied the geometric properties of characteristics of

p.d.eo and the integral curves of the corresponding Pfaffian equations in [49] by M. A. Nikolaenko. The methods considered in detail in [7]

are successfully applied by V. I. Bliznikas and

Z. Yu. Lupeikis to the study of systems of p.d.e, of the first, second, and third orders, The intrinsic objects of connections invariantly associated with s.d.e, are discovered [8], The investigation of the geometry of o.d.e, of the fourth order of I. F. Kovrenko, begun in [34], is continued.

In [35] she studies properties of the special class of the indicated

o.d.e, admitting the introduction of a consistent parameter on the integral curves,

S. D,

Kozlov [36] carried out a very detailed classification of second order o.d,e, with respect to the pseudogroup g2,4 (of the classification of E. Cartan [85]).

For almost all classes

there are found in finite form the right sides of the o.d~ Nirenberg [97], Curteanu [87], Dobrescu [88], E. Kh. Naziev and G. I. Keleinokova and B. D. Chebotarevskii

[48],

[74] are devoted to group invariance of s.d.e.

Examples of work of explicitly applied character, done by the methods of group analysis, are the papers of Frankel [90] (Maxwell's equation), Lefebvre

202

and Metzger

[94] (Berger's equa-

tion and the heat equation), Clyde and Hermann [86] (controlled systems), Brans (Einstein-Petrov equations) [80], Neuman [96].

In this connection, cf. also [51].

Affine properties of s.d.e, are studied in his papers by N. I. Vilis, K. S. Sibirskii [17], D. B. Dang [18], Margulescu [95], Arrowsmith [77]. There are posed and solved problems of reduction of d.e. to a certain canonical form in the papers of A. S. Udalov [73], Kinosita [93], Kijowski, Smolski [92]. V. A. Truppov, treating systems of p.d.e, of the first and second order as finite equations ofhypersurfacesin the corresponding spaces of tangent elements, obtains partially affine-invariant classes of d.e. of the first order and the equations of the intrinsic fundamental object of a hypersurface, of a corresponding s.d.e, of the second order [71, 72].

V.

L. Izrailevichuses an analogous treatment [21]. One should note specially the investigations which have appeared in recent years of A. M. Vinogradov, V. V. Lychagin, and certain other authors [13-15, 44-47].

The return to the

initial ideas of the geometric treatment of d.e. of S. Lie on the basis of contemporary differential-geometric and algebraic concepts made it possible to get an interesting and constructive method of investigating d.e.

In particular, there appeared new possibilities of

classifying d.e., of getting conservation laws, among them those induced by symmetries, of finding exact solutions of concrete equations, describing physical processes, There exist problems in explicit form not containing in their formulation references to the geometry of d.e., but either reducing to it, or interpretable in its terms.

In view of

the unreality of a precise isolation of the circle of such problems, we restrict ourselves to some examples.

In [37], V. A. Kondrashenkov interprets certain special cases of the geometry

of three-parameter families of curves in three-dimensional analytic space with the help of linear o.d.e, of the second and third orders.

Of earlier papers we mention the study of 2-

webs in three-dimensional space, reducing to the geometry of o.d.e, of the second order (cf. Bol. [78], Dubourdieu [89])~ As to the bibliography accompanying the present survey, we note that it is considerably abbreviated due to the omission of abstracts of reports, of which there is an expanded account in later papers of the authors. 4. We concern ourselves with the cycles of investigations of the geometry of a system of p~d.e., which figure in many bibliographies, but of which there is lacking a systematic account of the methods applied and results achieved in the surveys known to us.

These works

are carried out by the invariant method of E. Cartan, G. F. Laptev, and A. M. Vasil'ev.

The

previous investigations of E. Cartan on this theme [81-84] were not (at least up to the appearance of the collected works of E. Cartan) widely known.

The idea of this method, in con-

nection with g.d.e., was formulated in very short form by A. M. Vasil'ev in [i0] and then realized in [Ii], where he studied the geometry of a system

$321 (three first-order partial

differential equations with three unknown functions and two independent variables; this symbolism will be used in this section and later) with respect to a pseudogroup of analytic transformations of the space of dependent and independent variables (in all cases below, as the initial fundamental group we shall mean precisely such a group).

Three d.e. single out

203

from the ll-dimensional manifold of two-dimensional tangent elements an 8-dimensional submanifold.

Relations distinguishing this manifold can be described in invariant form, as

linear relations between invariant forms, not dependent on the entire manifold. ficients of these relations

form

equations of the submanifold,

The coef-

the field of fundamental objects of the submanifold

and correspondingly,

(the

the fields of objects, admit extensions).

Various versions of the choice of principal forms of the submanifold with subsequent canonization of frames, lead to an invariant classification and to invariants of a system Further,

$321 .

there is singled out a class of systems $32~ , containing all quasilinear systems of

the given type, a test is proved for the reducibility of a system to quasilinear also to linear) form.

(and then

There are singled out quasilinear systems having intermediate inte-

grals, there are considered conservation laws of such systems, there is proved a necessary test for the existence of a conservation law.

Some classes of s.d.e, are interpreted in

terms of line geometry and the geometry of webs. examples,

taken from mechanics

To conclude there are given illustrative

(planar stationary flow of an incompressible ideal fluid in

a conservative force field; one-dimensional motion of a polytropic gas in an adiabatic process). These studies were continued by Kh. O. Kil'p, generalizing the theory to more general systems

Sm21 , for which there was found a canonical form, there was carried out a classifica-

tion according to canonical form [26], for Sm2 ~ the theory of G-structures

with constant coefficients by the methods of

there are found maximal invariance groups [27], there are consid-

ered conditions for the existence of intermediate integrals, a system to a linear one, the connection of laws [29].

For a quasilinear

tests for the reducibility of

Sm2 ~ with the geometry of webs, conservation

$421 there is established its connection with an isothermal

surface in R 3 [30] and with the theory of surface of zero Gaussian curvature in R 4 [31]. Simultaneously in [28, 32] there was examined the theory of systems yet unstudied classes of s.d.e.

Finally,

S~21 , touching on as

in [33] the constructed theory was considered from

the point of view of the contemporary theory of connections in generalized spaces. The rest of the survey of necessity will carry a considerably briefer character. Systems $231 were studied by G. M. Kuz'mina

[38-42].

For linear systems there is made

an analogy with Dirac equations.

There is established a connection between the reduction of

$23~ to one second-order equation,

the existence of a special class of conservation laws, and

the possibility of lowering the number of independent variables. Quasilinear systems

$421 under the condition of pairwise coincidence of characteristics

were studied by E. M. Kan (Shvartsburd), elasticity

applications were given to problems of the theory of

[22-24, 75].

L. N. Orlova [52, 53] considered a system of two p.d.e. and one of the second), carried out a classification,

(one equation of the first order

explained the conditions for existence

of intermediate integrals. Generalized conservation laws of equations kevichute

204

[54].

02~ ~ e

were considered by D. K. Petrush-

S. I. Bilchev (jointly with other authors) continues the investigation developed in detail in [3-5] of the theory of a quasilinear hyperbolic system S~2~ and its conservation laws

[6]. 5. Starting with this point, we shall present a survey of results obtained in the general theory of o.d.e, of arbitrary order n ~ 3 .

(To speak of the analogous general theory for

p.d.e, or even for s.d.e, is still premature.)

This survey is the direct continuation of the

survey paper [57]; the formulation of the problem,

terminology, and notation introduced in

[57] remain in force here too (all changes induced by the further development of the technical apparatus will be mentioned specially).

In references to [57] we preserve the numbering

of formulas and theorems in [57], adding here to the corresponding number a Roman one (for example, Theorem 8.1). the first time.

Part of the results were published in [57-69], part are presented for

One should note that on the character of the investigation (and, correspond-

ingly, the account) a quite essential influence turned out to be the tendency to create a differential-geometric theory with "inverse connection," i.e., applied to some degree to the theory of d.c.

This goal was clearly proposed by S. Lie himself and E. Cartan, but many

later investigators of g.d.e, no longer pursued this goal.

A consequence of this is, for

example, the requirement of "computability" of formulas (although investigation is also carried out by a coordinate-free method, upon introduction of coordinates, all functions appearing in the theory must be constructively expressable in terms of the coordinates and the right side of the o.d.e., examples of such calculations will be given), which, in its own right, entails the rejection in many cases of generally accepted abstract-theoretical constructi~ns and the corresponding notation (despite their convenience and esthetic attractiveness), a decrease of attention to slightly constructive theorems, etc. The well-known difference of the geometry of o.d.e, of even and odd orders compels one to seek a formulation, identically suitable for both cases.

Thus, for example, the geometry

of the o.d.e, y~'o=0 , in which the space of integral curves for even n is affine-symmetric, and for odd n is pseudoeuclidean, admits one description with the help of the introduced concepts of curve-norm and space-norm.

From now on we intend to adhere to this principal as far

as possible. And, finally, complicated calculations and proof of computational character always in such cases

will as

be omitted.

6. The goal of the present point is the introduction of certain unimportant changes in the differential-algebraic terminology and notation introduced in [57], with the subsequent considerable extension of the concepts of differential algebra to a considerably wider circle of objects. i. In view of the appearance of new, previously unexpected analogs, it turned out to be more natural to name the formal partial derivatives (f.p.d.) introduced in [57] covariant partial derivatives or in cases not admitting different interpretations, partial derivatives (and even simply derivatives). 2. In the basis

D(~n)

indices -- 1 will always be put in the first places.

3. We introduce the new notation

205

~----H~; a0_~=H-2;

s,t,r=O

.....

n--l,

which makes it possible, after an elementary reconciliation of signs

(which in no way in-

fluences the theory), to write in abbreviated form

d m = m x A H x + . . . . X , ~ , v = - - l , O . . . . . ~z--l. 4. In view of the essential differences of meaning, equations obtained by means of canonization will be distinguished by the special notation 5. We introduce a new index for the coefficient for

Up ~ 0 . o~2:(@ I).

Then equation (35.1) can

be written in the new notation

d u F + u - d ,la-o~ ~ ~- O ~ o ~ l ) = u j / + 2 t e~2+ttpxII ~ p 20,

(1)

and one can also call the coefficients

a:~k, u~+1) covariant derivatives of the component

respectively, with respect to

We also use analogous notation for (34.1)

H~, ~ 2 "

((p)) = U~ +I) ~2 @ U~ ~

uT,

(2)

but the coefficients here, it is clear, are no longer derivatives. 6. We extend the concept of i- and 2-degrees to all actually occurring objects, component

eF this is trivial, because O and o are functions of the row of indices:

9@); a ( ~ F ) ~ ( P ) .

The matter is somewhat more difficult with the forms

secondary forms (except ~I0 a~1 ) appear in at least one derivative

~ipq ,

For the @(uF)=

but since all

((~) , it is natural that

9 and ~ ascribe to this form values just like the corresponding derivative, which gives the formulas

~(%)--~-9+~.

~,

The next step consists of the extension of these formulas to all invariant forms of the pseudogroup G, after which we have

p (m)=p (~;)=o;

c, (m)=,3 (~oL)=n-s;

p (11-2) = p ( ~ o ) = 1;

o (II -2) = o~ ( ~ o ) = '~ + l;

p ('Io) = p (~

Definition I.

= 1;

,3 (O, Io) =,~ (o,~1) = ~.

For any two objects, having degrees 9 and a, the degrees of (any!) of

their products are calculated by the formulas

9=9i-[-92--1;

cI= ~I1-[- ~2-- r/..

This definition can be extended in the obvious way to an arbitrary number of factors. 7. The operation of covariant differentiation can be generalized also to principal forms

dlI = [ I v A I I for an arbitrary number of differentiations. 0)10 ----

Moreover,

206

(II~+1=O!) Then 6002 =

il__l

_i =Ji__

2"

These f o r m u l a s remain v a l i d 8, I f one t r a c e s

f o r any o b j e c t s

the process

with contravariant

index.

by which t h e i n d e x (+l) a r i s e s ,

9 ( ~ + 1 ) ) = O (U~);

then (3)

~ (U~§ a)) = ~ (U;) + 1.

9. One has the very useful practical theorem THEOREM i.

All formulas of the theory are homogeneous

in 9 and ~.

The proof reduces to the verification of homogeneity of the original formulas and the preservation of this homogeneity by all the operations performed. i0. To conclude we write,

in view of their importance,

the formulas

(37.1) in the new

notation:

--(s:%

%o)A +(1 2)~a

dII<=--~~ 2;%

dt"~ = ~ n--I

d~ = ~ 2

C~

+/~II~AII*; n(n +1) ttn-~,a~~176

2

+ 12 (n -~ l) ('~

+ R & ~11~AlI~

(/~n--1,3 ~]-1 ~-~/'L-l,n-l,3II0) -[-q zl0)fl]~'

-] n (n + 1) m~2A (tt-l,n-~,a II~

AIIO-+-Q_-I~II~ArF;

llI) -~-Q~,-1,srlsAII-1

+Q~176176

2 2 ~ n (n2}.a d~~2 __ r176162176 + 1) (0022A (2ft. 1,n_l,a[[1_~/.tn_l,ali2)~

-

where similar terms are purposely not given and all the coefficients

R~v, Q~]K

functions of the components of the expansion of the characteristic derivatives ((~--l, 0)),((n--1, I)), ((n--l,2)), ((--I,~--I, I)), ((--I,n--1,2))

(4)

are known ((s)),((--l,s)),

with respect to the principal forms

and the form ~2" 7. The relations between the components

uF

in the general case are very complicated

and we restrict ourselves to the necessary minimum. tions reduces to the following: help of (30.1) one can eliminate

The general principle of such calcula-

we seek expressions aF 9

eF~eF(aF) , from which then with the

Thus, for example, m

U~tn=a~tnq-~tR~n+U~Rtn

m

for p = 3 k

(u~t=O),

or k

where

Yfst are contants,

u~

are the coefficients of the expansion with respect to princi~

pal forms of the corresponding basis derivative. relations between

Eliminating from such relations

as~n are known), we get the formulas required.

amples of such calculations,

astn (the

We can only give some ex-

choosing them from the most important ones

Ust _l = Id,-lst --/,~s, t-I--Us-It "q- IGmR~-; 4- lZmtRs~-1 q- gE'stltK_ 1, where

207

[u~+t,~R~-~; s + t ~ n - 1 ; U~_l=l--Un_2,e @ Un_l,mm~_l; s + t = n , n + ~; e = l , 2 ; [0; s + t > n + l ; ,

(5)

un-l,e,-l= --~-2,2@u~-l,mR~-l; e, e:t, e2= I, 2; k

~n-~,e,o=O; m -- u me~ae,,n-1--,~n--l,e, (~n--l,e,+e2,0--~ e,+e~,mRo,n-l); m Un-l,e,,e,=Ue,,e . . . . 1@un_l,mRe,,e=--Ue,mRe~,n-I m Un--,,e,j=Un--1,],e+U.--l,=(R~--~); referring

j>2,

to t h e r e m a i n i n g f o r m u l a s , as to f o r m u l a s o f t y p e ( 5 ) , by a n a l o g y .

Definition

2.

Of t h e components

u~t, u-lsx

we s h a l l

s a y t h a t t h e y from t h e b a s i c o b j e c t

uT. Careful consideration of formulas of type (5) leads to theorems considerably facilitating the investigation. THEOREM 2.

All coefficients of structural forms can be expressed in terms of the com-

ponents of the basic object and their partial derivatives. There also holds a much stronger theorem. THEOREM 3.

All coefficients of the system (2) can be expressed in terms of

uZ

and

their partial derivatives. The theorem is proved by means of a rather involved ordering of the equations of the system (2) and subsequent induction.

Since (i) are differential consequences of (2), Theorem

3 extends also to all coefficients of (i). There is also valid a theorem, in some sense converse to Theorem 2. THEOREM 4.

All components u F can be expressed in terms of

This same fact can also be formulated in the following way: eF and

R~

R~

and their derivatives.

the systems of functions

, taken for one and the same equation, are functionally equivalent.

formulated by the author in [58].

The value of the theorem is quite great, because it ex-

haustively solves the problem of the relation between intrinsic g.d.e. tions of structure

Theorem 4 was

(4)] and extrinsic

[defined by the system (2), (i)].

[defined by the equaIt is known from

classical differential geometry that extrinsic geometry always includes intrinsic, but does not necessarily coincide with it.

There are also known elementary examples of different solu-

tions of this problem. In the present case Theorem 4 asserts namely the equivalence of extrinsic and intrinsic geometry, which in principle makes it possible to use only the equations of structure We, however, turn to systems erations.

(4).

(2), (i), but this will be dictated by purely technical considr

We note, finally, that all the calculations,

described in the present point, can

be carried out to the end explicitly. Remark.

The components

uF

are not independent, but satisfy compatibility conditions,

which can also be calculated in the way already described.

Thus, for example, strictly speak-

ing, one should eliminate from the number of components of the basic object, the components U-lst , since the formulas their derivatives.

(5) show that

U-lst

can be expressed in terms of

and

But since this would evoke repeated complication of the computational

part of the theory, we shall, as before, consider which will not lead to any complications.

208

ust, u-l,-1.s

u-~st as appearing in the structure of

uF ,

8. In the following investigations, and construction of invariants, Direct calculations give for

the principal goal of which will be the detection

an essential role will be played by the coefficients

uF~+l).

u[: Us, t, (+b = - - ~Vs+ligs+l, t - - )vt.~ll.Zs, t+l,

l l - l , s , t , (+I) = - - },-I/~st - - /21--1 , - - I . % ( + I )

=

--

We choose an arbitrary coefficient up(+1}k,....

~s+ill-1

,--1 ,s+l;

ks+l/s

u]

,s+I

}~t+lls

,t --

S (n--s) 2

~s

and we consider the sequence

u F, uf(+~).....

Under each differentiation with respect to the index (+i) the quantity p remains

unchanged ~ increases by one.

But for fixed p the quantity ~ cannot increase without bound:

there exists a Oread: ~n--3; p = 1; [ p ( n - - 1 ) ; p>l.

c~=

The number J=Gmax--O

will be called the degree of a N in k .

This name is explained by the

fact that it turns out to be possible to introduce a second parameter

k such that

~u--

up3+~ ) =

L. 0~ "

From everything recounted above follows Lt~+l)]+1

/lp(+1)/+2=

. . . =

THEOREM 5.

All coefficients up are polynomials in ~ of degree j.

COROLLARY.

The coefficients

Remark.

u-b-1,~-1; u(n-l)h

are independent of % .

The formula for defining j is always valid, but

(Tmax can for objects of a dif-

ferent character also be defined by another formula. 9. For an arbitrary component

ur7 its vanishing in the general case is not invariant,

since it leads to the dependence

u#~+~ + Up-~U~=0, k

equivalent for

u~+11~=0

one should require Up--O

, then

u~

with the canonization

u~+1)=0.

.

Hence, for the invariant of

up=0

If this equation holds either identically or by virtue of

is a relative invariant

(this definition is somewhat broader than the usual

one), while if not, then we consider the system all the preceding discussion,

uF=0

etc.

up=uF(+1)=O

, with respect to which werepeat

From the preceding Sec. 8 it follows that this process is

always finite~ i.e., after a finite number of steps we arrive at a system up(+~/=O;

having invariant meaning. object, and

~+l)m, m = l

present case

Then the collection

..... k,

l=0

.....

k,

Up ..... Up(+1)k_~ will be called the invariant

the envelope of the component

.

We note that in the

up(+l)k_~ is always a relative invariant.

The concept introduced admit considerable generalization. components

uF

{V~} , where

are defined.

We consider a collection of

V~ are arbitrary rational integral functions

We shall say that

{V~}

eF,

for which p and.

form an invariant object if the system 209

Vi~O has invariant meaning.

One has the obvious test for invariance of an object:

the satisfac-

tion of the system Vi(+1) ~0, by virtue of

~=0

.

Analogously one can also define the envelope both of each component

Vi and of the object

{Vi} in the large.

We distinguish an important special case:

if the components

V~+1=Ai+iVi(+1 ) , then we call the object linearly ordered.

V i can be ordered so that

The system

u~+1)m

already con-

sidered has this property for example. We consider from the point of view expressed certain properties of the components of the basic object

u~.

I. The components relative invariant. II. All

u-1,-1, s

If

form a linearly ordered invariant object,

u-~,-bn-I=O

, then

u-l,-I, n-2

ust form an invariant object, u,_~,n-1

Un-l,n-2 and U-l,n-bn-1

invariant object. the components

ust (a>~k).

u-1~t (g~k--l)

is a

becomes a relative invariant,

is a relative invariant.

become relative invariants,

The components

u-1,-1,n-1

etc.

All components

etc.

For un_~, n-~=0 ,

ust (o~k)

form an

form an invariant object compatibleL~with

Upon the equality to zero of any invariant object from ]lUstl[ ,

IIu-lstlI the remaining components

form an invariant object,

etc.

i0. The constructions made above have broad generalizations,

placing at our disposal a

practically unbounded arsenal of relative invariants and invariant objects. i. We consider an arbitrary minor of order r of the matrix

Ilu~tl[ 9

By the (+l)-extension

of the minor we shall mean a minor of the same order, for which all rows

(or columns),

ex-

cept one, coincide with the rows (columns) of the minor taken, and one row (or column) has numbers one greater than in the original minor. THEOREM 6.

The derivative with respect to (+i) of a minor of order r of the matrix

IIu~tll is a linear combination with constant coefficients of its (+l)-extensions. COROLLARY I.

i) All minors whose

(+l)-extensions are equal to zero

are relative in-

variants. 2) All minors

bounded to the right-below by zeros

(or of bounded matrices) are rela-

tive invariants. 3) Any minor with all its (+l)-extensions forms an invariant object. 4) If a minor of order r is a relative invariant, minors of order

s~r

has invariant meaning.

The analogous theorem also holds for

l]u-lstI[ , with the complication that this matrix

can be considered only compatible with the matrix turn out to be mixed minors whose rows

dV+ V extension of this equation gives

Hustll , and in the (+l)-extensions

(columns) are taken from different matrices,

2. For an arbitrary relative invariant,

210

then the equality to zero of all its

satisfying the equation +

=

there can

Vn-1

whence it follows that

is a relative invariant,

Vs(s~2)

form a linearly ordered in-

variant object. Remark.

Calculations

show that for u~-1,3=u-1,~-1,~=0

all

V~ (s=0 ..... n--l)

form a

linearly ordered invariant object. 3. We consider a collection of similar components V~ and their envelopes. k~0

there exist constants A ~ such that

variant,

A~V~(+I)m,m = 0

and

.... ,k--l,

A~V~(+I)~--0

V=AiV~(+~)k_~ is

a relative in-

form an invariant object.

ii. We introduce new differential-algebraic cases.

, then

If for some

characteristics,

We consider an arbitrary component V, having

p(V)

and

more convenient in certain

o(V) , and we assume

d V + W [(1 -- p) COlo+ (~ -- n) co~] = 0 (rood II z, o~2), and, writing

a=p--1;

b=n--o,

we shall call ~ the 1-weight and b the 2-weight of the component V. components

VI

Then for the product of

(al, bi)'V2 (a2, b2)

a=a1+a2; b=biTb2, I

i.e., a and b behave like certain exponents

(as will be shown below,

they actually can be so

interpreted). If for

Vl(a~,bl)

and

V2(a2, b2)

one has

j al bl ~0~ a2 b2 jI then we shall call V~ and V2 balanced.

Similar components are always balanced, but it is

clear that the converse is false. By an index of equilibrium we mean an index, differentiation with respect to which gives a component,

balanced with respect to the original.

For such an index ~ we have

_--1; a=O, ~-~; a~O, where ~

must be negative integer.

a

Completely analogously,

the concept of weights can also be defined for differential

forms. 12. A relative invariant becomes absolute for

a=b=O (0=1, a=n). Among the uP themselves,

there are no absolute invariants.

For the construction of absolute

invariants we shall assume that the operations described are realizable divisor is not equal to zero, etc.). values

a,b(p,~)

(for division,

the

We note that as a result of these operations we can get

, outside the range of the original definitions.

The following facts have completely elementary algebraic justifications:

211

I. For any two unbalanced relative invariants, one can always choose numbers

~i and k 2

k, will have preassigned weights ~ and b, not simultaneously such that the invariant V = V~I 9V u equal to zero. 2. From any two balanced relative invariants one can always construct and absolute invariant

V~V~'.V~L

3. From three arbitrary relative invariants one can always construct an absolute invariant

V~'.V~.V~L

Constructions essentially analogous to that described ferential forms:

can be carried out also with dif-

one can construct forms, whose exterior differentials do not contain

and ~i ; forms, whose exterior differential does not depend on ~

~I0

, etc.

13. The presence of invariants allows one to make an invariant classification of equations, which, by virtue of the constructed analytic apparatus, is a classification of differential-geometric structures, invariantly associted with an equation,

It is only pos-

sible for us to recount the basic principles of such a classification and to give the brightest and most interesting examples of the classes obtained and the relations between them. nothing special is said, then we shall always start from a structure

If

y(g2,6) (here and later,

after the symbol Y, denoting a connection, in parentheses we shall indicate the fundamental group, generating element, and other information, clear from the context). We choose an arbitrary relative invariant I.

Then the relations I = 0 and I # 0 divide

the set of equations into two disjoint, invariantly distinct classes.

These classes are far

IK=const=/=

from equivalent; the inequality I # 0 makes it possible to carry out canonization

0 (reductive subclass), the equation I = 0 does not give such a possibility (nonreductive subclass).

It is necessary to note that this division of subclasses makes sense only in the

presence of relative invariants, and the case of their absence is also possible, when all invariants are absolute (absolutely invariant structures); in this case the formal classification loses interest, because each absolute invariant can in principle give a continuum of invariantly distinct classes.

But at our stage of understanding of reductiveness and nonreduc-

tiveness it turns out to be quite useful. In practice the nonreductive classification is technically considerably easier for us to get from the fundamental mass of facts, because almost every time passage to the reductive subclass can be reduced to the reductive restriction of the nonreductive class, requiring that the relative invariants defined are not equal to zero. One can give the following methods of nonreductive classification: a) Differential-algebraic classification, based on the numerical characteristics p, o, ~ . b) Structural classification, essentially using the equations of structure of and actually reducing to the classification of connections

Y(g2,6)

y.

c) Tangential classification, at the base of which are located the properties of tangent manifolds.

Some facts of tangential classification were already given in [57].

It is not

possible for us here to consider in detail this classification, but we note that in principle all classes obtained by the use of a) and b) also admit tangential interpretation, Remark. delineation 212

The list a-c) is, of course, quite conditional; in fact, to give a precise of

methods is difficult.

The fact is that essentially there is one classifica-

tion -- one is concerned in practice with the most convenient description of a class in one terminology or another. 14. We fix equation

P-P0

U(p0)=0

-

The set {~F}' P(P)=P0

will be denoted by

U(p0) ; correspondingly the

will be the abbreviated notation for the system

uK:0,

~ y C U (P0) 9

In

general,

later, in parentheses after the symbol U we shall write the characteristic of the

set {~7}

; the equation U = 0 has analogous meaning.

i. The equation THEOREM 7.

U(p)~0

always distinguishes an invariant subclass.

From U(2) = 0 follows

U (/)=0, I>3.

2. From U(1) = U(2) = 0 it follows that the structure considered is flat Remark.

From

U({)=O,

[>3

it does not follow that

(all u ----0).

U(~)=0,~>/.

3. One can introduce even narrower classes

u u (p)-- u 0 > po) = o,

~o > 3.

P>Po

We shall call such structures covariantly polynomial, of the degree of a polynomial. 4. For fixed p:p0 o~0) :0

we take

In particular,

~:~0,

k) = 0 ,

p=p0

will be the analog

admissible for the given p0; then the equation Such constructions

U(p0,

can be generalized by a

a description in this language is possible for any invariant

object, which is the envelope of some 5. For

p0--1

We call the class U(2) = 0 covariantly linear.

distinguishes an invariant subclass.

set of methods.

the number

we take k, 0 < k < p 0

uF,

and wider generalizations are possible~

, and we consider an equation of the form

which distinguishes an invariant subclass.

U(p0, ~ - i ~

If one sets k = 1 (all components u~

with

at least one index n -- 1 are equal to zero), then the index n -- 2 starts to play an analogous role, etc.

In general, systems of type

U(flo,~ m + 9 9 §

distinguish invariant sub-

classes. 6. Somewhat modified construction: k) : 0 .

we fix k and we shall consider for all

9~k, U(~_~

We shall call such a structure covariantly polynomial with respect to the index n -- 1

of degree k -- I, and if k = i, then covariantly independent of n -- i. Generalizing,

just as above, everything said to ~m,.-.,~n-I , we get a structure,

covar-

iantly polynomial in the indices m, ..., n -- 1 of total degree k -- 1 (and for k = I, covariant!y independent of m, ..., n -- i). 15. i) In the structural classification,

besides structures

consider structures obtained by fixing a tangent element

Y(g2,6; t~_, %), we shall also

th (stationary substructure

yh),

which in the general case will also be spaces with connection. 2) The collection of all torsion-curvature thus, for example,

for a flat structure

Y

forms of a connection will be denoted by ~ ;

we shall write

9=0

teness in the separation of forms into principal and secondary, forms ~s define the true torsion, forms ~-i 95, ~i, ~ 0 lection of components

the forms

let us agree to say that the

9~i,~I0 the true curvature.

will define the conditional curvature. R~v(R[x )

In view of some indefini-

will be denoted by

The collection of

In the necessary cases the col-

R~iRS), where the contravariant index can

be fixed or can run through a certain numerical set. 3) If some form ~ does not occur essential in ~ , then this will be denoted by which also generalizes to a system of forms

{~N}:Q\~h

9\~

,

nhG{~N} ; to the exterior product of 213

{nx}:~\nhA~;

forms from from

ft.

~h nh6{nN} , and can also be used for subsets and separate forms

The geometric meaning of such cases will be explained as they arise.

4) We consider as an illustration the case

Q\~202

, for which it suffices to set

Un--i, 3 = U--I, n - l , 3 ~ 0 .

This c o n d i t i o n

means t h a t

are justified of

~\~0~

in considering

a connection

leads to a stronger

THEOREM 8. leads

in the torsion-curvature

COROLLARY. Q \ H

?(E2,6; -i

v(g2,6; t ~ - l ) .

~02e

is absent,

But c o n s i d e r a t i o n

i.e.,

we

of the consequences

conclusion.

The a s s e r t i o n s

to a c o n n e c t i o n

forms t h e form

~\~0~

and

~ \ H n-I

are equivalent,

so t h a t

e a c h o f them

tn-2).

implies

~\~0~,

~ \ H ~-i .

Such an i n v e s t i g a t i o n

can be c o n s e c u t i v e l y

continued. 5) F o r t h e f u r t h e r

classification

§

form

~s).

On i t

it

is

convenient to distinguish

the subobject

R[r

(or

one can impose t h e r e q u i r e m e n t s

+

a)

[ = n - - 1 , . . . , m ; f o r m = 0 we h a v e u , t = u _ l s t = 0 , so t h a t we s e t f u r t h e r

Qs\Nz,

we h a v e t h e a n a l y t i c

conditions R~m=O,

b) ~s\HrAHt; r, t ~ k ; THEOREM 9.

m>0;

or analytically

max{t,

r}>m; t,r~k.

R~td~O;

The requirements

+

I

~sNHrmHt ; F, t > ~ ;

II ~sAH-tAH~ III

all

c) Thus,

u

A Kk-1=0;

(p>k--1)

are flat,

t h e r e becomes v i s i b l e

can d i s t i n g u i s h i n v a r i a n t s u b c l a s s e s + A) ~ \ H z , l>m; + B) ~ s \ H r A H t , r, t ~ k ; C) u n d e r t h e c o n d i t i o n s o n l y on

are equivalent. a rather

general

characterized

A), B) t h e t o r s i o n

scheme o f i n v e s t i g a t i o n

o f R~r :

one

by t h r e e p a r a m e t e r s p, m, k, w h i c h means

of the stationary

substructure

yp d e p e n d s

t~, i~<m--1.

In fact,

one can s i m u l t a n e o u s l y

realize

a l s o some c l a s s e s

of such structures

y (p~, mz,

6) Now the classification of ~s reduces to the investigation of distinct combinations of classes ~s and special cases of the object ~-~,-L~7) Conditions of a somewhat different character are imposed in the presence of the relation d~2 A ~2 =0,

(6)

which induces in ~ e space of E-frames a bundle whose fiber is distinguished by the fixed integral equation ~.~=0

We note that

I. Distinguishing a fiber gives a nonreductive structure. II. It is not canonization, but is connected in the end, with restriction of the holonomy group of the connection

214

?.

III. The invariance of such an operation is quite conditional, since on the one hand, the existence itself of the fibration, undoubtedly, has invariant meaning, but on the other hand, the choice of a fiber is arbitrary, because all the fibers are isomorphic. THOEREM i0.

Under the condition (6) the equation admits a structure

Y(g~,s; tn-1).

Analogous theorems also hold for the cases

d~o2A~o2--d~olA~o2--O, 2 2 __ 2 2__ I 2__0, d%2AOo2--dOolAOo2--doloAOo2-2

2

__

2

,2

__

9

(7)

and many others of the same character. THEOREM ii. the connection

y

Condition (7) is necessary and sufficient for the conditional curvature of to be equal to zero.

The general classification of connections

y

is now obtained by combining all the methods

listed. 16. The reductive classification of equations is in practice immense, and here we can touch on only one of the problems of this classification. For canonization one can use three secondary forms

~i' ~I0' ~ 2

after which

~02 = 8 n - - l , ~

where the lower indices are chosen so as to preserve homogeneity. case at a structure

We arrive in the general

Y(g6,1), where all coefficients are absolute invariants and further clas-

sification is impossible.

Such structures will be called absolutely invariant.

There arises

the question of the existence of absolutely invariant structures, invariantly associated with an equation, but different from

Y(E6,~) 9

Three types of such structures are well known:

i. Maximal group g2,s of invariance of the equation

y(n~=O.

2. Maximal group g5,5 of invariance of equations

yc~)=m~_~y~-2~+

.

.

.

+moy.

3. Maximal intransitive (with one invariant) groups y~=mn-2(x)r

E2,3

of equations

. .. +mo(X)y.

The investigation of the question turned out to be very laborious and led to the investigation of structures, where all relative invariants are balanced.

After reduction of one of

the balanced invariants to a constant, all invariants remaining relative must vanish, after these relations are investigated as to compatibility with the Bianchi identities.

The re-

sult is THEOREM 12. structure

If an equation cannot be reduced to a linear one, then it always admits a

y(g6,1).

The theorem was formulated in [59]. Remark.

Upon weakening the requirement of invariant association one can also give other

examples of absolutely invariant structures. 17. Below will be given the simplest examples of application of the constructed theory to the theory of d.e.

Considering that at the present time investigations of this kind are

215

comparatively few, the account will be given in somewhat more detail.

In the present point

we introduce the most general coordinates, placing ourselves thus within the boundaries of one coordinates chart and losing many advantages of the coordinate-free method.

This is

compensated for by a series of factors: a) We shall have to do with sufficiently narrow classes of equations, due to which the coordinate description can now become suitable. b) There will be investigated the possibility of introduction of coordinates,

invariant-

ly associated with the equation. c) The coordinate form helps to represent the structure of the functions

(coefficients)

occurring in the theory more intuitively. d) Upon the choice of concrete classes of equations there always turn out to be possible specific simplifications

of the analytic apparatus and computational formulas.

Later we shall always denote by x, y, yl,..., yh tegrable systems II-1=II~

the first integrals of completely in-

, where k runs through all values from 0 to n -- i,

and each coordinate chart for k=k0

is compatible with all coordinate charts for

Considering the structure of the differential equation

[the system

II"=0, s = 0 .... ,n--I

be equivalent to yln)=f(x,y ..... y(n-l)) ], one can write the expression for IP

IIs--~s(d!]s--~ts+~dx)-{-~ s

s=O, . . . , n - - l ;

k
must

in the form

r < S;

iI-l= __~_~o(dx_• w h e r e , as a l w a y s ,

y~=f.

To s i m p l i f y ,

(8)

we w r i t e

1

[the choice

i s e x p l a i n e d by t h e f a c t

~, SzI; ~

that

in

y(g~,~) t h e y r e m a i n f r e e s e c o n d a r y

parameters]. Calculations give

d In ~1+ o~ -- kgldz + (In ~l)-~II-~ + (ln ~)o Iio = O; k~

2 n ( n + l ) (~3Un--1 ' 3('0~2-~ lLn--1,2, ;LI]~') - - (In ~)o Iio +- ? zollO= O;

d•

a In

1

1 2

(8) a r e i n v e r t i b l e ,

explicitly

i.e.,

(9)

= O;

introduced

n e e d e d i n what f o l l o w s .

to coefficients

from them one can e x p r e s s

dx, dg, dg ~.

one can d e f i n e p and a n a t u r a l l y

~(v~)=n-~;

The f o r m u -

and f o r m s , t h e f o r m u l a s o f

x = g - t ; g=y0):

0(#)=~; 216

no + On

only the coefficients

symmetrically with respect

18. F o r t h e p a r a m e t r i z a t i o n generality

m + On

2

where t h e r e a r e w r i t t e n l a s o f (9) a r e w r i t t e n

+

v(k)=l;

a(k)=n;

(setting

for

~($~)= I+~L,-~;,; 9(• after which,

generalizing

~(;~)=n-~+~;

6 (• =n-k- 1;

to this case the formulas for p and 6 of a product it is easy to

see that all formulas remain homogeneous

in p and 6.

Then for an arbitrary function of the parameters

Ocp

, Oq~

Oqo

introduced,

having 9 and o.

Ocp

Ogg :Oq~\

i0~\

~f = 6 ( q O + l ; and, generally,

P( (~(

The requirement

o~

. . . . . ~);

)=v(~)§

8g ~, ... Oh,~t

OtqD ) ~ 6 (qo)-[- ~ (~,, . . . . . Og~,.... Og~tt

of homogeneity

is satisfied

for

[~t)"

9(f) = ~ ( f ) = 0

, whence

9 ( Og~, ~... Og~,f ) = P (~1. . . . . P~t); Or/~,

Og~t

19. In the present point there are explained efficients

occurring in the theory.

components V, for which

certain details of the structure of the co-

All constructions

carried out here are suitable for any

p, 6, 6max are defined.

i. Let

dV----V(ao~{o+bo~1)=O(modn ~', r so calculations

(according to the parametrization

introduced)

give

v=~--7- w, 1

where W no longer depends on

-

~, ~.

2. By hypothesis we know the degree j in

W=fJ%

~: 7"----~max--6 , whence

l = 0 . . . . . 7,

where ~t depend only on the principal parameters

V= j~o 3. For the simplest

(linearly ordered) 1

g~.

(~lr

object l~ I

Finally

Vo= ~---~: ~g~q-- (No~ ~o,);

V0 ..... V~

l = O . . . . . k;

217

(N~,,o,~+~); /=0, -.., k

V ~ = J - -~Lo- ,

v~: ~ ( N ~ f ' , o , , + ~ ) ;

1;

l=O . . . . . k--i;

(lo)

~ N k--,*0,k--1); O (Nk-,~XPok-/

Vk-, - - - - ~1

1 7 N~$ok; V k : ~a~--where

bi:bo--i, N~ Equations

are constants.

(i0) are solvable for the functions

~:90i

, so the system

~--0 is equivalent with the system V0..... V~ and j ects

~{ ..... ~

~0 ..... ~h and

Vti--O

and has invariant meaning.

are equivalent.

In this sense the objects

This assertion remains valid also for any subob-

V{..... V~ (kmi>O).

We note that although all ~{ are functions only of the principal parameters, in general, not invariant.

they are,

But actually, all invariants which can be constructed from the

object V{ can be constructed from ~{.

The operation of passage from V{ to an equivalent ob-

ject ~{ we call reduction of the object

V{ to principal parameters,

and the ~{ themselves,

reduced components of the object. 4. We dwell specially on V~ and V~-~ in order to illustrate intuitively the properties of relative invariants and invariant objects. i. To the relative invariant Vh corresponds a unique reduced component ~k. 2. If

Vk=~h:0

, then

V/,-1-~

N~

i.e., there is gained the structure of a relative invariant, etc. 3) If

V k ~ 0 (~k~0), then one can reduce

Vk~Ck=/=O, Vk_1~O, NOq~k

~bk----~=

after which, from (I0),

Ck ;

N~ ~k-' 9 N lk_l *k

4) We give some examples of the construction of absolute invariants. 4a) If invariant canonization

is carried out, after which

(ll) then V k becomes an absolute invariant of the form

fk~

N~ (v~) ~ (y') ~ (v~)

4b) Now if here there is also carried out canonization, by virtue of which

218

%=E(gu),

then all

Vi, i-----O .....

(12)

k, also become absolute invariants, whose expressions can be obtained,

substituting (ii), (12) in (i0). 4c) Suppose there are two balanced relative invariants V and V' (V' # 0); then v

:v,-N is

an absolute

,,

invariant.

4d) Under the conditions of Sec. 12, from three relative invariants one can get an absolute invariant

5) An important example of a linearly ordered invariant object is the subobject u-~.-1.~ of the object

uF; for it we have 1

6) The theory given above can be generalized also to a wider class of objects, namely, to objects having the properties

V.+~ = [~{Vj([~/= const); 9(Y]):p(Vi); For example, the components

Ust, t ~

a ( V ] ) = ~ ( V i ) + I. I[u,t[[

of the minors of the matrix

have such properties,

and there are others too. Thus, for

ust

(omitting the calculations) 1

As an example of the application of this formula, we give the calculation of the expression for the relative invariant [ = ]/Ln--2, n--2 /s n--1 l /~n--1, n--2 t~n--I,n--1

According to (I0), we have

Ui~_l,n _ l = T~n-~ N n-1, oo n--Ir

~n-3/NlO I n--I, n--2~n--I,

Urt--1. n - - 2 = " T

n--I;

tNOO tz--1,n--2~n--1,

n--1 T

rt--2);

~n-4

Un-~, n - 2 = - -~1 ~ - v * ~,ArU -2, We n u m b e r t h e

constants

00

Nut=l;

"N ~-2~25~-1, ~-1 T

s, t = n - - 2 ,

~.1,10 Vn--t,n--2~

~,n-I

I~ n--2~n--1, n - 2 T" N ~-2,

n--1 , after ,.

--~=Y;

11 Nn-2

n-2~?2;

which,

~176 ~ - 2 ~ - 2 , n-2). ,,-2, we g e t

10

Nn-2, n - z = 2 y ,

and then

219

-n-4 (V 2~n-1 .... 14- 2y~,~-1,~-2 + 1 ~1

n-2 rln-a(YxP,,-1,

~,,-2,

4-

,~-I

+ ~ n - 1 n-2,

I = ~

~n-a(y~,,_~, n-~ 4- ~,,-~, ,,-2)

~l~-2~,,-x,

n'~-~

I

n--1

~-a

~tz--2, n--l~n--1,

n--1

as was assumed by the theory. 7) All the constructions

given carry an entirely formal character, which allows us to

apply them to any (not necessarily nonlinear)

orderedness,

invariant)

for example,

to $~

object,

satisfying the property of linear

which will also be done below.

20. We calculate in explicit form the components I. For these components

(Imax=fg,

(or

~-i"

SO

i.e., the given coefficient

~_l=const. 2. E -1

We n o t e

occur.

a decisive

circumstance

Differentiating

in

the

present

case:

IIm and comparing the coefficient

in none of the of

d l I '~

does

lira A

IimA[[ -I with zero, we get:

re=l: (In ~ l<m
Summing all these expressions

ln~

~m_~-=O;

n ~n--~'-

(13)

~n--2 -- O.

--

from m = i to m = n -- i, we get

+ _ _ fn-1-- ~= O,

In ~ | ~

2

' 1

/_,

whence after simplifications

(ln~)_l Substituting

2 2~ yn_l" ='-fi"-~l ~IO ~nn (n__l)

(14) in (13), we get for all

~-i

the expressions

(14) of the same type

(15)

m 1 ~m--1 = A m , ~oq-Bmfn<,

where

Am 21.

are

constants,

We c o n s i d e r

B m = B m ( [ , ~1) a n d

an arbitrary

function

A m, B m c a n

be

F ( y u) a n d i t s

calculated

elementarily.

differential

d F = F ~ I I ~ - - OF 9 u - - OY ~ a y

Comparison of the two expressions

.

for dF, taking account of (8), gives the formula OF r . OF Fk=Ak0~mOao~, l > k

from which follows

220

>0),

I. The equations Fn_1=O

Fn_l-~O

2. For

(~ O F

=0)

OF

oun_~ = 0

and

are equivalent.

Fn-2=O

the equations

and

can also be formulated as the equivalence of the systems 3

For any

for Fn-1=.

/>0

9

=Ft+1=0

0e

Oyn_2 --0

become equivalent 9

Fn-x=Fn_2=O

from F l = O

follows

..

and

0--[-F =0

OF

OF

This 0

d-~rr----O-~7=r=v. (and conversely)

Oyl

'

the systems Fn_l = Fn_2 . . . . . OF OF Oyn_ 1 - - O y n _ ~ . . . . .

F~ = 0, OF Ogi = 0

are equivalent9 4. For

I=0

OF F0 there is present -~-

in

and the preceding equivalences do not general-

ize to this case. 5. For F_I the complete notation OF

1

OF

admits the abbreviation already used by E. Cartan F_~

1 dF dx"

6. As an immediate application of the formulas found, we get different expressions for

~tN--I~-1=

where

A n, B ~

=n

AmL+

are computable constants.

22. We g i v e e x a m p l e s o f c o n c r e t e A. The r e l a t i v e

invariant

calculations.

U~-l,n-1

is

c o n n e c t e d w i t h any component

R~-l,n-1

by the rela-

tion

R~m--l,n--I -----A~m--ll~n--l,n--1 9 Taking,

for

simplicity,

m = n-

1 (this

h a s no i n f l u e n c e n--1

f r o m (8) we g e t

n--I

~n--2,n--I

Differentiation of (15) with respect to n -

on t ~ e r e s u l t ) ,

A n - 2 l l n - 1 n-1.

1 gives

l.l.n--I,n--1 ~ Pn-1 ,n-l f n-I ,n-I,

or, using the relation of the previous point, Un--l,n--1

=Qn--l,n--1

O~f ~ (Oyn-9

221

(in

P~-1,n-1

and

Qn-1,~-1 there occur

$, D

and numerical factors, these coefficients can be

calculated). By virtue of what was recounted above, from

Un-~,n-1--~O follows

OV - 0 (Oyn_~)2--

and the equa-

tion assumes the form

y(n) = Mly(n-t) _~_M2 ' w h e r e M1 and M2 a r e i n d e p e n d e n t THEOREM 13. with respect

F o r an-~,~_l=O

lows

g(~-~).

the equation

a s s u m e s t h e f o r m , where t h e r i g h t

side is linear

t o g(~-l).

COROLLARY. The p r o p e r t y ter

of

(16)

used of derivatives

with respect

g~:~ ) r e m a i n s v a l i d f o r any number o f d i f f e r e n t i a t i o n s ,

t o t h e i n d e x n -- 1 ( t h e p a r a m e -

i.e.,

from u(n ~)k=0

always fol-

0~f = 0 , whence t h e r i g h t s i d e o f t h e e q u a t i o n i s a p o l y n o m i a l ( i n t h e u s u a l s e n s e ) (Oyn-1)k

i n g(~-~)

o f d e g r e e k -- 1.

B. A n a l o g o u s a r g u m e n t s and c a l c u l a t i o n s THEOREM 14. independent of From t h i s

For

u,-~,,-x=U,-1,,-2=0

lead to the conclusion

the equation

assumes t h e form ( 1 6 ) ,

where M~ i s

y(--2~. t h e o r e m one can d e r i v e

corollaries

analogus to the corollaries

The c h a i n o f s u c h a r g u m e n t s and c a l c u l a t i o n s 23. We c o n s i d e r

a structure

can be c o n t i n u e d f u r t h e r

o f Theorem 13.

also.

where dK•

-~ - - 0 ,

(17)

or

u~_~,3=u~_~,2,~=O

(s=l ..... n--l).

In the preceding classification this possibility was held repreatedly. The geometric meaning of (17) is that the equation

H-I=0

is completely integrable and defines on the plane a one-

parameter family I of curves, which we, by analogy with the theory of Lie groups, call a system of imprimitivity; a structure with the condition (17), and also the corresponding equation will be called imprimitive also

y(f) will denote an imprimitive structure.

Any curve

from a system of imprimitivity intersects each integral curve (at least locally) exactly once, This allows us to introduce on the integral curves a compatible parameter, requiring that each curve of a system of imprimitivity correspond to a definite value of the parameter. Imprimitive equations appear in many classes, i.e., are sufficiently diverse, while on the other hand many familiar equations having practical applications turns out to be imprimitive. For imprimitive equations one can make a classification analogous to the general one. consider an important special case.

In a primitive structure

dg-I=--~l The c o e f f i c i e n t

Un-2.2

is automatically

upon passage from one curve to another.

222

We

AH-I

2 n ( n + l ) Un-z'2H-IAH~

eliminated We set

upon f i x i n g

a curve from I , but changes

gu--2,2 =

0,

whence

~-a,n-l,s=O.

~n-2,s=O;

0

We shall call such a system I homogeneous and denote it by

I ; the corresponding structure

0

gets the notation y([) and the same name. 24. The simplest and most important examples of imprimitive structures are the so-called linear structures,

defined by the condition

u(2) =o, whence all

R~r

(except

R~,-:)

and the conditional curvature are equal to zero.

The system

~=~=~Io=O becomes completely integrable,

(18)

and we are justified, without losing any information,

in con-

sidering the geometry of the fiber, distinguished by (18). We get the equations of structure

alI*--(1--6~_~)II~+~AII-~+R~,_tlI-~AIIt

(t K s - - I);

(19)

d[[ -1 = 0 , where d

and g e n e r a l l y ,

after

S Rt,--1 =

S -1 R t , - - l , - - 1 ~[ ,

any number of differentiations, s d R , s, - i k : Rt,_lk+' II -1 9 0

We see that the structure (19) is y(l). s ture (19) for any Rt,-1 define a group.

It is most important that the equations of strutA simple change of invariant forms leads to the

equations of structure d[Ik ~- Y[k+1AIf-!; drl "-~ : H ~ r I I A I I - ~ ;

k
dl1-1 = 0 9 THEOREM 15. stant

For U(2) = 0 the equation

coefficients),

exist nonconstant equation

either

a transitive

g r o u p g6,2

or an intransitive group g2,~ (for nonconstant coefficients).

coefficients,

ll-i=0 .

admits

Both

(for

con-

If there

then they are functions of one i n v a r i a n t - an integral of the

g8,2 and

g2,3 are groups of invariance of linear equations.

25. We compute in explicit form the coefficient

z

and its derivatives, which turns out

to be quite useful later. Differentiating

dln~

in (19) and comparing to zero the coefficient of II2All-I , we get

k~ q_On~)-~=o, whence,

considering

(14) and (19), we shall have

k•

2~V,~_~,2 . n(n+ l)~n+~'

(20)

223

H f n-l'2 " 1 + g'Hfn_l, 2 ' Differentiation

H

-

-

2~2 n(n + I)~n+''

of (20) gives

d (Hfn_l,2)--(Hfn_i,2)2dy t (l + yltt f n_!,2) 2 whence,

in its own right,

--;%H f n_l,3 (1 +glHfn_l,2) 2 ;

z(+l)

Hfn-l'2'J 9 j=2, .,n--l; x . / = (1 + ylHfn_l,2) z " ""

H f n-l,2,1+ H (n + 2)-~- f n-I,2 + (H f n-l,2)' ~ (1 + ytHfn_i,2) 2 (we do not need the expressions Now, writing

for x0 and x-l).

(17) in coordinates,

we get dxAdxAdy:O,

or z = z ( X ,

y) , which means X(+I) = 0 ;

From (21), considering

~S-~O ( S = 1 . . . . , n ~

the correspondence Osf

not equal to zero.

will be considered below),

f=

THEOREM 16.

derivatives,

we have

j = 2 , . . . , n--l;

.=0;

02f

Oyn-loy20y 1

0~f = 0 Oyn-~Oy2

(21)

between ordinary and covariant

Oyn-tOy2dyJ

where L is a constant,

1).

2

(22)

=o,

Under the assumption

integration

that

0~f 5A0 Oyn-~Oy2

(the case

of (22) gives

g~ + K (x, g, v') y~-~ + N (x, y,. . yn-2). L y I + C ( x , y) "'

In order that the equation be imprimitive

it is necessary and sufficient

that it assume the form

y(n)= y " + K ( x , y, y') Lv' + C ( x , V) y ( ~ - ~ ) + N ( x ,

C (x, y), If(x, y, y'), N (x, y,..., y~n-2~)

where

V, V' . . . . , y(~-2)),

are arbitrary functions of the designated param-

eters. We restrict the supplementary lines x = c

choice of principal parameters

, curves of the system of imprimitivity.

x, y, taking as coordinate

Then one must have

I1-1 =--~]dx, which,

upon comparison

with

(8),

gives

•

, and this, 0~f

c)yn-t c)y2

224

=

O.

(23) in

its

own r i g h t ,

is

equivalent

with

Thus, considering the corollary also, the equation associated with a system of imprimitivity can be written in the form

9(~)=K(x, y, 9')9<~-~)+N(x, y, 9~,..., 9(~-2)). 26. In equations, argument is time.

describing any process,

in the overwhelming majority of cases the

Physical time is an invariant parameter,

chosen up to the origin of refer-

ence, and possibly up to the choice of unit of measurement. We shall consider this question only from the position of the projected theory; we examine the possibility of introducing a compatible parameter on the integral curves, chosen with minimal arbitrariness. We shall assume that the equation considered admits a homoge0 neous system I , and when necessary we shall refer the equation to it. A~ The requirement of preservation of the parametrization transformations

(23) restricts the possible

of the parameter x to an infinite pseudogroup

9~ = 2 ( x ) . B, For

~2 ~02=~i~0

we have d~-~=

2 -~" d ~ : --~o~A~ ,

2

-1.

,~2_ 2 ,2 d%2--%2A%v

~02AH ,

i.e., to the projective group, allowing us to choose the parameter up to a linear-fractional transformation. In the present case the equation

e~0

is completely integrable and taking the fiber

of the induced bundle, we get d H - I = --m~iAH-1;

dm~l=O ,

which corresponds to the possibility of choosing a compatible parameter x up to a linear transformation. Finally,

taking the fiber along which

e~2=e~1=O

, we get

dH-1=0, which

gives

the

In the

possibility

scheme given

of choosing there

are

the

packed

compatible

many c l a s s e s

parameter of the

C. We a p p r o a c h t h e p r o b l e m f r o m a m o r e g e n e r a l p o i n t 0 Y(g2,6; I ) , we a s s u m e t h e e x i s t e n c e of a relative invariant

x up t o a n a d d i t i v e

classification

of view. V with

carried

As b e f o r e , the

constant. out.

considering

properties

a) V ~ O ; b)

Then for the form

H-I~VH-1

d V - - V ~ I : V _ I H -1.

we shall have d~-l=O;

where x is

existence

also

the

invariant

parameter

~-~=--dx,

sought.

(or possibility of construction)

Thus,

the

problem

[for example,

reduced

to the

of a relative invariant V, having properties a-b).

Condition a) means simply the reductiveness of the class. one can give classes

has been

U(p>l)=0

Condition h) is stronger, hut

], where it is automatically satisfied.

More-

225

over, in view of the invariance under the given conditions of the system

V~=0, s = 0 ..... n--l,

one can distinguish a subclass where V will have property b). Among the u themselves,

there are no relative invariants with such properties (p= I, a = n ~

hence one must use the constructions of Sec. 12. It is not appropriate to carry out a detailed description of the cases in which it is possibl e to introduce an invariant parameter, classification of equations.

since in practice this has to include the whole

We restrict ourselves to several examples and general remarks.

I. Let, for example, u-1,-1, n-l~=O; U-b-bn-bs=O; s~O , so

V=~/tt-1,-I,n-I (One could t a k e any i n v a r i a n t w i t h II.

p= 1, changing t h e a l g e b r a i c degree c o r r e s p o n d i n g l y . )

If there exist at least two relative invariants

u O, uh~ with properties a, b), then

V = (uij~v \ukt]' where automatically,

p=l, y

is chosen so that

o(V)

III. In the presence of two relative invariants b), one can choose numbers

71, ?2

=n--I 11 and fe (0(ll)~i, p(12)~i,

satisfying a,

such that

V=I~I.Q=;

p(V)=l;

a(V)=n--1.

IV. There is an elementary sufficient condition for the possibility of introduction of an invariant parameter:

the existence of at least one absolute invariant,

satisfying the

equation

dT= V_I~-I; V_I=~O, upon differentiation of which, we get

dH-1=0.

27. We shall now assume that the equation admits an invariant parameter of the previous point.

It is impossible,

in general,

to perform a completely analogous procedure with the

form H0, but it can turn out to be possible to normalize

R ~ invariantly so that

d~ o= d (WHo)= 0 (rood H-~). This operation no longer defines y with the same definiteness as x, but one can always assume that y is defined up to some function of x. Besides the existence of the invariant V of the previous point, we require that there should exist in addition a relative invariant W with the properties

a' ) W@O; b ' ) dW--Wolo=WolI~

W-I~ -I.

Just as above, one can give classes of equations, where a, b) are either realized or can be realized. After performing normalizations

the corresponding structural formulas can always be

written in the form

dh-1 = 0 ;

226

d~0 = I~IAII-1.

(24)

THEOREM 16.

In order that an equation have invariant parameters x, y (in the sense for-

mulated above), it is necessary and sufficient that it be possible to perform normalizations leading to (24).

If (24) holds, then invariant parameters are introduced by the expressions

~-l= _ d x ;

HO=dv--v'dx.

Despite several tautological formulations, Theorem 16 is by no means devoid of positive content.

In particular, this relates to the case when (24) is obtained by another method or

holds by virtue of special reasons. LITERATURE CITED I. 2.

3. 4. 5.

6.

7.

8.

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12.

13. 14~ 15. 16.

17. 18.

19.

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CHRONOLOGICAL SERIES AND THE CAUCHY--KOWALEVSKI THEOREM A. A. Agrachev and S. A. Vakhrameev

UDC 517.955

We consider the Cauchy problem for a system of partial differential equations. We prove an existence theorem for a solution of this problem which is analytic in the spatial variable under the assumption of measurability and local integrability of the right side with respect to time only. The solution is represented in the form of a chronological series.

In this paper we consider the Cauchy problem for a system of partial differential equations 0t~, u (t, x ) =

: f ( t , x, u(t, x) .....

~ Ot k, (Oxl) ~,

) (Oxn) ~ . . . . .

0j by)u(t, x)i~=. =%(x), xeO~R ~,

j:O,

1. . . . .

(z)

m--1

with respect to an N-dimensional function u. Here f is some N-dimensional function depending on t, x, u and all partial derivatives of u of the form

Ok+J~l Ot k(Oxl) ~1 ... (Oxn)~~,, where

o~=(el .....

~.),

[o~l+k.<m,

k<m,

Is]=~x,+...+o~,.

The classical Cauchy--Kowalevski theorem asserts that if the functions

~, ~h, k=0, ! , . ,

m--l, are analytic in all free arguments in the corresponding domains, then there exists a unique analytic solution of this system, defined in some sufficiently small neighborhood of an arbitrary initial point (to,Xo)ERXG. Despite the fact that this theorem has been known for many years, the subject is sufficiently important that the question of the possibility of strengthening and generalizing it in various directions is constantly found in the field of view of specialists.

We shall not

Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 12, pp, 165-189, 1981.

0090-4104/83/2102-0231507.50

9 1983 Plenum Publishing Corporation

231