Integrable systems on Lie algebras and symmetric spaces

INTEGRABLE SYSTEMS ON LIE ALGEBRAS AND SYMMETRIC SPACES Advanced Studies in Contemporary Mathematics A series of book...

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INTEGRABLE SYSTEMS ON LIE ALGEBRAS AND SYMMETRIC SPACES

Advanced Studies in Contemporary Mathematics A series of books and monographs edited by R. V. Gamkrelidze, V.A. Steklov Institute of Mathematics USSR Academy of Sciences, Moscow, USSR Volume 1

Geometry of Jet Spaces and Nonlinear Partial Differential Equations I.S. Krasil'shchik, V.V. Lychagin and A.M. Vinogradov Volume 2 Integrable Systems on Lie Algebras and Symmetric Spaces AT. Fomenko and V.V. Trofimov Additional volumes in preparation

Lagrange and Legendre Characteristic Spaces V. A, Vasilyev Some Classes of Partial Differential Equations A. V, Bitsadze ISS,N:0884-0016. This book is part of a series. The publishers will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details.

INTEGRABLE SYSTEMS ON LIE ALGEBRAS AND SYMMETRIC SPACES By A.T. Fomenko and V.V. Trofimov Faculty of Mechanics and Mathematics, Moscow State University, Moscow, USSR Translated from the Russian by A. Karaulov, P.D. Rayfield and A. Weisman

Gordon and Breach Science Publishers New York

London

Paris Montreux Tokyo

Melbourne

Q 1988 by OPA (Amsterdam) By. All rights reserved. Published under license by Gordon and Breach Science Publishers S.A. Gordon and Breach Science Publishers

Post Office Box 786 Cooper Station New York, New York 10276 United States of America Post Office Box 197 London WC2E 9PX England

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Japan Private Bag 8 Camberwell, Victoria 3124 Australia

Library of Congress Cataloging-in-Publication Data Fomenko, A. T. Integrable systems on Lie algebras and symmetric spaces.

(Advanced studies in contemporary mathematics, ISSN 0884-0016 ; v. 2) Translation of: Integriruemye sistemy na algebrakh Li i simmetricheskikh prostranstvakh. Bibliography: p. Includes index. 1. Hamiltonian systems. 2. Lie algebras. 3. Symmetric spaces. I. Trofimov, V. V., 1952II. Title. III. Series. QA614.83.F6613 1987 512'.55 ISBN 2-88124-170-0

87-26798

No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publisher. Printed in Great Britain by Bell and Bain Ltd, Glasgow.

Contents

Introduction

1. Symplectic Geometry and the Integration of Hamiltonian Systems 1.

Symplectic manifolds

1.1.

Symplectic Structure and its Canonical Representation. Skew-Symmetric Gradient The Geometric Properties of Symplectic Structures Hamiltonian Vector Fields The Poisson Bracket and Hamiltonian Field Integrals Degenerate Poisson Brackets

1.2. 1.3. 1.4. 1.5.

2.

2.1.

Symplectic Geometries and Lie Groups Summary of the Necessary Results on Lie Groups and Lie Algebras

xi

1

1

1

4 8 11 15 17 17

2.2. Orbits of the Coadjoint Representation and the Canonical

Symplectic Structure Differential Equations for Invariants and Semi-Invariants of the Coadjoint Representation

22

3.

Liouville's Theorem

30

3.1. 3.2. 3.3. 3.4.

Commutative Integration of Hamiltonian Systems Non-Commutative Lie Algebras of Integrals Theorem of Non-Commutative Integration Reduction of Hamiltonian Systems with Non-Commutative

30

Symmetries

36

2.3.

Orbits of the Coadjoint Representation as Symplectic Manifolds 3.6. The Connection between Commutative and NonCommutative Liouville Integration

27

32 34

3.5.

46 47

Vi

CONTENTS

Algebraicization of Hami Ionian Systems on Lie Group Orbits 4.1. The Realization of Hamiltonian Systems on the Orbits of the Coadjoint Representation 4.2. Examples of Algebraicized Systems 4.

5.

Complete Commutative Sets of Functions on Symplectic Manifolds

2. Sectional Operators and Their Applications 6.

Sectional Operators, Finite-Dimensional Representations, Dynamic Systems on the Orbits of Representation

Examples of Sectional Operators Equations of Motion of a Multi-Dimensional Rigid Body with a Fixed Point and Their Analogs on Semi-Simple Lie Algebras. The Complex Semi-Simple Series 7.2. Hamiltonian Systems of the Compact and the Normal 7.

52 52 58 63

67

67 71

7.1.

Series

71

75

Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Ideal Fluid 7.4. Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Incompressible Ideally Conductive Fluid

7.3.

3. Sectional Operators on Symmetric Spaces 8. 9.

79 89

100

Construction of the Form Fc and the Flow XQ in the Case of a Symmetric Space

100

The Case of the Group S _'$ Spaces of Type II)

105

x Sj)/.5 (Symmetric

The Case of Type 1, III, IV Symmetric Spaces 10.1 Symmetric Spaces of Maximal Rank 10.2. The Symmetric Space Sn' = SO(n)/SO(n - 1) (The Real 10.

Case)

10.3. Hamiltonian Flows XQ, Symplectic Structures Fc and the Equations of Motion of Analogs of a Multi-Dimensional Rigid Body 10.4. The Symmetric Space S"-' = SO(n)/SO(n - 1) (The Complex Case) 10.5. Examples of Flows X, on S"-' (The Complex Case)

107 107 111

120 121 131

CONTENTS

4. Methods of Construction of Functions in Involution on Orbits of Coadjoint Representation of Lie Groups Method of Argument Translation 11.1. Translations of Invariants of Coadjoint Representation 11.2. Representations of Lie Groups in the Space of the Functions on the Orbits and Corresponding Involutive Sets of Functions

11.

12.

Methods of Construction of Commutative Sets of Functions Using Chains of Subalgebras

vii

136 136

136

138

143

Method of Tensor Extensions of Lie Algebras 13.1. Basic Definitions and Results 13.2. The Proof of the General Theorem 13.3. The Application of the Algorithm (21) to the Construction of S-Representations 13.4. Algebras with Poincare Duality

147

Similar Functions 14.1. Partial Invariants 14.2. Involutivity of Similar Functions

167

Contractions of Lie Algebras 15.1 Restriction Theorem 15.2. Contractions of 7L2 -Graded Lie Algebras

171

13.

14.

15.

5. Complete Integrability of Hamiltonian Systems on Orbits of Lie Algebras Complete Integrability of the Equations of Motion of a Multi-Dimensional Rigid Body with a Fixed Point in the Absence of Gravity 16.1. Integrals of Euler Equations on Semi-Simple Lie Algebras 16.2. Examples for Lie Algebras of so(3) and so(4) 16.3. Cases of Complete Integrability of Euler's Equations on Semi-Simple Lie Algebras

147 151

160 162

167 168

171 174

179

16.

17.

Cases of Complete Integrability of the Equations of Inertial Motion of a Mufti-Dimensional Rigid Body in an Ideal Fluid

The Case of Complete Integrability of the Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Incompressible, Ideally Conductive Fluid 18.1. Complete Integrability of the Euler Equations on Extensions f2(G) of Semi-Simple Lie Algebras

179

179 185 189 194

18.

198

198

Viii

CONTENTS

18.2. Complete Integrability of a Geodesic Flow on 18.3. Extensions of fl(G) for Low-Dimensional Lie Algebras Some Integrable Hamiltonian Flows with Semi-Simple Group of Symmetries 19.1. Integrable Systems in the 'Compact Case' 19.2. Integrable Systems in the Non-Compact Case. MultiDimensional Lagrange's Case 19.3. Functional Independence of Integrals

203 204

19.

20.

The Integrability of Certain Hamiltonian Systems on Lie Algebras

205

205 208 212

214

20.1. Completely Involutive Sets of Functions on Singular Orbits in su(m)

20.2. Completely Involutive Sets of Functions on Affine Lie Algebras Completely Involutive Sets of Functions on Extensions of Abelian Lie Algebras 21.1. The Main Construction 21.2. Lie. Algebras of Triangular Matrices

215 219

21.

22.

224

224 232

Integrability of Eider's Equations on Singular Orbits of Semi-Simple Lie Algebras

22.1. Integrability of Euler's Equations on Orbits 0 Intersecting the Set tHa, teC 22.2. Integrability of Euler's Equations x = [x, (J1abD(x)] for Singular a 22.3. Integrability of Euler's Equations z = [X,(QabD(x)] on the Subalgebra G. Fixed Under the Canonical Involutive

Automorphism a:G-+G for Singular Elements aeG 22.4. Integrability of Euler's Equations for an n-Dimensional Rigid Body Completely Integrable Hamiltonian Systems on Symmetric Spaces 23.1. Integrable Metrics dspbD on Symmetric Spaces 23.2. The Metrics dsob on a Sphere S" 23.3. Applications to Non-Commutative Integrability

236 236 244

247 253

23.

24.

Morse's Theory of Completely Integrable Hamiltonian Systems. Topology of the Surfaces of Constant Energy Level of Hamiltonian Systems, Obstacles to Integrability and Classification of the Rearrangements of the General Position of Liouville Tori in the Neighborhood of a Bifurcation Diagram

254

254 257 263

66

CONTENTS

24.1. The Four-Dimensional Case 24.2. The General Case

lx

266 270

Bibliography

281

Index

293

Introduction

There are at present quite a few integrability problems known in dynamics. The

solution of these problems is based on the existence of n independent first integrals in involution, n being the dimension of the configuration space (which

is equal to the number of degrees of freedom) of a mechanical system. Henceforth, these sets of functions will be referred to as complete involutive sets. In these cases, according to Liouville's theorem, the equations of movement are integrated in quadratures. We know that the existence of a complete involutive set of first integrals implies a consequent qualitative picture of the behaviour of trajectories in 2n-dimensional phase space. Every phase space can be stratified

by congruent surfaces of the level of first integrals into closed n-dimensional invariant manifolds. If these manifolds are compact and connected, then they are n-dimensional tori and the motion through them is quasipcriodic. This book sets out some new methods for integrating Hamilton's canonical equations. Common to all these methods is one overall idea: the realization of canonical equations in Lie algebras or symmetric spaces. Basically, the book sets

out new results obtained by the authors and by participants in the scientific research seminar Contemporary Geometry Methods, run at Moscow University under the direction of A.T. Fomenko. For the reader's convenience, classical information on Hamiltonian systems is included in the first chapter.

In classical mechanics the most widespread method for integrating Hamilton's equations is the Hamilton-Jacobi approach. We know that Hamilton-Jacobi equations enable us to solve the classic problem of finding geodesics in a triaxial ellipsoid. In contrast to the Hamilton-Jacobi method, instead of a non-linear equation in the partial derivatives (aS/at) + H(t,q,(aS/aq)) = 0, in applying the methods which are to be described in this book, we must solve the system of linear partial differential equations of the first order Y_k.J C'Jxk(aF/ax;) = 0. At the same time, as is well known, there exists an effective algorithm for solving such systems. This leaves us with

a purely algebraic procedure for finding the first integrals of canonical equations: if the FV solution of the system V is >k,j

0, then

F(x + Aa) is the first integral for any Ac l8, while all such integrals are found to be

in involution. Three fundamental themes are examined in this book. First and foremost we are concerned with constructing the algebraic embeddings in Lie algebras of the xi

xii

INTRODUCTION

Hamiltonian systems which are so well known in mechanics. We shall state that

these systems allow an algebraic representation. It has been shown that the general construction of a sectional operator for an arbitrary linear representation of a Lie group, permits us to realize many physically interesting mechanical systems on the orbits of the representations. Within the framework of the theory of sectional operators a construction is offered for symplectic forms

(non-invariant under the action of the group) on symmetrical spaces, with respect to which the systems constructed are Hamiltonian. The second theme of the book concerns effective methods for constructing complete sets of functions in involution on orbits of coadjoint representations of Lie groups. The third and final theme of the book is the proof of the full integrability, after

Liouville, of a fairly wide range of many parameter families of Hamiltonian

systems that allow algebraic representation in the sense mentioned. One important fact is that these systems happen to include some interesting mechanical systems, e.g. the equation of motion of a multi-dimensional rigid body with a fixed point in the absence of gravity, the inertial motion of a rigid body in a fluid, as well as certain finite dimensional approximations of the equations of magnetic hydrodynamics. The basic difficulty which arises here is the proof of the functional independence of the first integrals.

1

Symplectic geometry and the integration of Hamiltonian systems

1. SYMPLECTIC MANIFOLDS I.I. Symplectic structure and its canonical representation. Skewsymmetric gradient

We shall begin by studying an important class of smooth manifoldsthe so-called symplectic manifolds. They appear in many applied problems, for example in problems of classical mechanics, and it is therefore absolutely essential that they should be studied in order to

solve many specific problems. One of the ways of introducing additional structure on a smooth manifold is to define a skew-symmetric scalar product which depends smoothly on the point. This leads us to symplectic manifolds, whose geometry is substantially different from that, for example, of Riemann spaces. Since the skew-symmetric scalar

product (in the tangent spaces) is defined by a second-degree skewsymmetric tensor it is sufficient to define an exterior differential form of the second degree.

A smooth even-dimensional manifold M2 is called symplectic if it has defined on it the external differential second-degree form to = Y-; <j coi; dxi n dx', so that (1) this form is non-degenerate, i.e. the matrix of its coefficients I wij(x) jj is non-degenerate at each point, (2) this form is closed, i.e. dw = 0, where d is the operation of exterior DEFINITION 1.1

differentiation (see [24], [120]). This form co is sometimes called a symplectic structure on a manifold.

It is clear that form w defines in a tangent space TAM2" a nondegenerate skew-symmetric scalar product w(a, b) = Ei,j wi;aW where a = (a'), b = (hi), a, b e TAM. If the point xo is fixed, then, as is apparent I

2

A. T. FOMENKO AND V. V. TROFIMOV

from algebra, there exists a transformation of coordinates in the tangent

space Tx0M, generated by a regular transformation of coordinates x1, ... , x2n in the neighborhood of the point x0, so that the matrix 1w1i(xo)II takes the canonical form:

where D = 0

D

This scalar product evidently defines a canonical ide tangent space TxM with the cotangent space T *M (on this operation see

[24], [120] for more detail). Let us bear in mind that the dual space

T*M consists of covectors-linear forms in tangent space. The canonical identification of tangent and cotangent spaces, generated by the form w, enables us to define an important operation which is an analog of the operation for constructing a gradient vector field grad f on a manifold, provided the symmetric scalar product is defined on M (i.e. a Riemannian metric, see [24], [166]). DEFINITION 1.2

Let f be a smooth function on M and w be a

symplectic structure. The skew-symmetric gradient s grad f of the function f (the "skew gradient") is the name given to the vector field on M, uniquely defined by the relation w(v, s grad f) = v(f), where v is an arbitrary smooth vector field on M and v(f) is the value of v considered as a differential operator on the function f.

In other words, the first task is to examine the covector

(fix

' 8x

") e T*M

and then, relying on the canonical identification TxM and T,*M arising

from the form w, to construct the corresponding vector field for this covector field: the resulting vector field will be the skew-symmetric gradient. If we had used Riemann's metric for this identification, we would have obtained the vector field grad f. The simple nature of the s grad f construction is a consequence of the non-degeneracy of Co. Usually local coordinates on a symplectic manifold are denoted by , q,, where these coordinates are chosen in such a way PI,. , p", 41, .

3

INTEGRABLE SYSTEMS ON LIE ALGEBRA

that at a fixed point x0 the matrix Ilw;;(xo)II is written as:

0

0,

(-E

where E is the (n x n) unit matrix. But if the coordinates are arranged in then the matrix may be the following order pt, qI, P2, q2, ..., p,,,

written in block-diagonal form, as shown above. Given this special choice of local coordinates pI, ... , pn, q,,. . . , qn, the vector fields grad f at the point x0 can be written particularly simply. In fact, since

(L.3f\ EToM, x

aPt

then

s grad f =

aqn

of of - of of agl,...'agn' apt,..., aR,

In the coordinates pl, q1, ../. , p,,, qn we have: s grad f (xo) =

of , - of ... , of

\\0q1

- f)

aqnOpn apt Obviously, the Riemannian metric can also be reduced at a point to a canonical (diagonal) form by choosing appropriate local coordinates. In this sense both structures, Riemannian and symplectic, are similar. They do, however, exhibit one serious difference which becomes evident the moment we proceed to examine the neighborhood of the point x0. As is known (see [24], [48], [166]), Riemann's metric cannot generally be

reduced to diagonal form in an entire neighborhood by means of coordinate transformations, as a non-zero Riemann curvature tensor can prevent this. A symplectic structure, on the other hand, can always be reduced to canonical form by transforming coordinates in a small neighborhood of the point (the size of the neighborhood being defined by the properties of the form). PROPOSITION 1.1

Let o) be a symplectic structure on Men. Then for any

point xo e M there exists an open neighborhood with the local coordinates pt, ... , pn, qt, ... , q2, so that in these coordinates the form to can be written in the simplest canonical way : w = i= t dpi A dq;.

For proof see, for example [1], [46], [117], [120]. This theorem proves to be useful in many calculations connected with symplectic structures. Local coordinates, whose existence is affirmed in the theorem and which reduce the form w to canonical form, are sometimes called symplectic coordinates. It is clear that by covering the manifold M with

4


open neighborhoods of the form mentioned in Proposition 1.1 we shall obtain an atlas for M (see the definition of an atlas, for example in [24], [120]) an atlas that is also sometimes called symplectic. The simplest example of a symplectic manifold is the Euclidean space 182n(p1, provided with the form co = dpl A dql + + dp" A dqn.

,

qn),

1.2. The geometric properties of symplectic structures

Unlike a Riemannian structure, a symplectic structure cannot be given

on all smooth manifolds. As we noted in Definition 1.1, the most elementary limitation is that any symplectic structure must be even-

dimensional, since any skew-symmetric matrix of odd order is degenerate. A more substantial limitation is that any symplectic manifold is orientable. To prove this statement it is sufficient to define on

the symplectic manifold M2k of dimension 2k a nowhere zero differential form of degree 2k. n Aw Let w be a symplectic structure on M2k, then 0 = w A (the product is taken k times) gives a form of degree 2k on M2k. We shall show that f) # 0. Indeed, in every tangent plane TXM2k of M2k there can

be found a system of coordinates x' Co = x1 n yl +

xk y1

,

yk such that

+ xk n yk. Then, clearly,

f j = ! _ 1)[k(k-1)]/2kl x1 A A xk A y1 A ... A yk

0.

We should note that a symplectic structure does not exist for all evendimensional orientable manifolds. PROPOSITION 1.2

There is no symplectic structure on the even-

dimensional sphere Stn in the case of n > 1. Proof Let there be on Stn (n > 1) the differential form co, which defines

a symplectic structure on S2n. Then dw = 0 and since the space of cohomologies H2(S°) = 0 where p >, 3, then co = dot for a certain one A CO = dimensional differential form a. It is obvious that CO n A dot) (the external product is taken n times). On the d(a n da n

A co is not zero at any point other hand, as we saw above, 0 = CO n on the sphere Sen. Therefore f2 = f vol where f is some smooth function and vol is the volume form on Sen. Thus, using Stokes' theorem (see, A dot) = 0, but for example, [120]), we have j s2. 0 = f d(a A da A IfS=" f volt >, m J S2. vol = mv(S2") # 0, where v(S2") is the volume of sphere S2" and m is some constant, m > 0. This proves the proposition.

5


It is clear that the proof of Proposition 1.2 works for any compact manifold M such that HZ(M, l) = 0. For example, on a REMARK

compact semi-simple Lie group there are no symplectic structures, since

H2(6, R) = 0 for the compact semi-simple Lie group 6. We shall now pass on to examples of symplectic manifolds which arise

in various geometrical and mechanical constructions.

The first source of symplectic manifolds is smooth orientable closed Riemann surfaces, i.e. spheres with handles. Here we may take as a symplectic structure the standard two-dimensional Riemann volume form which is a closed non-degenerate exterior two-form. If the surface is

given parametrically r = r(u, v), then the form of the volume has the aspect co = EG - FZ du A dv, where E = (r,,, r.), F = (r,,, r.), G = (ru, rv) and r,,, r denote partial derivatives of the radius vector r along u and v respectively (see, for example [24]). Any sphere with handles allows an explicit parametric definition, for example, the equation 3

3

2

z2 + Lye + (a' - xz) fl (x -a i)2 fl (x + a.)2x2J = Ez i=1

a,

i=1

a, (i A j) where s is sufficiently small, describes a sphere with eight

handles. An analogous equation can be written for a sphere with n handles.

The second source for obtaining symplectic structures is from cotangent bundles. As a rule, the position space of a mechanical system

is a smooth manifold M. This is what we call a mechanical system's configuration space. From the mathematical point of view, phase space coincides with the cotangent bundle T*M of the manifold M (see [120]).

Points of the cotangent bundle T*M are pairs (x, ), where x e M, e T ,*M, i.e. is a covector at the point x. It is not hard to verify that T*M is a smooth 2n-dimensional manifold, n = dim M. The natural projection p: T*M - M is defined thus: p(x, ) = x. It is clear that T*M

is the total space of a vector bundle, its base being the initial manifold M, while the fiber p-1(x) over the point x is the cotangent space T*M. We shall define a symplectic structure (see for example [120]) on manifold T*M. To do so we shall first construct on T*M the smooth 1-form co"), Let a e Tq(T *M) be a tangent vector of the cotangent bundle T *M at the point y e T ,*M, see Figure 1. The differential mapping p : T *M - M

maps the vector a into vector p*a, tangent to the manifold M at point


6

P_ f

Fig. 1.

x = p(y) = p(x, ). Now we shall define the differential 1-form on the space T*M in the following way: co("(a) = y(p*a), i.e. the value of the form is equal to that of the covector y on the vector p*a. Finally, for the 2-form we are seeking let us take the external differential form co"),

i.e. w = da ' The form we have constructed is closed and non.

degenerate, i.e. T*M is transformed into a symplectic manifold. We now adduce a coordinate description of the symplectic structure created. Let

U be a coordinate neighborhood in M. Using the mechanical interpretation, we denote the coordinates in U by q', ... , q", n = dim M.

Let us examine U c T *M; these are covectors whose point of apposition is in U. We can examine the basis fields a/aq',... , a/aq". Let the covector values on these fields be pl, ... , p.: we may take (q1, ... , q", pt, ... , p") as the coordinates in T*M in the neighborhood U. co, the 2form constructed in these coordinates, has the classical form:

w=dp1 Adq'+...+dp" Adq" (see [120]). As an immediate corollary we find that T*M is an orientable manifold for any M. Not all symplectic manifolds can be obtained by this method. (a) T*M is not a compact manifold, and (b) the form co which gives a symplectic structure is exact, i.e. co = da for a certain 1-form a. The third source of symplectic manifolds are Kahler manifolds. Let MZ" be a complex manifold (see [24], [48]), on which a Hermitian scalar product rl) is given. Let us examine It is apparent that w is a skew-symmetric non-degenerate 2-form. In order to obtain a


7

symplectic structure on MZ", it is essential to have the equality dw = 0. This requirement is not met in an arbitrary complex manifold with a Hermitian metric. DEFINITION 1.3

A complex manifold, provided with a Hermitian

metric, is called a Kahler manifold if the imaginary part co of the scalar product 1) is a closed differential form (dw = 0). Thus, all Kahler manifolds are symplectic. The converse, however, is not generally true (see [188], [182]). One classic example of a Kahler manifold can be found in the complex projective space CP". We have the

natural holomorphic mapping n: c"+'\0 - CP". We may examine C""\0 the covariant 2-tensor

f

Ek"=0 / 1zkzk)z

J\ > ZkZkE k

dzk ®dZk/

k

where z0, .

. .

(zkdzk)®( zk®dzk/},

, z" are the standard coordinates in C"+I

There exists on CP" the Kahler metric F such that 7r*F = P, where f is the form defined above. PROPOSITION 1.3

This statement is derived from the following four evident properties of

tensor F:

a) The restriction of P to the mapping fiber of n : C""\0 -+ CP" equals zero.

b) The tensor P is invariant in relation to the natural action of the group C* = c\0 on Cn+1\0: z(z0, ... , z") = (zz0.... , zz"), z E C*. c) The restriction of P to the orthogonal complement of a fiber with respect to the flat metric in C" is positive definite. d) The differential 3-form d(Im F) on CP" is invariant under the mappings induced by unitary transformations A of the space C"+1, AEU(n+ 1). The metric F constructed on CP" is called the Fubini-Studi metric (for details see, for example, [1]). We are now able to construct a rich store of symplectic manifolds.


8

DEFINITION 1.4

Any/ subset of the form

V(f1.... JN) _ {P = (ao:...: a.) E CP": f1(aO:...: a") _ ... = fN(aO:...: a") =

o},

where { f1, ... , fN} is any set of homogenous polynomials in the ring C[X0, ... , is called an algebraic variety in CP".

If grad f (i = 1, ... , N) are linearly independent, then the algebraic variety V(f1,... , fN) is a complex manifold embedded in CP", this being an immediate consequence of the theorem on implicit functions. Let

j: V(f1, ... , f,) - CP" be the inclusion of the complex manifold V(f1,... , f ,) into complex projective space CP", II v(f,,.,,, f.^,) = j * Im F where F is the Fubini-Studi metric. The differential form SZ v( f '...'fN)

gives a symplectic structure on the manifold V(f1,... , fN). This statement results from V(f1i .

.

.

,

fN) being a complex submanifold

of CP". The construction that we have set out gives us examples of compact Kahler manifolds.

DEFINITION 1.5 We shall call a bounded open connected subset in the space CN a bounded domain.

Any bounded domain is a Kahler manifold (see [48]) and therefore a symplectic manifold. 1.3. Hamiltonian vector fields

DEFINITION 1.6

A smooth vector field v on a symplectic manifold M

with the form co is called a Hamiltonian field if it has the form v = s grad F where F is some smooth function on M which is called the Hamiltonian. In special symplectic coordinates (p;, qt) the Hamiltonian vector field

is written as (8F/8q;, -8F/8p;) (see above). Hamiltonian vector fields (sometimes called Hamiltonian flows) allow of another important description in the language of the one-parameter groups generated by them from diffeomorphisms of the manifold M. Let v be a Hamiltonian field and ( be a one-parameter group of diffeomorphisms of M, represented by translations along the integral trajectories of the field v. This means that group 6" consists of transformations of g, operating on


9

3c=,V(O) Pxo

Fig. 2.

M thus: g,(x) = y, where x = y(0), y = y(t), y is the integral trajectory of field v passing through point x at the moment of time t = 0, see Figure 2. In other words the diffeomorphism gt moves point x for time t along trajectory y. Since the form co is defined on M the diffeomorphism g, transforms this form into a new one (g*w)(x) = w(g,(x)). Consequently

the derivative of the form w is defined along the vector field v, i.e. d/dt(g*w). A vector field v on the symplectic manifold M is called locally Hamiltonian if it preserves the symplectic structure co on M, i.e. the derivative of form win the direction of vector field v is equal to zero: d/dt(g*w) = 0. To put it another-way+-,form w is invariant with respect to DEFINITION 1.7

all transformations of type g, generated by field v, i.e. is invariant in relation to the operation of the one-parameter group (i°. The term "locally Hamiltonian field" owes its derivation to the following: A smooth vector field v on a symplectic manifold M is locally Hamiltonian if, and only if, there exists for any point x E M a neighborhood U(x) of this point and a smooth function Hv defined in this neighborhood so that v = s grad Hv, i.e. field v is Hamiltonian in the neighborhood of U with the Hamiltonian Hu. PROPOSITION 1.4

For proof, see for example [1], [87]. It is clear that any Hamiltonian field on M is locally Hamiltonian. The reverse is not true, i.e. a field that allows a representation of the kind s grad Hv on the neighborhood of U may fail to allow a global representation in the form s grad F where F is some smooth function which has been defined on the entire manifold. In other words, the local Hamiltonians Hv defined on separate

neighborhoods do not always "slot together" into one function F defined on all of M. In any case we shall be studying mainly the Hamiltonian fields defined on the entire manifold and having the form s grad F where F is a Hamiltonian defined on all of M.

10


One of the most important examples of Hamiltonian flows is a geodesic flow. Briefly recapitulating its definition, let M be a compact closed n-dimensional Riemann manifold, i.e. a covariant tensor field of 0, 0 degree two g, is given on M so that (a) g1, = gji, (b) S. (see [24], [166]). We shall define g'i by the requirement that g'Jg;k =k The Riemann metric g;j defines a scalar product in the cotangent bundle p) = n e T*M. We have a canonical symplectic structure on T *M. We can examine the Hamiltonian H(x) =,L g''p; p, on T *M and its corresponding Hamiltonian flow x = s grad H(x) with respect to the canonical symplectic structure on T *M. Insofar as H(x) is a first

integral of this flow the unit cotangent bundle S = {x H(x) = 1} is invariant under the flow s grad H.

The restriction of the flow z = s grad H where

DEFINITION 1.8

H = 129''p; pj to the invariant surface S is called a geodesic flow on the Riemannian manifold M.

Metric g;j gives a natural diffeomorphism T*M ~ TM, which is linear in each fiber (the classical raising of indices). The following theorem is valid: THEOREM 1.1

Under the natural isomorphism T*M -> TM the

geodesic flow trajectories map into trajectories which consist of vectors tangent to the geodesic lines in M. An individual transformation gt maps a pair (xo, po) to a pair (xt, pt) = gt(xo, po) where the geodesic line must be taken through x0 e M in the direction po in order to obtain xt. xt will then be at distance t along the geodesic line from the point x0 whereas vector p, is tangent to this line at x, and has the same direction as po. Let M be a surface in 1183 given locally in the graph form z = f(x, y); (x, y) will then be a local system of coordinates on M. Let p, py, be the corresponding coordinates in T*M. In this case the Hamiltonian of the geodesic flow has the following form:

H_(1+ f2(x,Y))Ps-2TsfP1p +(1+fZ(x,Y))Py 2(1 + fz + f2) T *M = l 2(x, y) © R2(Px, pr)

In one sense geodesic flows are universal Hamiltonian systems: according to the Maupertuis principle any Hamiltonian flow with the Hamilton function H = i Y al(q) p; pj + V(q), where a'' is a positive


11

definite matrix, coincides with the geodesic flow on the manifold of constant energy H(q, p) - It for the metric ds2 = (h - V(q))a;; dq' dq' (see for example [1], [29]). 1.4. The Poisson bracket and Hamiltonian field integrals

The Poisson bracket of two smooth functions f and g on the symplectic manifold M is the name given to the smooth function {f, g} defined by the formula DEFINITION 1.9

w;;(s grad f)'(s grad g)'.

{f, g} = w(s grad f, s grad g) _ i { f, g} satisfies the relations: (1) the operation { f, g} is bilinear; (2)

the operation { f, g} is skew-symmetric, i.e. { f, g} _ - {g, f }; (3) the Jacobi identity {h, { f, g}} + {g, {h, f }} + { f, { g, h}} = 0 holds for any functions f, g, h. Thus the infinite dimensional linear space C°°(M) of smooth functions F on a smooth symplectic manifold M is naturally an infinite Lie algebra

over the field R. We should note that the emergence of this algebra is entirely due to the presence of the "skew gradient" operation, since the

operation of taking the ordinary gradient does not generate a Lie algebra. This shows yet another distinct quality of skew-symmetric scalar products (symplectic structures) as opposed to symmetric ones

(Riemannian metrics). In the pages that follow we shall often be concerned with various finite-dimensional subalgebras in the Lie algebra C°°(M). We may construct a natural mapping at of the Lie

12


algebra C°°(M) into the Lie algebra V(M) of all smooth vector fields on

manifold M. This mapping can be defined thus: a(f) = s grad f. The mapping a: C°°(M) -+ V(M) is a homomorphism of Lie algebras, i.e. a{ f, g} = [a(f ), a(g)]. This means that the operation of LEMMA 1.1

taking the Poisson bracket changes, under the mapping a, into the operation of taking an ordinary commutator of the two vector fields a(f), a(g).

The proof follows from Proposition 1.5, and the definition of the operation s grad. The image of the Lie algebra C°°(M) in the Lie algebra V(M) is a subalgebra which we denote by H(M). Its elements are vector fields on M, which can be represented as s grad f, i.e. (in our terminology

hitherto) Hamiltonian fields. Accordingly H(M) is a subalgebra consisting of Hamiltonian fields on M. This means in particular that the

usual commutator of two Hamiltonian

fields is

once more a

Hamiltonian field though the Hamiltonian is obtained as the Poisson bracket of the two initial Hamiltonians of the fields being commuted. We should note that the mapping a: C°°(M) - H(M) is an epimorphism, but not a monomorphism, inasmuch as a has a non-zero kernel. If the manifold is connected the kernel consists of functions

which are constant on the manifold. Consequently Ker a is onedimensional and H(M) = C°°(M)/Kera = C'(M)/W. Of course the subalgebra H(M) of the algebra V(M) depends on the choice of symplectic structure on M. If in fact we change the form co, the subalgebra H(M) will also be changed: strictly speaking we ought to write it as H.(M), but we shall take the subscript co for granted and omit

it. The Poisson bracket of functions f and g has the following simple interpretation If, g} = (s grad f) g, i.e. it coincides with the derivative of function g along the vector field s grad f. This follows from the definition w(v, s grad g) = v(g) = E; v` af/ x`. This simple observation is extremely useful when studying the integrals of Hamiltonian fields. DEFINITION 1.10 The smooth function f on a manifold M is called an integral of a vector field v if this function is constant along all integral trajectories of field v. In other words the derivative of function f in the direction of field v must equal zero, see Figure 3.

Let v = s grad F be a Hamiltonian field on M, and f be a smooth function which commutes (in the Poisson bracket sense) PROPOSITION 1.6


13

Fig. 3.

with the Hamiltonian of this field, i.e. { f, H} = 0. Then the function f is the integral of the field v. -

Proof If we calculate the derivatives of f along the field v, we have The v(f) = w(v, s grad f) = w(s grad F, s grad f) = {F, f } = 0. proposition is proven. The function F-the Hamiltonian-in particular is always an integral of the field v, since {F, F} - 0 owing to the skew symmetry of Poisson's bracket. Thus any Hamiltonian field has at least one integral, this being

the Hamiltonian. From this we get the following property of Hamiltonian fields: on a manifold the field's integral trajectory cannot be everywhere dense, for it always stays on the hypersurface which is defined by the equation F = const. Therefore if a field has an integral trajectory which is dense everywhere in a given open domain, this field cannot be Hamiltonian. In this way a Hamiltonian field preserves the foliation of the manifold n hypersurfaces F = const.

If f and g are two integrals of a Hamiltonian field v = s grad F then their Poisson bracket { f, g} is also an integral of the PROPOSITION 1.7

field.

Proof From Jacobi's identity we obtain:

IF, { f,g}} _ -{g, {F, f}} - {f, {g, F)) = 0. This proposition does allow us in principle to arrive at new integrals

of a Hamiltonian field provided that two such integrals are known. Nevertheless this method of constructing integrals all too frequently ends in failure since the function {f, g} may turn out to be functionally

dependent on the initial functions f and g (i.e. we shall not find out anything new). Here we have run up against the problem of finding as

14


great a number as possible of integrals of a Hamiltonian field. The point

is that each of these fields defines a system of ordinary differential equations of 2n order on a manifold M2 , since in the local coordinates x', ... , xZ" the field v is written as x' = v'(x', ... , xZ"), where v' are the components of v, while z' are the derivatives with respect to time t. The integral trajectories of the field v are the solutions of the corresponding system. Thus, if a field v has an integral f we lower the system's order by one, restricting the field to the hypersurface f = const along which the flow v "runs." Let us suppose now that field v has two integrals f and g

and that f and g are functionally independent on the manifold. It is convenient to formulate the condition of independent in the terms of the

functions' gradients. What is actually happening is simple: if the two fields grad f and grad g are linearly independent at each point of a subset U c M which is open and everywhere dense, then f and g are functionally independent on the manifold. Moreover we shall later be dealing with polynomial functions on algebraic manifolds and for that reason we require the set U to be open and dense in M. When working with smooth functions it must be borne in mind that in this case a pair of functions may be functionally independent on one domain in M and yet dependent on another domain in M, see Figure 4. These situations are of course impossible for polynomials. If f and g are two functionally independent integrals then they define

two different foliations of manifold M into hypersurfaces: f = const and g = const where in the "general position" case these hypersurfaces intersect transversely in submanifolds Q of dimension 2n - 2. Since f and g are integrals of v, field v is tangent to the surfaces Q211 - z, see Figure

5. Consequently, field v may be restricted to the family of invariant surfaces of dimension 2n - 2 and we have lowered the system's order by

two. In the general case if we have found r functionally independent integrals of a field we have lowered the order of the initial system by r and reduced the problem of finding solutions of the system to a problem

Fig. 4.


15

Fig. 5.

on surfaces of dimension 2n - r that are invariant with respect to the field. In the "ideal case" 2n - 1 functionally independent integrals ought

to be found in order to find solutions of the system. In this case the common level surface of the set of the functions fl,... , would be one-dimensional trajectories coinciding with the initial system's integral trajectories.

One more function is formally required in order to give the move-

ment of points over these trajectories. In other words we should have completely integrated the system in the sense that we should have represented all its solutions as the common level lines of the functions f1,. , f2,,- I which we know. In real mechanical systems, however, such a situation is so rarely met with that it is not practical to count on such a set of independent functions existing. In this connection we are sometimes forced to make do with "partial integrability" of the system.

It is desirable to find a set of independent functions fl, ... , f, on a manifold so that their common level surfaces (over which the flow v runs) might be arranged simply enough, for example, for all, or almost all, of them to be diffeomorphic to some well-studied manifold. In fact the problem of describing the system's solutions can be broken down

into two stages: (1) first we produce r integrals thus allowing us to describe their common level surface Q2, `'; (2) we then try to describe the system's motion (i.e. the motion of points along the integral trajectories)

on these level surfaces. One remarkable fact is that many concrete systems have sets of integrals which do allow us to realize the theoretical program described above. One of the best known results of this sort is Liouville's theorem. 1.5. Degenerate Poisson brackets

So far we have been examining only non-degenerate Poisson brackets (if

16


are local coordinates, then det{x' , x'} 0). Degenerate brackets are also of some interest. They are connected with foliating symplectic structures, as the following statement shows. X`

Let a (degenerate) Poisson bracket be given on manifold TM be the corresponding Hamiltonian operator: = {'p, o}(m), m e M. Then (1) the distribution m -> I',.(T*) in M is integrable and (2) the operator F induces a symplectic structure on the distribution's integral manifolds. LEMMA 1.2

M. Let F: T*M

Here is an example of a degenerate Poisson bracket arising in a concrete geometrical situation (see [189]). THEOREM 1.2

Let M be a manifold having a torsion-free affine

connection and R(M) be the principal frame bundle over M, while n: R(M) -+ M is its projection, then an n-parameter family (n = dim M) of closed 2-forms w e W(R(M)), dw = 0 exists on R(M).

Proof On the manifold R(M) are defined the structural differential forms w', w; E f21(R(M)); n*(5) = w'(4)e;, where E TX,Q,) R(M) and if = d/dtJ,_0(x(t), (e;(t))), then De;/dt = where V/dt is the

covariant derivative arising from the affine connection on M. These forms satisfy the structural equations: dw' + wk A wk = zS,gw° A wq A CO", dw; + ws n wj = where Sam, is the torsion tensor and R' pq the Riemann curvature tensor

[120]). The definition gives us the value co a;w A w E f22(R(M)), aj = const. We claim that dw = 0. In fact by using structural equations we arrive at: (see

dw =

[a;(dwj A wj - w; A dwJ]

= 2 Z a, Ri,. wp A wq A w' -

1\

2

ai S'nq wj l w° A CO.

Since Spq = 0 Ricci's identity RJ,pq + R,.qi + Rq jp = 0 is valid, and therefore dw = 0. REMARK An analogous structure may be carried out for any principal

0 bundle. Using the form co to define the skew gradient s grad f we find s grad f

17


can be defined exactly modulo the elements from the form w's kernel. Form co gives a bundle mapping A: TM -p T *M, A(s)(y) = co(s, y), s, y being sections of the bundle TM. Let f, g c C°°(M) be smooth functions so that df, dg e Im A, it will then be possible to define correctly their Poisson bracket {f, g} = w(s grad f, s grad g). This definition does not depend on the choice of representatives for s grad f If df 0 Im A then the definition gives us the value t f, g} = 0 for any g e C°°(M). In this case, therefore, C°°(M) is a Lie algebra and Lemma 1.2 can be applied to it. As

the final result we obtain a foliation of space R(M) into symplectic submanifolds.

2. SYMPLECTIC GEOMETRIES AND LIE GROUPS 2.1. Summary of the necessary results on Lie groups and Lie algebras

Let (h be a smooth manifold on which the group structure is given. (f) is then called a Lie group if the mapping 6 x (5 -+ (5 given by the formula (a, b) -+ ab-' is smooth. DEFINITION 2.1

Let v c TQ(5, then, dispersing v by means of left shifts over the entire group 6, we shall obtain a vector field on (5. To be more exact we can define Lo(g) = ag for ac-T9 and, since (La)-1 = La this is a diffeomorphism of the group Cf). We may put _ (dL0)e(v), La: (5 _+ 6.

The vector field constructed is left-invariant:

bab

for all

a,bcCC).

The space of left-invariant vector fields is a Lie algebra G under the commutator (bracket) product of vector fields. This Lie algebra is finite-

dimensional: dim G = dim (5. It is called the Lie algebra of the Lie group 6. If is a left-invariant vector field on (5, then generates a certain globally defined group of diffeomorphisms. The smooth homomorphism

: R' -+ 6 is called a one-parameter subgroup of the Lie group 6. It is easy to verify that the left-invariant field generates a one-parameter subgroup of (5. A Lie algebra can thus be defined by one of four equivalent means: (a) one-parameter subgroups; (b) tangent vectors at the unit of the group; (c) left-invariant vector fields; (d) left-invariant actions of the group R. All

one-parameter subgroups can be gathered into one universal mapping

18


Exp: G -+ 6 where Exp(tX): R --+ G - 6 gives a one-parameter subgroup with velocity vector X at the unit e e 6. Group 6 acts on itself by means of the conjugation (91,92) -' 91929,.1 . This operation then induces a linear operation on

DEFINITION 2.2

the Lie algebra Ad: 6 -+ GL(G), which is called the adjoint representation of the group 6: Ad9 = The group's adjoint representation induces that of the Lie algebra: ad = d(Ad)e: G -+ Hom(G). The equality ad,(Y) = [X, Y] is valid, where [X, Y] is the commutator in the Lie algebra G. The requirements of Hamiltonian mechanics demand a different representation. Let G* be the dual space of G, i.e. the space of linear mappings f : G - R. We can

define the coadjoint operation Ad*: 6 - GL(G*): (Ad9 f)(x) _ f (Ad. _ 1x). This representation's differential is called the Lie algebra's ad*: G - Hom(G*). The equality coadjoint representation: (ad* f)(Y) = f ([X , Y]), X, Y E G, f E G*. We shall be dealing with the

orbits or coadjoint representation Ad*. There is a natural symplectic structure on these orbits and any homogenous symplectic manifold is the orbit Ad* of some Lie group (see [156]). We should note that orbits of Ad* and Ad are different in the general case. The simplest example we can cite of this phenomenon arising is as follows: Let 6 be the group of affine transformations of the straight line x -+ ax + b, a 0, a, b E R. This group allows a matrix realization: a # 0, a, b e R1.

There is no difficulty in showing that

G= TQ6=

ac6, EG.

and

If

a= (0

1),=(0

02)

then:

Ada = CY1 0

alb

q1 q2 SIa2)=(0 0).


19

The explicit form of the coadjoint representation Ad gives Jnl =FY q2 = b2a1 - 1a2

and the orbits are therefore constructed as indicated in Figure 6. We can now look closely at the orbits of coadjoint representation. We choose in G the basis 1

el

0

0

0)'

e2 =

0

1

0

0)

but in G* the conjugate basis fl, f2 : f (e) = 5;;. In this basis we have the coordinates x1, x2. Simple calculations show that Ad* (0

(x1,x2) _ (x1 - x2a2,x2a1), 12)

therefore the orbits Ad* are arranged as shown in Figure 7. In particular

group 6 gives an example of representations Ad and Ad* not being equivalent.

If there exists on G a non-degenerate scalar product (X, Y) so that (Adg X, Adg Y) = (X, Y) then the adjoint and coadjoint representations are equivalent, i.e. they have identical orbits. PROPOSITION 2.1

Fig. 6.

Fig. 7.

20


The theorem of the classification of such algebras may be found in [183], or in: C. R. Acad. Sci. Paris, 301, No. 10, Ser. 1 (1985), 507 510. In particular all the Lie algebras that we call semi-simple satisfy this condition. The bilinear form B(X, Y) = tr(adx ad,.) is called the CartanKilling form. The characteristic property of semi-simple Lie algebras is that the form B(X, Y) is non-degenerate on G.

Recall the structure theory of complex semi-simple Lie algebras

(see for example [47], [50]). The maximal Abelian subalgebra H c G such that adh for all h e H is a semi-simple linear transformation of G, called a Cartan subalgebra. If X e G is an arbitrary element then we use G(X, 0) to denote the subspace of elements in G that commute with X. The element X e G is called regular if dim G(X, 0) is

minimal. If X E G is a regular element, then G(X, 0) is a Cartan subalgebra in G; this subalgebra is denoted H(X). Regular elements form in G an open, invariant and dense subset. If G is a semi-simple Lie algebra then any Cartan subalgebra is commutative. Let G be a semi-simple Lie algebra over the field of complex numbers

C. We fix a certain Cartan subalgebra H. A linear form a(h) on H is

called a root if there is an element E. e G, E, # 0, such that [h, E,] = a(h)E, for any h E H_ Let G2 be an eigen-subspace corresponding to a. Then G = H ED Y-,#o G1, where the sign Q signifies the straight sum of linear spaces. In the semi-simple Lie algebra G all subspaces G' are one-dimensional in the case of a # 0 (over the field C). We know that [G', GO] c G'+0, i.e. [E., EO] = N,8E,+0. If a + f # 0 then E. and E. are orthogonal with respect to form B(X, Y). Vectors E,

and E_, are, however, not orthogonal. The restriction of the form B(X, Y) to H is non-degenerate, if r = dime H (the number r is called the

rank of G) then there exist r linearly independent roots of the algebra G relative to H. The complete number of roots is generally speaking

greater than r and the set of all roots is therefore not linearly independent. If a, fi, a + fi are non-zero roots, then [G', GO] = G'+p;

the only roots proportional to root a # 0 are 0, ±a. Roots a can be represented by vectors Ha EH. Since B(X, Y) is non-degenerate on H, for every a e H* (H* is the dual space of H) there is a unique element

H' e H such that a(h) = B(h, Ha) for all h e H. Then, if a # 0, then [E X] = B(E X)H' for X e G and B(a, a) # 0. We denote by Ho c H the subspace generated by all the vectors HQ with rational coefficients. Ho is a "real" part of H. It turns out that dimo Ho = dime H = 2 dimR H (here 0 is the field of rational numbers).

21


Furthermore, the restriction of the form B(h, h') on Ho is positive definite and takes rational values (h, h' E Ho); a(h') E 0 wherever a # 0. In particular a(h') is a real number if h' e Ho. In future we shall use A to denote a set of non-zero roots of G. Let H,,. . . , H, be any fixed basis in

Ho. If A, ,u are two linear forms on Ho then it is said that A > p if 1(H;) = µ(H;) given i = 1, 2, ... , k and A(Hk+l) > u(Hk+l). We should not forget that if 2, y are roots, then 1(h'), µ(h') are real numbers for any h' e Ho. Thus a linear ordering is defined in the set A. The root a e A is called positive if a> 0, i.e. a(H1) = 0 given i = 1, 2, ... , k and a(Hk+1) > 0. Root a's positivity means in itself that the first of its nonzero coordinates is positive.

The linear ordering is not unique: from now on we shall suppose

that the basis H1..... H, (r = rk G) is fixed. We shall denote the set of positive roots by A +. Then A = A + u A - where A+ n A- = 0, and there is also a one to one correspondence between 0+ and A- which is given by the involution a -+ -a. It is clear that if ac-A' then (-a) e A-. The positive root a is called simple if it cannot be represented as the sum of two positive roots. If r = rk G = dims H, there then exist exactly r simple roots al,... , a, which form a basis in H over C and a basis in Ho over Q. Moreover each root /3 E A can be represented in the form /3 = > m;a;, where m; e Z are integers of the same sign; if m; 3 0 then fl e 0+ and if m; c 0 then l E A-. The system of simple roots al , ... , a, is usually denoted by II. The system A + is defined uniquely by

the system 11. If we let V+ = >,>o G2, V- = E, 1 we have: d

f)

t=o

do-1

d dt

d dt

0

dTn -

i=0

dn -1

F(Ad& p(,

r=odT n-1 t=0

d" ds" S=0

LEMMA 2.8

F(AdEx P TCAdEx Pr4 f)

1

F(AdEx., f ),

r)4

f)

s = t + T.

If the function F e A(G*) then

F(Ad* pf4 f) = F(f) +

(- (p(s))"F n=1

n!

(f).

t".

Proof results from the expansion of F(Ad*,,, f) as a Taylor series using Lemma 3.7. PROPOSITION 2.2

Let function F e A(G*), then

a) F is an invariant of the coadjoint representation of group % if, and only if, X1F = 0, i = 1, ... , n, n = dim G; b) F is a semi-invariant of the coadjoint representation of group 6

corresponding to a character x

if,

and only if, X,F = -AiF,

i = 1, ... , n = dim G, 2i = dx(ei) and dx is the derivative of x at the group 6's identity element.


29

Proof a) As F(Ad Ex ,, f) = F(f), so d/dtl t=0 F(Ad*Ex P,{ f) = 0 and therefore according to Lemma 2.7, when n = 1 we find that X; F = 0, i = 1, ... , n. Conversely, if X,F = 0, then (p(i )F = 0 and therefore [-tp(i )]"F = 0, when according to Lemma 2.8. F(AdExpt4 f) = F(f ). Since Chi is a connected group, F(Ade f) = F(f) for any g e (5. b) From the equality F(AdExp,4 f) = X(Exp t)F(f) it follows that d

d dt

F(Ad,*pt4f)=dt t=o

t=o

X(Exp tj)'F Cr)

X*()F(f). Conversely, [(- (p(s))"]F = therefore, in accordance with Lemma 2.8:

and therefore [(-

F(Ad ExP, f) =

and since X(Exp ti;) =

11 +

n1

[X*()]" tnl . F(f) ni.

J

our theorem is complete.

Lastly, in order to find the invariants, the system of differential equations Cxk OF/8x; = 0, i = 1, ... , n must be solved, or for the semi-

invariants-C xk OF/ax; = 1F, i = 1, ... , n. For the methods of solving these systems see [106], [117]. The semi-invariants' system's solution cannot generally be found for every character. In terms of the operators X; a criterion of invariance of a subspace W c A(G*) with respect to the operators Ada , g E 6 can be given. PROPOSITION 2.3

Let W be a finite-dimensional subspace in A(G*) and

f e A(G*), then for any g e (5 (% being a simply connected Lie group having Lie algebra G) f (Ad,* x) e W if, and only if, X, f c- W, i = 1, ... , n = dim G.

Proof If f e W then from the fact that f (Ada x) E W for any g e 6 it follows that d/dtl t = 0 f (Ad* P,{ x) e W since any finite-dimensional subspace is closed and then, according to Lemma 2.7 Xi f e W. Conversely, it is enough to check that f (Ad* x) E W for g = since the connected Lie group is generated by any neighborhood of the identity element, and a sufficiently small neighborhood of the element is generated by one-parameter subgroups. We have W (-cp(s))" f (Ad* P t4 x) = f (x) + n=I f (x) n! and since W is closed, f (AdE.p,4 X) E W.


30

REMARK

Let p: 6 - End(V) be an arbitrary

finite-dimensional

representation of the Lie group 6 in a linear space V A function F: V -* l is called invariant if F(p(g)x) = F(x) for all g c- 6, x e V. Let dp(e;) f; = a fk where f is a basis of V and e; a basis of G (the Lie algebra

of group 6): F will be invariant if, and only if, axk(8F/ax') = 0, which is proved in exactly the same way as in the case of p = Ad*.

3. LIOUVILLE'S THEOREM 3.1. Commutative integration of Hamiltonian systems DEFINITION 3.1

One says that two smooth functions f and g on a

symplectic manifold are in involution if their Poisson bracket equals zero.

As we have seen, full integration of a system requires that we should know 2n - 1 of the system's integrals. Actually for Hamiltonian systems it is enough to know only n functionally independent integrals (where 2n is the dimension of M) that are in involution. In this case each integral "can be reckoned as two integrals," i.e. it allows us to lower the system's order each time by two units at a go, instead of one. Moreover, in this case the initial system is integrated "in quadratures."

Suppose that a set of smooth functions fl , ... , f in involution, i.e. { f , f } - 0 when 1 < i, j < n, is given on a THEOREM 3.1 (Liouville)

symplectic manifold MZ". Let M, be the common level surface of the i.e. M, = {x e M: f(x) = i;;, 1 < i n}. Let us imagine functions are functionally that on this surface of the level all n functions fl,. .. independent (i.e. the gradients grad f , 1 < i n are linearly independent in all points of surface Me). Then the following statements hold :

1) The surface M, is a smooth n-dimensional submanifold, invariant under each vector field v; = s grad f , whose Hamiltonian is the function f-

2) If the manifold M, is connected and compact then

it is

diffeomorphic to the torus T" or, in the general case, a connected nonsingular manifold (which need not necessarily be compact) being a quotient of Euclidean space 68" over some lattice of rank < n if all flows s grad f are complete.


31

3) If the level surface M, is compact and connected (i.e. if it is an n-

dimensional torus), then in one of its open neighborhoods regular curvilinear coordinates s1, ... , s", q,. .. , qp" where 0 < T; < 2n (the so-

called "angle coordinates"), can be introduced, such that (a) the symplectic structure co in these coordinates is written in the simplest + ds" A d(p", which is equivalent to saying way, i.e. w = ds1 A dcpl + that functions s 1 , ... , s,, cpl, ... , (p" satisfy the following correlations: {s;, sj} = {(p;, (pj} = 0, {s;, (pj} = d;j; (b) functions sl, . . . , s" are coordinates in the directions transverse to the torus and are functions of

the integrals fl, ... , f", i.e. s; = s,(fl, . . , f"), 1 < i < n; (c) functions x S1 where Ti is the (pl, ... , (p" are coordinates on torus T" = S' x angular coordinate on the i-th circle Si, 0 < T, < 2n; (d) each vector .

field v = s grad F, where F is any one of the functions fl,. .. takes the when written in the coordinates cpl, ... , q,, on form (p; = torus T": i.e. the field's components are constant on the torus and the

field's integral trajectories describe the quasiperiodic motion of the system v, that is they give a "rectilinear helix" on the torus T". Here functions q;, 1 < i and = ad, X> = . Let c- G* be a given covector. We examine in the algebra G the subspace H4 = Ann(e), which consists of all vectors X so that ad* = 0. Subspace H is called the annihilator of the covector . We shall say that e G* is the covector of general position if its annihilator's dimensionality is minimal. We

shall call the dimension of the annihilator of a covector in general position (see [26]) the index of the Lie algebra G. If G is a semi-simple

algebra this definition coincides with the rank of the Lie algebra G (= the dimension of a Cartan subalgebra).

34


3.3. Theorem of non-commutative integration

Using some ideas from Marsden and Weinstein [75] we shall formulate a theorem which naturally generalizes Liouville's theorem and which is proved by A. T. Fomenko and A. S. Mishchenko in [88].

Let a set of smooth functions , fk, whose linear span is a Lie algebra G with respect to the Poisson bracket, i.e. {J, f } _ Y9_, CQ f9 where C9 are constants, be given on a symplectic manifold

THEOREM 3.2

A_.

M2". Let M, be a common level surface in general position of the functions (f), i.e. M4 = {x e M : f(x) = i, 1 < i < n}. We shall suppose that on this level surface all k functions fl, ... , fk are functionally independent. We shall also suppose that the Lie algebra G satisfies the

condition dim G + ind G = dim M, i.e. k + ind G = 2n. Then the surface M, is a smooth r-dimensional submanifold (where r = ind G), invariant under each vector field v = s grad h, where h e H. Further, let v be one of the following Hamiltonian fields on M: (a) either v = s grad h, where the Hamiltonian h is an element of the algebra of integrals G and lies in the annihilator H4 of the covector which defines the level surface M,; (b) or v = s grad F is the Hamiltonian field on M for which all the functions in algebra G are integrals, i.e. 0 = {F, f } for all

f c G. Then, as in the case of Liouville's "commutative" theorem, if manifold M, is connected and compact it is diffeomorphic to the rdimensional torus T' and on this torus the curvilinear coordinates cp, , ... , cp, can be introduced, such that vector field v, being written in i.e. these coordinates on the torus, takes on the form (p. = the field's components are constant on the torus and the field's integral trajectories define the quasiperiodic motion of system v, i.e. they give a

"rectilinear helix" on the torus P. The proof will be given below. In the special case, when the Lie algebra of integrals G is commutative, the condition dim G + ind G = dim M becomes the condition that k + k = 2n, since the rank here is

G = dim G = k. Thus k = n and we get Liouville's "commutative theorem." In many concrete examples the Lie algebra of the integrals turns out to be compact and non-commutative. As can be seen from Theorem 3.2, the system's motion proceeds through tori T', whose dimensional r is equal to the index of algebra G. In the semi-simple case the rank r of the algebra G (= ind G) is less than its dimensional and moreover in all fundamental cases it may be considered that r v",


35

where k is the dimensionality of G. Thus, for example, in the case of the

series A,,-,, when G = su(n), we have r = n - 1, k = dim G = nz - 1, i.e. rank G z dim G. This means that r < k and since r + k = 2n, then r < n = 1 dim M. In other words the motion of the system v = s grad F proceeds through tori whose dimension is less, and substantially less, than half that of the manifold. This shows that Hamiltonian systems with non-commutative symmetries, i.e. which have a non-commutative algebra of integrals in the sense we have outlined above, are very much degenerate; that is, their integral trajectories (in the general position case) wind densely everywhere round tori of low dimension r. This is what distinguishes such systems from those which satisfy the conditions of Liouville's "commutative theorem" whose motion proceeds through

tori of half the dimensionality, i.e. r = n = z dim M. Thus the "noncommutative theorem" 3.2 allows us to integrate systems with strong degeneracy, a degeneracy all the stronger, the smaller the index of the algebra of integrals of the system. Such types of system, being systems

"with degeneracies" on the initial manifold, may turn out to be "Liouville type" systems on a given submanifold K in M. What is more, an interesting link can be found between commutative

and non-commutative integration. For example, if a Hamiltonian system has a non-commutative algebra of the integrals (with the condition dim G + ind G = dim M, then in many cases it also has a commutative algebra of integrals of half the dimension. Furthermore, the following proposition proven in [88] holds. THEOREM 3.3 (Fomenko, A. T.; Mishchenko, A. S.) Let v = s grad F be a Hamiltonian system on a compact symplectic manifold M, and let it be completely integrable in the non-commutative sense, i.e. having a Lie

algebra of the integrals G so that dim G + ind G = dim M. Then this same system will be completely integrable in the ordinary Liouville commutative sense, i.e. it also has a second commutative Lie algebra of the integrals of G', for which dim G' = i dim M.

Here we are assuming, of course, that the additive generators of both algebras G and G' are functionally independent almost everywhere on

M. It is clear that these algebras are not isomorphic if G is noncommutative. For this reason on a compact manifold the Hamiltonian system "with degeneracy" which is to be integrated has yet another commutative "general type" algebra of integrals. From the geometrical point of view such systems have an extremely simple structure. Let { T'}

36


be a family of r-dimensional tori where r < n; over these tori the system's trajectories move, forming in general position helices on them which are everywhere dense. Then (see Theorem 3.3) these low r-dimensional tori can be organized into greater tori of dimension n, i.e. half the dimension of M, and the system's trajectories move over them. These greater tori

T" are level surfaces of the second algebra of integrals, now a commutative algebra, see Figure 10. We should note that such a system's trajectories cannot be everywhere dense on the big torus T", since this torus' stratification into tori T' of low dimension is arranged locally as the direct product of a torus T' with a given complementary submanifold of dimensional n - r, see Figure 10. Theorem 3.3 can be proved from Proposition 3.9 by using the results of Chapter 5 and applying the classification of the (finite-dimensional) subalgebras of the Lie algebra C°°(M) (with respect to the Poisson bracket) for the case of

compact manifolds M; it is known that any finite-dimensional subalgebra in the Lie algebra C°°(M) with respect to the Poisson bracket) on a compact symplectic manifold is reductive. So far no analogous result has been proved for non-compact manifolds. Hypothesis: any Hamiltonian system on any symplectic manifold which is completely integrable in the non-commutative sense is also completely

integrable in the Liouville commutative sense. The expansion of Theorem 3.3 to non-compact manifolds, it may be thought, arouses interest because many concrete Hamiltonian systems are realized in the form of flows on non-compact manifolds.

Fig. 10.

3.4. Reduction of Hamiltonian systems with non-commutative symmetries

We shall describe a simple and elegant construction which allows us to convert a Hamiltonian system which has a group of symmetries into a Hamiltonian system on a symplectic manifold of lower dimension (J. Marsden, A. Weinstein [75]). This procedure is called the reduction of a

Hamiltonian system. As one of the applications, we shall prove Theorem 3.2.


37

Let a Hamiltonian vector field v = s grad F with algebra of integrals G

be given on a symplectic manifold (Mzn, (o), and let the algebra's additive generators be k (almost everywhere) independent smooth functions fl, ... , fk. Let 0 be the corresponding simply connected group operating on M by symplectic diffeomorphisms (i.e. those that preserve (o). For our purposes it is easier to look at the following mapping cp. We shall identify each point x e M with the linear functional (the covector) co., on the algebra G. We shall take the value tps(f) = f (x),

where f e G. Thus q' is an element of the space G* dual to G. Consequently we have defined a smooth mapping qp:M -' G*. Let E G* be an arbitrary operator. Then its full inverse under the mapping (p, is a common level surface M4 of the integrals f1, ... , fk which generate algebra G. LEMMA 3.1

image

Proof According to the definition T

{x e M : f (x) = (f)},

where f e G. Since (f) is an additive basis in G, then f = E+=, a; f, i.e. If (f) _ I, 1 , = All fields s grad f that are generated by elements f of the annihilator are thus tangent to the corresponding level surface M,. PROPOSITION 3.2 The equality (Tx M4) n (s grad f ; f e G) = Hx = (s grad h; h e H,)

is valid, (see Figure 13).

Proof We have shown above that

(s grad h; h eH4) c TxM, n

(s grad f ; f eG). We shall prove the converse. Let x e TxM4 and X = s grad f, where f oG. It must be proved that f oH4. We {f g}> - 0 for any g e G, since { f, g}(x) = 0. examine f ), g> = This last equality arises from the fact that t f, g}(x) = (s grad f )glx = X(g) = 0 since X e TM4, and all g e G are constant on the level surface f) = 0, f), g> - 0 given any g e G. This means that M. Thus

i.e. f e Ann = H4. The argument is proved, see Figure 12.

Fig. 12.


39

We also consider the group Sj with Lie algebra Hf, i.e. Exp H4 c 0. COROLLARY The level surface M4 is invariant relative to the action of

the group 6 on the manifold M. We examine form co on M and assume that 6 = O) M4 is its restriction to the surface M. The action of the subgroup .5, on M, generates at each

point x e M, the plane Hs c TM,, formed by vectors s grad f, where f EH4, see Figure 13. In other words plane Hx is generated by the subalgebra H.

Fig. 13.

PROPOSITION 3.3

The kernal of the form w (the restriction of the form TxM.

w to W coincides with the plane Hx

Proof We shall first prove that Ker w

H. Let X = s grad h where

h e H,, X e Hx c T,,M,. We need to prove that X lies in the kernel of the form co, i.e. that w(X, Y) = 0 for any vector Yin the plane TxM. In fact, w(X, Y) = w(s grad h, Y) = Y(h) = 0 since vector Y is tangent to the

level surface, while function h, being an element of the algebra of integrals G, is constant on the level surface. We shall now prove the converse, i.e.

that Ker w c H. Let w(X, Y) = 0 for any vector

YETxM,. The vector X must be represented in the form X = s grad h for a given function h EH,. We then consider the form co as a skewsymmetric scalar product on the tangent plane TxM and we use (TxM4)' to denote the orthogonal complement to plane TXM, in TxM relative to the form co. Since the form is non-degenerate, the equality dim(TTM,)1 = dim M - dim TM4 = dim V = k is valid. We should

bear in mind that in the case of skew-symmetric scalar products the space TM need not necessarily decompose into the direct sum of T^ and (T M4)', since these planes can have a non-zero intersection. It is clear that Kerco = T. M, n (T,, M,)'. We shall prove that (sgrad f;

40


f e G) = (TTM,) -. In fact, if Y E TM, then uo(s grad f; Y) = Y(f) = 0,

since f = const on M,. Thus (s grad f; f c- G) c (TXM,)1. Further, dim(s grad f; f e G) = k = dim G. This equality is a consequence of the linear span of the gradients (grad f ; f c- G) having dimension k (see definition of G); the skew-symmetric scalar product is non-degenerate and the skew gradients' linear span also has dimension k. Lastly we note that dim(TAM,)1 = k, therefore (s grad f; f e G) = (1 M,)1, see Figure 12. The argument has been proven.

Let us now bring all these facts together and study the geometric picture of the mutual interaction of the submanifolds we have described. The fundamental objects are: (a) the level surface M,, dim M, = 2n - k; (b) an orbit 05(x) of a point x, dim (5(x) = k; (c) an orbit S>,(x) of a point

x under the action of the subgroup .54 = Exp H,. It is clear that Tx6(x) = (s grad f ; f c G), T.-5,(x) = HX = (s grad h; h e H,). It follows

from this that (5(x) r M5 = SJ,(x), see Figure 14. We note that the dimension of orbit 1),(x) equals that of Sao and is equal to r.

Fig. 14.

Let us look at the action of the group 6 on M and assume that in a small neighborhood of the surface M, this action has a single type of stabilizer subgroup, i.e. that all orbits of the group 0 close to an orbit 6(x) are diffeomorphic to it. Let us examine the projection p: M - M/6 of manifold M on the orbit space M/(5 = N. This space need not be a smooth manifold and it may have singularities. What is important to us is that space M/6 is a smooth manifold of dimensionality 2n - k in a small neighborhood of the point p(5(x) E M/0i. In reality, if (fi is for example a compact group and operates smoothly on M then the union


41

of the set of orbits in general position which are diffeomorphic to each other is an open and every where dense submanifold in M; therefore

space N is a 2n - k-dimensional manifold everywhere, with the exception of a subset of measure zero. We should bear in mind that the space (manifold) N need not necessarily be symplectic since it may be, for example, odd-dimensional. The projection p restricted to the surface

M,, projects it onto the surfaces Q4 = M,/.5,. Therefore space N is stratified by surfaces Proposition 3.2.

see Figure 15. Here our argument is based on

Fig. 15.

The manifolds Q4, i.e. the quotient manifolds of the level surfaces M, under the action of the subgroup .5,, are symplectic manifolds with a non-degenerate closed form p, which is the projection of the form w on M, under the mapping p: MC - Q4. Here p*p = th _ PROPOSITION 3.4

w/M4.

The proof ensues from Proposition 3.3, since the kernal of the form w

on TM, coincides with the plane H,, c 1 M4.

Let us now go back to study Hamiltonian systems on M. Let v = s grad F be a system with algebra of integrals G, i.e. {F, G} = 0. Since the Hamiltonian F commutes (in the Poisson

bracket sense) with all elements of G, F is therefore invariant with respect to the group 6. In actual fact (s grad f )F = { f, F} = 0, f e G. In particular, the subgroup .5,, in acting on M, also maps function F to itself. Thus we have defined a natural projection of the vector field s grad F onto the space N = M/Qi. At the

42


same time the vector field s grad F is tangent to the surface M4 and is also projected onto a certain field E(F) on the quotient Q4, since the field

s grad F is invariant with respect to .54. Thus space N is stratified by symplectic manifolds Q4 and a vector field E(F), which is tangent to all

surfaces Q4, see Figure

16, is

defined on N. Finally the triad

(MZ", s grad F, co) corresponds to a new triad (Qr, E(F), p). '

,f(x)

Al Fig. 16.

The vector field E(F) is Hamiltonian with respect to the symplectic form p on the manifold Q4 for the Hamiltonian function F, which is equal to the projection of function FI M4 onto the manifold Q4, i.e. E(F) = s grad p(p*FI Md. PROPOSITION 3.5

The proof follows from Proposition 3.4 and the invariance of

Hamiltonian F under the group action. The correspondence (M, s grad F, w) - (Qs, E(F), p) constructed above is what we call the

reduction of the initial Hamiltonian system s grad F. With this reduction we get a new Hamiltonian system on the manifold Q4 of dimensional 2n - k - r, lower than that of the initial manifold M, viz. 2n, while dim Q < dim M, = 2n - k. It could emerge that the reduced system on Q4 turns out to be simpler than the initial system on M. We shall assume that the reduced system has been successfully integrated. This will then allow us to increase the number of integrals which the initial system s grad F on M had, by "pulling" these integrals back from the manifold N onto the manifold M.


43

Let G be a finite-dimensional algebra of integrals of system s grad F on M, satisfying all the conditions enumerated, and let E(F) be the reduced system on the manifold N = U4 Q4, a Hamiltonian Let Gbe a linear space of functions on system on each submanifold the manifold N, such that their restrictions to the submanifolds Q4 form a finite-dimensional algebra of integrals of the flow E(F). The space of functions G E) G", where G" = p* G', i.e. G" = {gp, g e G'}, p: M - N is then a Lie algebra of integrals of system s grad F, while [G, G"] = 0. PROPOSITION 3.6

Proof Let g be a given function on space N; its inverse image gp under the mapping p: M -* N is then a function on M, obviously invariant with respect to the action of (( on M. But this means that the function gp

is in involution with the whole of the initial (integrals) algebra of functions G. Thus every new function which happens to be an integral of the reduced flow E(F) on N gives a complementary integral gp of the

initial Hamiltonian flow s grad F on M. That these complementary integrals are independent of the functions of algebra G follows from their gradients being non-zero in the direction of the submanifolds Q4 which lie (locally) in the level surface M4; at the same time the gradients of the functions in G are orthogonal to M. The proposition has been proved.

Proof of Theorem 3.2 Let v be one of the systems outlined in formulating the theorem, i.e. either v = s grad F, {F, Q} = 0, or v = s grad h, where h c Ann = H,. Let us examine the reduction above. Insofar as the supplementary condition dim G + ind G = dim M, i.e. k + r = 2n, has now been met, the described

dimension of the surface M, equals r. The dimension of the orbit S5,(x) which is contained in M, is also equal to r (according to the definition), see Figure 15. From this it follows at once that M, = .5,(x), i.e. in the terms of Theorem 3.2 the level surface M, is the orbit of a point x under the action of Sj,, and its Lie algebra is the annihilator of the covector which defines the given level surface. In particular, dim Q4 =

2n - k - r = 0. In the given case, therefore, the reduced system's structure is particularly simple. Since Q4 is a point, flow E(F) is zero, see Figure 17. Here space N has dimension n; since M4 is a level surface of the algebra G of integrals of the flow v, this flow is tangent to it in both the cases (a) and (b) (see the formulation of the theorem), i.e. M, is a rdimensional submanifold which is invariant with respect to all fields of the type s grad h, h e Ann(e) and s grad F, {F, G} = 0. It remains to be proved that the level surface is an r-dimensional torus in the case when


44

M, is compact and connected. To do so we shall need an auxiliary argument. PROPOSITION 3.7

Let c e G* be a covector in general position. Then its

annihilator Ann is commutative and so too, in particular, is the subgroup .5, (see for example [26]). Let us examine the coadjoint operation of the group 6 = Exp G on the coalgebra G*. We denote by the orbit which passes through point e G*. Since dim H, = r and

is in general

dim G* = k, dim 0*(c) = k - r. Since the covector

are also diffeomorphous to it, it may position the orbits close to also be taken that a sufficiently small neighborhood U of point is fibered into homeomorphic sheets, see Figure 18. We will denote a local section of the fibration of U by the orbits of the action of (b by X0. We

take advantage of U being representable as the direct product of the base and a fiber (i.e. a part of the orbit), see Figure 18. Let h(q) be a smooth

function on U which is constant on the orbits. We argue that a(c, dh(g)) = 0, where dh (the differential of h) is interpreted as an element of the dual space (G*)* = G, i.e. dh(c) e G. In other words, we 0 e Ann(e). We must verify that are arguing that for any g e G. We have g>

gj>

g), dh(f)) = 0,

g) = ad, lies in the plane TO* tangent to the orbit 0* of the point , while function h is constant on the orbits and since the covector

constant, in particular, on the orbit 0*, see Figure 18. If we take section X0, which is a smooth surface of dimension r transversely intersecting

the orbits close to the orbit O*(i;), we can consider on it the set of r independent functions h1, ... , h, and extend them into smooth functions on the entire U, by extending them from the section X0 with values that 0, 1 < i < r. Thus are constant along the orbits 0*. We get

45


Fig. 18.

E Ann(e), 1 < i = 0.

We may consider the arbitrary direction rl in the neighborhood U and differentiate the function 0 along this direction, i.e. we examine

0 d b(c) = d dhf()]> +

_ +

[dq

dh;()] )

dh;()J) + )

,

xl>,

x">W , x`>,

x">) 9

xl>, .. x

,

xi>} x">)

[xi, x']>

(2)

Formulas (1) and (2) show that the correspondence f - f* is an isomorphism of infinite-dimensional Lie algebras. If we take into account the property (FJ) we get the proof of our proposition. There are fairly abundant series of examples of Lie algebras which satisfy the condition (FJ). For example, as we shall see below, all semisimple Lie algebras V belong to them; for the commutative algebra Fo, the functions f ( + Aa), E V* are to be taken, where f is any function constant on the orbits of the coadjoint representation. Broad classes of soluble Lie algebras and certain semi-direct sums of Lie algebras also satisfy the condition (FJ) (see op. [89], [134], [126], [127], [10], [129], [123], [105]). Finite-dimensional Lie algebras of integrals V may be used likewise,

when they are not functionally independent, so long as the action of annihilator fj in the coadjoint representation has only one orbit type (see also 23.3). One example we can cite is a geodesic flow in the phase space of linear elements on sphere S" with standard Riemann metric (see

[88]). It will be easiest to consider that the sphere is contained in Euclidean space R" + 1, so that the space of linear elements L(S consists of pairs of vectors (x, y), Ixl = 1, x 1 y, i.e. (x, y) = 0. It is apparent that the dynamic system indicated is invariant under orthogonal transformations A E SO(n + 1). The system's equations may be written

in the following form: x = y, y = -x. Let (xo, yo) E T*S", then the tangent vector E T xu.yu)(T *S") may be given as a pair = (x, y), x I x0,

y 1 yo. Here the symplectic form co on the pair of tangent vectors S2) = (xl, Y2) 1 = (X141), S2 = (X242) takes the value (yl,x2). Then the function algebra V- so(n - 1) corresponding to the action of the group so(n + 1) consists of functions of the form:

50


fjx, y) = (x, cy),

c c- so(n + 1), (3)

{ff,,f'2} = f[c,.cz]-

Later (see §16) we shall show that the Lie algebra so(n) satisfies the condition (FJ). It is thus possible to construct polynomials Pk(f) which are pairwise in involution. It is therefore enough to make explicit the which are pairwise functionally maximal number of polynomials

independent on the manifold T*S". Formula (3) gives a mapping cp: T*S" -+ V*, which when written as a matrix has the form qp(x, y) =

xy' - yx', where x, y are understood as column vectors, and the operation y - y` is matrix transposition. The mapping co is equivariant to the action of the group SO(n + 1). Space T*S" is foliated

into submanifolds-the orbits of the action of the group SO(n + 1), which orbits may be parametrized by the length of the vector y, (x, y) E T* S" alone. The mapping cp maps the different orbits of space T*S" into different orbits, as the non-zero matrices A and 2A are not

equivalent when 2 # 1, 2 > 0. We shall show that the mapping (p, restricted to an orbit 0(x, y) T*S" has a Jacobian matrix of rank equal to 2n - 2. To do so we have only to take the point (xo = (1, 0, ... , 0), Yo = (0, 1, 0, ... , 0) and calculate the rank of the Jacobian matrix of the mapping at this point. The matrix 9(x, y) has the form II cp" II = T (X' y), (p'' = x' y' - y'x'. Then, given 2 < i < j, the partial derivative functions (pU at the point (xo, yo) equal zero. Further,

ail'=0, cx k

a ki

ask'=61, cy

2i

=o,

2 Fig. 20.

In the general case, naturally, there are not enough of the integrals indicated for full integrability of the Euler equations. In order to obtain full integrability the type of linear operator must be restricted. 4.2. Examples of algebraicized systems

A) The realization of equations of the inertial motion of a rigid body on the Lie algebra so(3). In this case the embedding of the system R3 which we are seeking exists and is simply arranged. It is enough to identify l3 with the space dual to the Lie algebra of orthogonal group SO(3), i.e. with the space so(3) of skew-symmetric real 3 by 3 matrices. This Lie algebra is isomorphic to R3 with the vector product serving as the Lie operation. The isomorphism is given by the formula:

f((J)1,(J)2,(0s) = I

0

-u)3

co3

0

-w2

(1) 1

w2

-w1 I

.

0

As we know, the orbits Adso(3l are spheres; the ring of invariants is generated solely by the function xl + x2 + x3. Let e1, e2, e3 be a basis of


59

= so(3), e', ez, e' be the dual basis, while D: (l83)* - l3 is the linear operator, D(e') = d'jej. We shall assume that d'i = d". Let us suppose that operator D is diagonal: ii

0

1/A

D=

1/B 1/C

0

then Euler's equations corresponding to the Lie algebra so(3) and this operator have the form (see the example in 4.1):

rl -

1)

XI =

C B

Xz =

A C )X1X3

xzxa

C1-1

1

X3 =

B

11 - A)X1XZ.

(3)

Equations (3) on G* = so(3)* = (1183)* are equivalent to the equations of motion of a three-dimensional rigid body with one fixed LEMMA 4.3

point, where the equivalence is given by the operator D: (l')* -+ R3.

Proof If we rewrite the equations (3) with the help of the operator Don so(3). We have 1

1

1

Yl = AX1,Y2 =

B

X2,Y3 = C X3

therefore x1 = Ayl, xz = Byz, X3 = Cy3, then equations (1) in the variables y; will take the following form: Ay1 = (B - C)Y2Y3

Byz = (C - A)Y1Y3

If we put (y1, yz, y3) = (p, q, r) we shall obtain classical Euler's equations for the motion of a solid body (see [140]):

AP = (B - C)qr Bq = (C - A)pr

Cr = (A - B) pq

(4)


60

where A, B, C are the body's moments of inertia with respect to the axes Ox, Oy, Oz respectively.

For integrating these equations enough integrals have previously been presented: F, = xl + x2 + x3 is the invariant of Ad*, while ( X 12

FZ

x22

x3

2 A+B+C 1

is the energy integral. In mechanics textbooks there is usually a change to coordinates y;, ending with the equations (4) having the following integrals:

F = A2yi + B2y2 + C2y3 = A2p2 + B2q2 + C2r2 is the integral of the moment of momentum, while the energy integral is

+ Bq2 + Cr2). F2 = i(Ayi + By22 + Cy2) 3 = 1(Ap2 2 We have thus constructed a realization of Euler's equations of inertial motion of a rigid body in a Lie algebra. Its uses are evident in this example: the very fact that it is realizable makes it possible to achieve integrability. Equations (2) can be rewritten on so(3) in the familiar commutator form. LEMMA 4.4

Equation (4) on so(3) is equivalent to the following system

pX = [cpX, X], X e so(3) and qp(X) = IX + XI,

22

and

-A+B+C 2

,

'2 =

A-B+C 2

,

X13 =

A+B-C 2

Proof is to be found by direct calculation. In this form the equations may be generalized to the n-dimensional case.

DEFINITION 4.7 We shall take the equations qpX = [QpX, X], X e SO(n) and p(X) = XI + IX, where


61

0

while A, + A, :?6 0 for any i, j, to be the equations of motion of an ndimensional rigid body with a fixed point. Later we shall give an invariant description of the operator q,(X) _ XI + IX (see 7.2). B) Realization of Toda's chain. First we shall show that the canonical symplectic space l 2n(q;, p.) together with co = dp1 A dq1 + + d p. A dq will allow realization in the Lie algebra of upper-triangular

matrices. We introduce the following notations:

are the upper-triangular matrices, 0

*

*

the lower-triangular matrices. We have a natural non-degenerate pairing a: T+ x T_ - 68, a(XY) = tr XY. This pairing lets us establish a

canonical isomorphism (T+)* = T_ and (T_)* = T+. We shall be examining the Lie algebra of upper-triangular matrices T+, then P

0

S

PZ

0

0

Pn-1

0

Sn-1

P.

It is easy to see that A = Uc A, where

62


P1 S1

0

P2

0 sn -1

P.

and A, is the orbit of coadjoint representation of the Lie group corresponding to T. We construct a mapping f : V?;Z" -+ A: PI

0

St

P2

0

0

Pn-1

0

Sn - I

Pn

f(x1,...,x,, Y1,...,Yn)

where

p1=Y1,sj=ex'-xj+1,1=1,...,n,j=1,...,n-1.

LEMMA 4.5

The mapping f : l2n -+ A gives the realization of a + dp" A dqn) in the Lie

symplectic manifold (l82n, dp1 A dq1 + algebra T+ of upper-triangular matrices.

Proof The equality {h1, h2} - f = {h1 - f, h2 - f } can be checked by direct calculation.

By a Toda chain we mean a Hamiltonian system on T*P" R"(Y1,

,

yn) O l"(x1,

, 1

xn) with the Hamiltonian: n-1

n

H = - Y_ Yk + Z akexk-xk+, 2k=1

k=1

The space T* l8" was realized by us in the Lie algebra T+ . This realization turns out to provide a realization of the Toda chain as well. To do so it is

sufficient to take the function H1 (F) = z tr(F + a)2 where F E A and

/0 a=

0

a1

0 ET,.

n-I 0

as the function H1 on A. Direct calculation shows that H = H1 o f. The


63

first integrals of the system s grad H1 on orbits of Ad* are given, for example, in [107]: Fk = tr(F + a)k. 5. COMPLETE COMMUTATIVE SETS OF FUNCTIONS ON SYMPLECTIC MANIFOLDS

As the preceding material shows, the following is a natural statement of

the problem: how does one find a commutative set of independent functions n in number on a symplectic manifold MZ"? In other words,

how does one find on MZ" an algebra which is commutative (with respect to the Poisson bracket) and of dimension n, whose additive basis

would consist of smooth functions, functionally independent almost everywhere on M? For brevity's sake we shall call these sets of functions

F(M) complete commutative sets. For the time being we are not concerned with integrating any Hamiltonian systems. If a complete commutative set is discovered on M, we automatically get a series of completely integrable Hamiltonian systems on M: it is enough to look at the fields sgradf where f is a function from set F(M). Then the ndimensional space of functions F(M) is a set of integrals for a system with Hamiltonian f. This approach to the problems of Hamiltonian mechanics raises the basic problem of constructing the greatest possible number of various complete commutative sets on symplectic manifolds.

The greater the store of such sets, the more examples we get of completely integrable systems (in the commutative sense). In the class of smooth functions, as A. V. Brailov has noted, the task of constructing a complete involutive family is solved very simply. To be more precise, the following proposition is valid. PROPOSITION 5.1 There exists on any symplectic manifold a complete involutive family of smooth functions which are functionally independent almost everywhere on the manifold.

Proof Each point has a neighborhood U(p,, q,) such that w has a canoni-

cal form w=dp1 ndg1+ +dp" ndq". Let f =pi +q; . It is apparent that {J, f } - 0 for any i, j. Further, let f = °= f = (p? + q?). We shall now define the smooth function g(t): l - R such that g(t) 1 1

when t 1< E1, g(t) = 0, t 1> eZ and g(t) > 0 is a monotone function of t when E1 < t < e2 i see Figure 21. We shall put h(p, q) = g(f (p, q))-this


64

Fig. 21.

function is now defined on the entire manifold (el , ez are chosen sufficiently small). As {h, f } = 0, it is obvious that {hf , hf } = 0 for all i, j. The functions hf are defined on the entire manifold and they are zero

outside a certain disk. By covering manifold M" with a denumerable number of disks we can now stitch the functions hf constructed above together into a set of functions k1, . . , k" on M", so that each k; is .

distinct from zero on an open subset that is everywhere dense. We claim

that hk,, ... , hk" are functionally independent. If F(hkl.... , hk") = 0 then

F(g(k1 +...+k")kl,...,g(ki +...+k")k")0 and, if function

F(g(x1 +...+ x")xl,... g(xl +.+

0

we arrive at k1,. .. , k" being functionally independent. Thus we are now left with only one thing to check, that F(g(x1 + ... + x")xl,...,g(xl + ... + x")x" # 0.

If this function is identically zero, this means that g(xl + + x")xl, ... , g(x, + + x")x" are functionally dependent. We shall write out the Jacobian of this system of functions: -

1))


65

= gn + gn - 1(x1 + ... + xn) ag ,

insofar as all invariants of the matrix X2

n

...

x2

n

apart from the trace are zero. If J - 0 we shall find that g satisfies the differential equation g + x(ag/ax) = 0, i.e. g is a linear function, which is untrue. There is one important way of making the original question bear more precisely on the problem: does a complete commutative set consisting of algebraic (rational) functions exist on any algebraic symplectic

manifold? There are probably topological obstructions that prohibit

our constructing such a set on an arbitary algebraic symplectic manifold. The most natural class of functions in which to seek complete

commutative sets on algebraic manifolds is the class of polynomial, rational or algebraic functions (see also §24). At the present time complete commutative sets have been discovered

on an important class of symplectic manifolds-on the orbits of the coadjoint representation of many Lie groups. In [88], [91] by A. T. Fomenko, A. S. Mishchenko hypothesis A has been formulated: let G be an arbitrary finite-dimensional Lie algebra, in which case there exists on the coalgebra G* a linear space of smooth functions whose restrictions

to the general position orbits of the coadjoint representation of the group 6 = Exp G on the coalgebra G* form a complete commutative set

F(O*) on these orbits 0*, i.e. any pair of functions f, g e F(O*) is in involution with respect to the canonical form o; the additive basis

fi,... , fk in F(O*) consists of functions which are functionally independent almost everywhere on 0*, and the dimension of the space F(0*) equals half that of the general position orbit, i.e. k = dim 0* _ z z(dim G - ind G). This hypothesis has been proved for all semi-simple and reductive Lie

algebras by A. T. Fomenko and A. S. Mishchenko in [89], [90] and also for many classes of non-compact real Lie algebras (see for example

[126], [127], [10], [134]). The complete commutative sets that were

66


discovered in verifying this hypothesis turned out to contain the Hamiltonians of important mechanical systems, which makes it possible to integrate them fully. We shall dwell on this in greater detail below.

The significance of hypothesis A is not limited to the opportunity to produce integrable systems. Its validity would lead to that of Theorem 3.3 not only for compact but for non-compact manifolds as well, i.e. in

this case any Hamiltonian system which is integrable in a noncommutative sense would automatically be integrable in a commutative sense too. To be more precise, the following argument is valid, as we proved in Section 3.6. Let a Lie algebra G of functionally independent functions, where dim G + ind G = dim M (see Theorem 3.2) be given on

a symplectic manifold M; then, if hypothesis A has been tested for algebra G, another commutative algebra of independent functions of G'

will now be found, so that dim G' = dim M. We now go back to

i

systems v, for which an embedding into a finite-dimensional Lie algebra G exists (see Definition 4.6). If hypothesis A is valid for algebra G then

there is a complete commutative set of functions on orbits in general position (see above) 0* c G*. Furthermore, if the Hamiltonian f, where

v = s grad f, belongs to the family F(O*), we then get a complete commutative set of integrals for v. Knowing the complete commutative

sets on orbits in G* allows us in principle to integrate Hamiltonian systems embedded in G*. One of the ways of integrating systems on Ii" is thus the following : (a) we try to represent the system as Hamiltonian on orbits in G* given an appropriate choice of algebra G; (b) if there exists such an embedding of the system we try to find a complete commutative set of functions, that contains the system's Hamiltonian, on 0* c G*. There are a number of methods that allow us to construct complete commutative sets on the orbits. One extremely effective means of doing

so has turned out to be a method based on the idea of shifting the invariants of the coadjoint representation along a covector in general position. REMARK

At the present time there is a great deal of active work

carrying on PoincarFs classic researches on the questions of the non-

existence of supplementary first integrals. On this topic see op. [59], [60].

2

Sectional operators and their applications

6. SECTIONAL OPERATORS, FINITE-DIMENSIONAL REPRESENTATIONS, DYNAMIC SYSTEMS ON THE ORBITS OF REPRESENTATION

As we have shown in Section 4, it is possible to construct a system of non-linear differential equations corresponding to a Lie algebra G and a linear operator C: G* -+ G on G*. These equations represent wellknown mechanical systems as special cases of the construction. Besides, we can apply to this system, viz. x = adc*(X)(x) the analog of HamiltonJacobi theory (demonstrating the fact that this system is based on group theory). The system of equations constructed is a Hamiltonian one on

all orbits of the coadjoint representation. If we do not have any conditions on the operator C, we cannot say anything about complete integrability of the equations x = ad*c(X)(x). For complete integrability of

the system, we need a special construction for operators C. We shall describe this in the present section. For a Lie algebra so(n) we can obtain

an invariant description (i.e. in terms of the Lie algebra so(n)) of the

"rigid body" operators cp(x) = XI + IX. As an example, let us construct the "rigid body" operators for some semi-simple Lie algebras first mentioned by A. T. Fomenko, A. S. Mishchenko in [89]. In [89] the use of the root expansion G = H Q V+ G) V- and of invertibility of the operator ada on the space V+ O V- was essential. If a Lie algebra is non-compact, there are not any natural analogs of the root expansion. Therefore, we need to introduce some new ideas which would enable us to include the case of non-compact Lie algebras. It turns out that in the non-compact case there are analogs of "rigid body" operators ipa,b,D (see [32, 33]). Let us consider them. The following general idea of "sectional operator" was introduced by A. T. Fomenko. 67

68


Let H be a Lie algebra, .5 be the corresponding group; p : H - End(V) be a representation of H in the linear space V; a: Sa -+ Aut(V) be the corresponding representation of the group; let O(X) denote the orbits of the action of the group S on V, X e V. If we introduce the linear operator

which we shall call a "sectional operator" Q : H - V, the vector field

go = p(QX)X will arise on the orbits. Note that the vector field 9 = p(Q(X))(X) is defined for any smooth mapping. However, here we shall not consider such a general case because for all applications Q is

taken to be a linear operator. Also, having defined such opertor, we sometimes can define a symplectic structure on the orbits. The special class of those sectional operators which form a many-parameter set with

two basic parameters a e V, b e Ker 0a; &h = (ph)a is important for applications. For example, in the particular case H = so(n), p = ad the field )to coincides with the equations of motion of a multi-dimensional rigid body with a fixed point (in the absence of gravity). Thus, let a be any point in general position, i.e. the orbit corresponding to a has the

maximal dimension. Let K c H be the annihilator of the element a, K = Ker Oa, ¢q being defined above. If a is a point in general position, then the dimension of K is minimal. Let b e K be an arbitrary element. Let us consider the action of pb on V. Let us denote Ker(pb) V by M.

Let K' be any algebraic complement to K in H, i.e. H = K + K', K n K' = 0. The choice of K' is not unique and hence, the set of parameters in the construction follows from the possibility of varying

this complement. It is clear that a c M. From the definition of K' it follows that the mapping 0a: H -- V transforms K' into some plane (pa(K') c V monomorphically. Since OaK' = 4aH, the plane &&K' does not depend on the choice of K', being defined uniquely by the choice of the element a and the representation p.

Let us suppose that there exists an element b such that V can be expanded as the sum of two subspaces M and Im(pb), i.e. V = M Q Im(pb). For example, we can take the semi-simple elements in K as b. Let us denote the planes which are

formed by the intersection of (.K' with M and Im(pb) by B and R' respectively. Thus, we have obtained a decomposition of O .K' as a direct sum of three subspaces B + R' + P, B and R' being uniquely defined. At the same time, the complementary subspace P can be chosen in several ways, introducing a new set of parameters. Let us consider the action of pb on Im(pb). The pb maps Im(pb) into itself isomorphically

(see Figure 22 where pb is invertible on Im(pb)). Let (pb)-' be the


69

Fig. 22.

operator which is inverse to pb on Im(pb). Assuming R = (pb) -'R' we obtain Pb: R -+ R'. The space R is uniquely defined. Let us consider in Im(pb) the space Z which is the algebraic complement of R on Im(pb). Then Im(pb) = Z + R'. Let T be the complement of B in M. Thus we have constructed a decomposition of the space V as a direct sum of four planes V = T + B + R + Z; R, B, M, Im(pb) being uniquely defined, whereas the choice of Z and T is ambiguous and therefore introduces a new set of parameters. Given a scalar product in V, Z and Tare uniquely defined as orthogonal complements. Since K' is isomorphic to 4aK',

K'=$+A

1B,R=Oa'R,P=Oa'P.

Thus, we have defined a many-parameter decomposition of the algebra H as a direct sum of four subspaces K + B + R + P. Let us define the sectional operator Q : V -> H, Q : T + B + R + Z -

K+P+I +P by setting D 0

0 0 a1

0

0

0

0

0

0

ga1p(b)

0

0

0

0

D'

B - B being the operator D: T - K being any linear operator, inverse to 46, on the subspace B, /a 1p(b):R - R, p(b): R - R', Oa 1: R' -+ A, D': Z -+ P (see Figure 22). Thus the operator Q is of the form Q(a, b, D, D'). Let us construct the dynamic system X"Q = p(QX)X, X e V. We choose a as the point in general position in V because in this case the dimension of K' is maximal, i.e. the operators to 'p (b) and 0. 1

70


are defined in a maximal space. Let us note some important examples of the construction defined.

If V = H*, p = ad*: H --,, End(H*), then 0a -'p(b) _ a 1 adb . For example, taking as H the non-compact algebra so(n) Q R" which is the Lie algebra of the group of motions in the Euclidean space, we find (see below) that the system IQ = adQ)X)(X) transforms into the equations of

inertial motion of a rigid body in an ideal fluid. Given the obvious identifications of H and H*, we obtain K = K*, Z = 2 = 0, R = R. Let (5/.5 be a compact symmetric space. Then, the Lie algebra G may be expanded as a sum of spaces H + V, H being a stationary subalgebra; V

being a subspace tangent to 6/.5; the subalgebra H acting on V coadjointly. Then, the expansion V = T + B + R + Z which defines

the sectional operator, is such that T is a maximal commutative subspace in V, a e T, R = R', Z = 0, b E K, qaK' + T = V = T + B + R. Provided that C: V - H is a sectional operator, we obtain , q) = B(CX, e T,T0 on the orbits

the exterior 2-form FC(X,

0(X)cV. There are wide ranges of symmetric spaces and sectional operators for which this form defines (almost everywhere on the orbit) a symplectic structure which is non-invariant to the action of the group. Let us return to the general case. Let , q e Tx 0 be tangent vectors. Then, the vectors e K'(X) such

that pi;' = , pit = q exist and are defined uniquely. Let us take the sectional operator C: V -+ H. Let us define the bilinear form Fc = B(CX, [c', q']), [ ', q] a H, CX e H. This form is defined on the orbits and is skew-symmetric. Also, we have defined the flow x"Q on the orbits.

PROBLEM A For what operators C is the form FI. closed and nondegenerate on the orbits? PROBLEM B For what C and Q is the flow XQ a Hamiltonian one with respect to the form Fc?

It turns out that in the case of symmetric spaces there are complete answers to these questions (A. T. Fomenko).

For example, let us consider the symmetric space SU(3)/SO(3). Then, the equations IQ on the plane B coincide with the Euler equations of motion for a three-dimensional rigid body with a fixed point and arbitrary inertia tensor. Note that among the systems IQ = adQ)X) X

which we have constructed, there are the Hamiltonian equations of


71

motion of a rigid body with a fixed point (for any n, not only for n = 3). To confirm this, it is enough to take the semi-simple group Sa as a

symmetric space. Then, it can be written in the form Sa x sj/15; the involution a:.5 x .5 -+ .5 x Sa is defined as a(x, y) _ (y, x). The corresponding expansion of the Lie algebra G = H + H is of the form V = (X, -X),XEH;H = (X,X),X E H (both H and its image in G are denoted by the same letter), aV = - V, aH = H. It is easy to verify that the form constructed above is transformed into a canonical symplectic structure on the orbits of the coadjoint representation; the field 9Q (D' = 0) being transformed into the "rigid body" equations which we wanted to obtain. Thus, we have found a "multi-dimensional" series of dynamic systems which contains the equations under study. They are of interest to us due to the fact that they are also defined on non-compact Lie algebras, being at the same time the natural analogs of "rigid body"like systems.

7. EXAMPLES OF SECTIONAL OPERATORS 7.1. Equations of motion of a multi-dimensional rigid body with a fixed point and their analogs on semi-simple Lie algebras. The complex semi-simple series

Let G be a semi-simple Lie algebra, B(X, Y) be the Cartan-Killing form,

f be a smooth function on G. Let us associate with this function a dynamic system on the cotangent bundle T*(5 of the group by extending f to a left-invariant function F defined on the whole space T*(fi. Since T*6 is a symplectic manifold, we can obtain a Hamiltonian system on T*6 taking F as the Hamiltonian. This system is left-invariant and can be divided into two systems, one of which is defined on the cotangent space at the identity element of the group which is isomorphic to the Lie

algebra G and is usually called a system of Euler equations. These equations can be described simply. Let grad f be the field on G which is

dual to the differential df, i.e. B(grad f, ) = (f ). Then, the Euler equations can be written in the commutator form I = [X, grad f(X)]. The case of geodesic flows for the left-invariant metrics on 6 is of particular interest. Here, f is a non-degenerate form on the algebra G; grad f(X) is defined by a linear operator in G, i.e. grad f(X) is of the form cpX; cp: G -+ G being self-conjugate.

72


Let G = so(n) be the Lie algebra of an orthogonal group, let the diagonal real matrix 0

A.#2j,

I=

i0j.

Let us consider on so(n) the operator O(X) = IX + X1. Then, we call

the equations i/iX = [X, >/iX] the equations of motion of an ndimensional rigid body. Let us write them in an explicit form using standard coordinates in so(n). Let us represent so(n) as the algebra of skew-symmetric matrices (real-valued) X = (xi,) then i(X) _ ((A, + 2)x1 ). It is clear that

_ A; - 2; + 2iI 4i

xjq xqj.

Assumingyn = 3 we can obtain

2 + Al X 13X 32

A 12

1

X 23

433

X13

_ 23-2j 3

+ % X12X23, 1

+ A22 X21 X13

As we have explained above, so(3) can be identified with l . Then, these

equations coincide with the classical equations of motion of a 3dimensional body (see above). It is for this reason that we have called the equations 1yX = [X, OX] (for any n) the equations of motion of a multi-

dimensional rigid body. Let us choose I in such a way, that 2; + 2; # 0 for any i, j. Then, the operator >' is invertible on so(n) and the inverse

operator 0

1

= cp

is of the form (pX);; = [1/(2; + A;)]Xj. Let us

substitute the coordinates Y = OX in 09 = [X, OX] then, the equations of motion of rigid body are transformed into Y = [0 -'Y, Y]

Multiplying it by (-1) and denoting Y by X, we obtain the Euler equations

X = [X, cpX],

cp: so (n) - so (n)

being a linear self-conjugate operator. In what follows, we shall consider

namely this form of the equations. In coordinate notation, we have


73

X. X A

q[=.,1 (A

Y_ xigxgj L

9=1

+ Aq)(2q + Ad

1

1

.+q

Aj + I,q

Above, having assumed n = 3 we obtained the embedding of this system in the Lie algebra so(3). A similar embedding exists for any n. 1, vector field I = [X, (pX], (p = OX = IX + XI is tangent to the orbits of the coadjoint representation of the group so(n) on its Lie algebra so(n). This field is Hamiltonian on all orbits.

PROPOSITION 7.1

The

Proof Let 0 be an orbit, the point X a so(n) = so(n)* belonging to the orbit. Then, TXO = {[X, Y], Y E so(n)} therefore [X, (pX] a TXO. We have X = s grad F, F(X) = <X, (pX>. This completes the proof. Let us define the analogs of the equations of motion of a rigid body on an arbitrary semi-simple Lie algebra. We have a many-parameter set of

the operators (p: G -+ G not only for the complex semi-simple Lie algebras but also for their real compact and normal forms. Thus, all systems X' = [X, pX] are completely integrable on the orbits in general position and hence their integrals define complete commutative sets of functions on both semi-simple Lie algebras and on their real forms. Let G be a complex semi-simple Lie algebra; G = T Q V+ Q V- be the root expansion of the algebra. Let a, b a T, a : b be two arbitrary regular elements of the Cartan subalgebra. Let us consider the operator ada: G - G. It is clear that adalT =_ 0, ad.: V' -+ V', ad.: V- -+ V-, i.e.

this operator preserves the root expansion of G over C. Indeed, ado E. = a(a)E0 for any a a A. We assume that a and b are in the "general position," i.e. a(a) 0, a(b) 0. Then, the operators ad, and adb are

invertible on V+ Q V- = V, namely, ada 1 Ea = Ea/a(a). Let us define the linear operator (pa,b,D: G - G as follows (according to the general rules in Section 7): (pa,b,D(X) = cp,,,,X' + D(t) = ad-' adb X' + D(t),

X = X' + t being the uniquely defined extension of X to V and T; D: T -, T being any linear operator symmetric with respect to the Killing form on T. The parameters a, b and D belong to the operator (Pa,b.D It is clear that (pa,b,DE= = [a(b)/a(a)]E0. In the Weyl basis (Ea, E_ H,) the operator 9 is defined by the matrix


74

= Y'a,b,D

Aq

ab. V -' V;

a(b) na = a(a) ; 7

q = dim V± = (the number of the roots a > 0). The operator cpa,b,D is symmetric with respect to the Killing form if a, b, D satisfy the conditions given above. PROPOSITION 7.2

Proof Let us denote the Weyl basis in V by (ei). It is enough to verify that B(q ei, e) = B(ei, epee) for any i, j. We can assume i 0 j. Remember

that the plane T is orthogonal to the plane V = V+ Q V-. As W transforms V and T into itself and D is symmetric on T, it is enough to check that pa,b is symmetric on V. Since Ea (a 0) are the eigenvectors of (P, BI a(b) Ea, Ef I = BI Ez, fi(b) EP)

=0

(if a + #

0),

B(E,,, Eft 0.

Ifa+/3= 0, then _ (-a)(b) a(a) (-a)(a) a(b)

This completes the proof.

In the "general position" case, the operator qp on V has q distinct eigenvalues which are multiples of two. The operator gyp: V - V is an

isomorphism of V with itself. Remember that V + is a nilpotent subalgebra. Since V + is generated by the vectors E a > 0, cpl " is symmetric with respect to the Cartan-Killing form B(X, Y). The eigenvalues of this operator in the "general position" case are distinct:


75

AI, ... , AV We shall call this series the normal nilpotent series of the

operator cp: V+ - V. According to our construction, each complex

series corresponds to one normal nilpotent series. The operator pp: G - G maps the subalgebra V+ ® T into itself, (pi v' ®T being an isomorphism of the space with itself. All eigenvalues of the operator (pi v. ®T are distinct and the operator is symmetric. This series of the operators is called a normal solvable series. In the Weyl basis, the operators q , and Cpl v' ®T are expressed by the matrices

(PI V+ EDT =

0

Thus, we have constructed Hamiltonian systems X = [X, cpa b p(X)] for each semi-simple Lie algebra (A. T. Fomenko, A. S. Mishchenko). These systems are analogs of the equations of motion of a rigid body and

are completely integrable (see below). In particular, we obtain the equations on the Lie algebra so(n) (S. V. Manakov).

7.2. Hamiltonian systems of the compact and the normal series

We shall consider the set of Hamiltonian systems on the arbitrary compact real Lie algebra using real forms of the complex simple Lie algebras. Every complex semi-simple Lie algebra G has the compact form Ga (see e.g. [47, 50]). Remember that

{E,+E_ , i(E,-E_,), iHQ} = W+Q+iT0. As in the previous section, we define the symmetric operator cp: G. -p G,'

which defines the Hamiltonian system X = [X, cpX] on G. which

of G. by the orbits of the coadjoint representation. Let a, b c iTo be elements in general position. Since preserves the foliation

ada E, = [a, E,] = i [a', E,]

(a = ia', a' E To),

ada E, = ia(a')Ea, a(a') being real. Hence,

76


ada(E,, + E_a) = a(a')(i(E, - E_a)),

ada(i(Ea - E_a)) = -a(a')(E2 + E_,). Thus, the operator ada: W+

W+ rotates the vector E. + E_2 into a

vector which is proportional to i(Ea - E_j and vice-versa. The operator adb acts similarly; it is invertible on W+ due to the choice of a e iTo. Then, all vectors Ea + E_a, i(E8 - E_a) are eigenvectors of the operator Pa,b = ada 1 adb: W+ -. W+ with the eigenvalues a(b)/a(a) _ a(b')/a(a'), a = ia', b = ib', a', b' e To. Similarly for the operators on the subspace W-. Let us define the operator tpa,b,D: G. - G. as follows: tpX = ip(X' + t) = tpa,b(X') + D(t) = ada 1 adb X' + D(t), X = X' + t being the uniquely defined extension of X to Ga = W+ $ iTo, Y e W+,

t E iTo; D: iTo - iTo being an arbitrary linear operator which is symmetric on iTo. In the basis ((E8 + E_a), i(E,, - E_8), iH') the 2

operator tp is defined by the matrix

/11

0

01

Ea+E_a 0

A9

A1

0

Y'a,b,D,

i(E2 - E_a) iH,

\

0

0

A

D/

a = a(b)/a(a) being real; q = dim W. PROPOSITION 7.3

The operator tp : G. - G. is symmetric if a, b, D

satisfy the conditions given above.

Proof The arguments are similar to the proof of Statement 7.2. The only point we need to check is the orthogonality of the basis chosen in

W. Remember that iTo is orthogonal to W. Then, we have B(Ea + E_., i(E, - E_2)) = 0. The orthogonality of the rest of the vectors is known. In the general position case, the operator tpa,b: W + - W + has distinct eigenvalues which are multiples of two. Let us construct a similar set of Hamiltonian systems on some simple

compact real Lie algebras which correspond to the classical normal compact subalgebras in the complex semi-simple Lie algebras. In any

77


compact form let us consider the subalgebra G which is called a normal

compact subalgebra. This subalgebra is generated by the vectors E, + E_ a c A. Since all these vectors are eigenvectors corresponding to the operator cp of the compact series, we get the normal series if the operators are restricted by the subalgebra G,.. These operators coincide with (p,b: G. - G., cpX = ado ' adb X, X e G,; a, b e iT0, a(a) # 0, a(b) # 0. In the basis (E, + E_,) the operators (p are defined by the matrices

1

0

(Pb=

,

q=dimW+.

0

Note that here a, b f G,,, i.e. to define the operators of the normal series we need the elements of some extended algebra. This is the difference between the normal series and the complex and compact ones, in the

latter case the elements a and b belong to the algebra itself. Not any compact semi-simple algebra can be represented as G. in some compact real form G. c G. A complete list of all these simple Lie algebras is given

below. The algebra

G,,

coincides with the fixed points of the

automorphism z : G -- G,,rX = X when the latter is restricted to G.. Let P c G be a subspace which is orthogonal to G in G,,, r = - 1. Then, the following commutativity relations are evidently valid: [G., G.] C [P, P] c G,,, [G., P] c P. Thus, the symmetric space 6 /6n is defined. Then, the space P is identified with the tangent space of the latter is embedded canonically in 6 as the Cartan model (see [48, 68]). Let us write down all the normal forms using the standard notations for the corresponding symmetric spaces (see [48]).

FORM Al G = sl(n, C), G. = su(n), G = so(n), aX = X, n > 1. The algebra G. is given in G. as the subalgebra of real skew-symmetric matrices.

FORM BDI G = so(p + q, C), so(p, q) is the Lie algebra of the component of the unit of the group SO(p, q). The algebra so(p, q) is X1 XZI, realized in sl(p + q, l8) by the matrices (X2 all X; being real, X3

X1, X 3 being skew-symmetric with the order p and q, X2 being arbitrary. Then, G. = so(p + q) D so(p) Q so(q),

p > 1 ,

q > 1,

p + q # 4.

78


The normal forms correspond to the following values p = q and p = q + 1, i.e. G. = so(p) ® so(p) and G. = so(p) $ so(p + 1). FORM CI G = sp(n, C), n > 1, sp(n, l8) is the algebra \Xs

XX i

)x1

being real with order n, X2 and X3 being symmetric. Then, G. = sp(n), G. = u(n), the embedding G. - G. is given as follows:

A + iB -.

( B B), A + iB E u(n), A and B being real.

These are all the normal forms G. c G,, where G. is a classical simple Lie algebra, i.e. one of the forms A, B,., C,,, D,,. Apart from these forms, there are also several normal forms which are generated by the special Lie algebras (these we omit). In conclusion, we show that among the Hamiltonian systems of the

normal series there are the classical equations of motion of a multidimensional rigid body with a fixed point (see 7.1). Let us consider the algebra so(n) which we represent as the normal form in the algebra su(n) (see above). Let us embed su(n) in u(n) in a standard way and consider

two regular elements a, b of the Cartan subalgebra iTo in u(n) (not in su(n)!). Let ial

0

0

iaa

ibl

0

a=

= diag(ial, ... , ia ),

=

b = 0

a,, b; E l8; a,

±a; b,

ib,,

±bl (i

j). Then, the operator <pa,,: G,,

G

acts as follows: a(b)

(Pab(E,, + E-.,) =

a(a)

(E, + E-a).

Since each root a is given by a pair of indices (i, j), i.e. a = a;, (see e.g. [11]), each eigenvector E, + E which corresponds to the pair (i, j), is multiplied by the eigenvalue ti,j _ (b, - b;)/(a, - a;). Thus, when cp acts, the basis skew-symmetric matrices

79


0

1

-1

0

Eij=Tj - Tji= are multiplied by ),,j. Therefore, the Hamiltonian system X = [X, c,X] is of the form

_

"

Xij = L9=1 EXigX4j(Agj - Ai9) =

b4 - bj Xi4X4j

fl=1

aq - aj

bi - b9 ai - aq

Let a = -ibZ, i.e. ap = bp. Hence, "

1

Xij = E xigxgj (aj -+a., 9=1

1

- ai + aq

Then, when a = ibZ we obtain the system of equations of motion of a rigid body with a fixed point which is already familiar to us (see 7.1). Moreover, among the operators cpab of the normal series there is the

classical operator iliX = IX + XI, I being a real diagonal matrix. Indeed, assuming b = -ia2, we obtain b; (PabEij =

0 = (Pa,_ia2;

ai -

bj

Eij = (ai + aj)Eij,

J.

(pabX = IX + XI

(I = -ia).

Thus, we have included the classical Hamiltonian system of the motion of rigid body with a fixed point (without potential) in a many-parameter set of similar Hamiltonian systems which are naturally defined on the simple compact Lie algebras. 7.3. Equations of inertial motion of a multi-dimensional rigid body in an ideal fluid

We shall give explicitly an embedding of such equations in the noncompact Lie algebras of the group of motion of Euclidean space. Then, the system will be a Hamiltonian one on the orbits in general position and will be completely integrable (see works of the authors [123], [124], [125]). We will construct completely integrable linear operators using the general construction of the sectional operators from Section 6. Let '(n) be the Lie group of rigid motions of I?'. It is known that B(n)

80


is the semi-direct product of the group SO(n) and the commutative subgroup T of translations, the latter being a normal divisor in the group d° (n) and isomorphic to n-dimensional Euclidean space. A matrix representation of the group 41(n) is of the form

(x1,...,x")EI.

The action of the group SO(n) on the normal divisor 1 coincides with the standard action of SO(n) in R". Hence, the Lie algebra E(n) of the group d(n) is a semi-direct sum so(n) ®, III", cp: so(n) - End l" is the differential of the standard representation of SO(n) in III"; l8" is regarded

as a commutative subalgebra. In a matrix notation E(n) is of the form

0

0

The commutation operation in the Lie algebra E(n) is of the form

[x + , Y + q] = Cx, YA + x(q) - Y( ), being the results of the action of the matrices x and y on the x(q), vectors q and respectively; so(n) acting on 1B" in the standard way. Let us identify the space E(n)* which is dual to E(n) with E(n). Let us define a non-degenerate scalar product in E(n) (non-invariant). We have E(n) = so(n) Q R" as linear spaces. Let B(X, Y) be the Killing form on so(n); (X, Y)e be the Euclidean scalar product in U8". We can assume ((x1,YI),(x2,Y2)) = B(xl,x2) + (Y1,Y2)e,

xl , X2 e so(n), y1 , Y2 E R". Let us regard all subspaces in E(n)* as subspaces of E(n) using the canonical identification which was given above.

Let us calculate explicitly the form of the operation ad* under the isomorphism E(n)* = E(n). Remember that we denote ad* (x), e G,

x x G* as a(x, ).


PROPOSITION 7.4

Let

E so(n),

x c: li',

81

S E so(n)* = so(n),

ME(P")* = R",

+xeso(n)6),R"=E(n), S+MeE(n)*=(so(n)(DR")*=so(n)g I". Then,

a(S + M, + x)l,o(") = [S, f] + 2' (Mx' - xM`), a(S + M, c + x)JRn = -CM. M, a e 11" are expressed as column vectors, (t) denoting transposition. The proof follows directly from the definition of the operation a(x, g). Let G be an arbitrary Lie algebra. Remember that the Euler equations on G* are the system of differential equations x = a(x, C(x)), C: G* -+ G

being a linear operator, a(x, ) being the linear functional described above. The flow z = a(x, C(x)) flows along the orbits of the coadjoint representation Ad* of the group 0. On these orbits, the Euler equations

are the Hamiltonian ones with respect to the canonical symplectic structure. Let us apply the above construction to the construction of sectional

operators for the coadjoint representation of the algebra E(n). We obtain a many-parameter set of sectional operators Q: E(n)* - E(n) for

which the Euler equations are completely integrable Hamiltonian systems on the orbits of the coadjoint representation of the group f(n). Let us consider in E(n)* (and hence, in E(n)) the subspace

K=

[(n+1)/21-2

O

l(E2k+1,2k+2) O Ire" C E(n),

k=0

E,, being elementary skew-symmetric matrices, e; being the standard orthonormal basis in R". The corresponding subspace in E(n)* we denote by K*. PROPOSITION 7.5

The subspaces K and K* allow the invariant

description K = Ann(x1) = { E G, a(x, ) = 0}, G = E(n) and K* = E G*,

x2) = 0; xI E G* and X2 E K

G being elements in general

position.

The proof follows from Proposition 7.4. Let us denote the orthogonal complement to the subspace W in E(n) or E(n)* with respect to the scalar product B(X, Y) + (Z, R)e by W.

82


LEMMA 7.1

x

Let a E K*. Let us consider the mapping 0.: E(n) --+ E(n)*,

x) E E(n)*. Then, &I( 1 c K#1, K1 being the orthogonal

complement to K, K*1 being the complement to the subspace K*. The proof is obvious. If a is in general position, then K = Ker ¢a and

0a: K1 -- K*1 is an isomorphism. Therefore, the inverse mapping K*1 - K1 is defined. We have the splitting as a direct sum of linear spaces E(n) = K1 p K and E(n)* = K*1 ® K1. According to the

general method, let a e K*, b c- K, a being in general position. If z = x + y e E(n)*, x e K#1, Y E K*, then Q(a, b, D)z = 0o I adb* x + D(y); D: K* - K being arbitrary.

Now, we can write down the general equations XQ = ad*,(X) on G* = (so(n) @ D8")* = so(n) $ p", Q(a, b, D) being the sectional operator which we have constructed and which is a non-compact analog of the operators cpa,b,D which describe the motion of a rigid body. In our case Q(a, b, D): E(n)* -+ E(n). Therefore, XQ is defined on the space G*. The equations XQ = ad**(x)(X) can be written explicitly as

S` = IS, f] +

2(Mx' - xm')

11%M =

(1)

E so(n), x e R" being functions of the elements S, M, their dependence being defined by the operator Q(a, b, D), i.e. + x = Q(a, b, D)(S + M). Here S + M E E(n)*, + x c- E(n). Since the operator Q(a, b, D) is

defined explicitly, it is easy to obtain an explicit expression for the dependence of

and x on S, M, a, b, D.

PROPOSITION 7.6 (Fomenko, A. T.; Trofimov, V. V.) The system of differential equations ii = ad,*,(a,b,D)x(X) on the coalgebra E(n)* (written

explicitly as the system (1)) is a Hamiltonian one on the orbits of the coadjoint representation Ad*('(n)). Moreover, when n = 3, the system coincides with the equations of inertial motion of a rigid body in

an ideal fluid. Thus, this system allows an embedding in the noncompact Lie algebra E(n) in the sense of Definition 4.6.

Therefore, we can say that the equations (1) describe the inertial motion of a multi-dimensional rigid body in an ideal fluid for any n. Before we prove Proposition 7.6, let us remember the classical equations

of inertial motion of a 3-dimensional rigid body in an ideal fluid (for details, see e.g. [39]). Let us introduce a moving frame of reference. Let ni be the components of the linear velocity of origin a moving frame and


83

w, be the components of angular velocity of rotation of rigid body. Then, the kinetic energy of the rigid body and fluid system is of the form T = 2(Atl(otwf + B1Juiuj) + Cuuw1uj,

AlJ, B.J, C1J being constants which depend on body form and density of body and fluid; the summation from 1 to 3 done over subscripts which occur twice. For example, let the body have three mutually orthogonal axes of symmetry. Then, it is easy to check that all factors A1J which have

i not equal to j are equal to zero and T = i(211uz + 222Uy + A33U2 + Aaawz + )55wr + A66wz)

For a sphere with radius b: All = 21rpb3/3, A22 = A33 = 27rpb3/3, A44 = 1.55 = A66 = 0. Thus, in this case 3

T=-3b (uX+uy+uZ). Let N = (yl, y2, y3); y1 = OT/8w1, K = (x1, x2, x3), x; = 8T/8u,,. Then, inertial motion of a rigid body in an ideal fluid can be described by the equations dN

dt =Nxw+Kx U dK

dt =

(2)

K x c o,

U = (U1, U2, U3),

w = (w1, w2, (0 3)

The kinetic energy of a rigid body is an arbitrary positive definite homogenous quadratic form in six variables u1i w1. Consequently, it is defined by the 21 factors (A,J, B1J, C.J) = (A1J). The equations (2) have three classical Kirchhoff integrals in the general case. Therefore, for complete integrability of the equations (2) to be attained, we need one

additional integral, functionally independent from them. Let us remember the classical Kirchhoff integrals. Taking the scalar product of the first equation in (2) by K, we obtain K(8K/8t) = 0. Consequently,

const = K2 = KX + Ky + K. Taking the scalar product of the second

equation by N and of the first by K and adding them, we obtain N(8K/et) + K(8N/et) = 0. Therefore, NK = KXN., + KYNY + K=Nz = const. Evidently, the third integral is the energy integral T = const. Indeed, taking the scalar product of the


84

second equation by U and of the first by co and adding them, we obtain

U a +w

a =0.

(3)

On the other hand, T is a quadratic form in ux, uy, uZ, o.

wy, wZ.

Therefore, according to Euler's theorem of homogeneous functions we have

aT OT aT aT aT aT 2T=ux-+uy+uZ+wz +wy awy -+wZaux NY

auZ

aWZ

awx

= UK + wN.

.

Differentiating this equation with respect to t and using (3) we obtain 2

di =K

at

We have dT

OT auz

aT auy

dt - aux at + NY at + aT awX

OT auZ

auZ at

OT awy

OT awZ

+ awx at + awy at + awZ at

=K Thus, 2(dT/dt) = (dT/dt). Hence, dT/dt = 0, which completes the proof.

Since the dimension of orbits in general position of the coadjoint representation of the Lie algebra ca(n) is equal to 4, four independent integrals (two of them define the orbit and two are not constant on the orbits) are enough for complete integrability of the system. Recall the three classical cases when this fourth additional integral is available (for a complete review see e.g. [39]). The first general solution of the equations of motion of a rigid body in a fluid was obtained for the function T (kinetic energy), using which it was possible to obtain the fourth integral, given the first three. Halfen suggested a solution of the first case of Clebsch, i.e. when the integral is, generally speaking, a linear homogeneous function of the variables x;, y;. In 1978 Weber studied the second case of Clebsch, i.e. when the fourth

integral is a homogeneous quadratic function of x,, y;, with some

85


particular conditions on the arbitrary constants. Ketter obtained the general solution of the latter problem. V. A. Steklov discovered the third kind of kinetic energy with equations (2) integrable explicitly.

1. First case of Clebsch be expressed as

Using variables xi, yi, the kinetic energy can

T = 2b11(xl + x2) + 2b33x3 + b14(x1Y1 + x2Y2) + b36x3Y3 2

+ 2b44(Yi + Yi) +

2b66Y3

The equations of motion (2) allow the fourth linear integral with the variables x; yi. With this energy, the body does not change its form

rotated around the z-axis through an angle of n/2. Assuming b14 = b36 = 0, we have a rotating body. 2. Second case of Clebsch energy

The equations of motion with the kinetic

T = b11xi + b22xi + b33x3 + b44Yi + bssyi + b66 Y3

and the expression for the coefficients b22

- b33 + b33 - b11 + b11 - b22 = 0 b55

b44

b66

allow the fourth integral which is a homogeneous quadratic function. The rigid body considered is symmetric with respect to three mutually orthogonal planes. 3. Case of Steklov of the form

2T =

The equations (2) are defined by the kinetic energy b11xI + 20

bssb66x1 Y1 + Y b44Y21

bii being defined by the relations

bll =

0,2 b44(b52 5

2

+ b66),

2

b22 = Orbss(b66 +

b42

4),

b33 = Q2b66(b44 + bss)

Apart from the three classical integrals, these equations allow the natural homogeneous integral of the second order with respect to xi, yi. Here, or is an arbitrary constant, the symbol E means summation over the three expressions obtained by cyclic permutations of the groups of subscripts 1, 2, 3; 4, 5, 6.

86


Let us prove Proposition 7.6. LEMMA 7.2

Let us consider the mapping >t': so(3) -* l83 which maps

the element X = xE12 + yE13 + zE23 to the point (z, - y, x). Then, fi[X, Y] = ->y(X) x ci(Y), qi(X) x >y(Y) being the vector product in O. Further,

fi(M)e - xM`) = M x x,

M=-

M, x E 111;3,

x M,

e so(3).

Proof If Z = + x e E(3), Z = S + M e E(3)* then, as shown in Proposition 7.4, a(Z, z) = (y, X);

X=

Y = [S, f] + Z(MX' - xM`) a so(3);

The vectors M and x being written as column vectors, the Lie algebra so(3) is given by the skew-symmetric matrices. This completes the proof.

Let us write down the operators Q(a, b, D) constructed above for the simplest three-dimensional case when E(3) = so(3) @ R3 (in this case many higher-dimensional effects are absent which facilitates writing in an explicit form).

K = K* _

K1=K*1=

0

a1

0

0

-al

0

0

0

0

0

0

2

/0

0

x2

ul

0

0

X3

u2

-X3

0

-X

2

Let

f=

0

0

f2

ul

0

0

f3

u2

0

0

-f2 -f3

)eK*`.

Then (-a, U2

0

0

- a1f3 + 2u1a2

0

0

a1f2 + 2u2a2

a1u1

0

0

a1f3 - 2u1a2 - f_a1 - 2u2a2


87

Let 0

b,

0

0

0®0

0

0

b= (_b1 0

0

0

X2

0

0

x3 ® y2

-x2

-x3

0

x=

,

bz

Then

b1x3 -j'bzy,

0

0

0

0

x36, - jb2 v1

-x2b, - jb2.v2

NX) =

.

0

0

-b2x2

-b1x2 - 3fb2y2 ® -b2x3 0

0

Therefore 0

0

4e ' adq

0

0

f2

0

0

f3 ® U2

z,

U,

=

0

0

-b

u, 1

z2

2

0

f

f2

0

0

-a1

a1 U2

'2

b2

1=

2a1 f3 + u1

z2 =

-2

-b2a2 2 2b,a,

62

bz

0

U1

b2

f3 - u2

b2a2 + 2b,a1 bzz

Finally Q(a, b, D)

0

f,

f2

-fi

0

f3 ® uz

-f2 -f3

0

U1

U3 a,

0

aft + flu3

a, 62

62

a1 U2

62

2 b2

f3 - u,

bZ

b2a2 + 2b1a1 -, u1 ® -2 a,fa22-

0

U3

b2a2 + 261a1,

a,

62 u2

U2

0/

62

b2

\

2

}'f1 + bu3

a, fi, y, b being the constants which define the operator D: K*

K. The

kinetic energy is of the form <X,Q(a,b,D)(X)). The matrix of this quadratic form is as follows

88


I -2a

0

0

0

0

0

Y

0

A

2

2a 1

0

0

0 b2

0

0

a,

0

al

0

0

0 v

-bzaz + 2b1a1

al bz

z bz

-2a1

al

0

b2

b2

2-P

0

0

bz

b2

0

0

- bzaz + 2b1a1

0 b2

0

0

0

S

l

It is clear that 1z

det A=

-a4b-4

r(T - f3J + 26a) ,

i.e. we can change the sign of det Abby the appropriate variation of e.g. the operator D. Writing A = bz z(bzaz + 2b1a1) we obtain the diagonal form of the form T

T= -2a(fi

-

-Au1-

2

Y

4a al

z

U3 +

/z

z

f3

b2A

z

+61 u3

2a1

al

- .i.Iu+bzA f2

\

z

+ (2a1)2 /

2

f3

z

(fz - 2b2) +

2ai(b1 - 1)

b z.lf3

1l2

)

2

f3

from which we can see that the form T is not positive definite. The system of differential equations on E(3)* for the kinetic energy T = X(Q(a, b, D)(X)) is integrable explicitly since it is of the form

Xl = -Q:xzY3 - Y +

ab1

\l

Jxzx3, z

zz =

al

(b2

+

xlx3 + axlY3,

89


X3 = 0,

al at + b2) YzX3 (b2

yt = -aY2Y3 1

yz = aY1Y3 +

-y

2X1 Y3 +

R

al

+

A\

-y

2)XZY3

+2+2

J

xzx3,

a

Ylx3 - 2 + 2

X1X3,

Y3=01 x; = OT/au;,

A = (b2a2 + 2b1a1)/b22 ,

yi = aT/aw;,

u; being the linear velocity, a , being the angular velocity of rotation of the rigid body, 0

X cE(3)*:X =

-Y2

-y3

0

Yz

-Y

1

Yi

© x2

0

(XX,3

If the dimension is greater than three, the explicit formulas become too complicated and therefore we do not consider them.

7.4. Equations of inertial motion of a multi-dimensional rigid body in an incompressible ideally conductive fluid

Let us consider the classical equations of magnetic hydrodynamics for a nonviscous ideally conductive fluid (see e.g. [64]). av

- + (rot v) x v = p `(rot H) x H - grad II ; at

aH = rot(v x H), at

n= II(x, t) being a uniquely defined function in the bounded region D with a smooth boundary. II is defined by the condition that av/at be a non-divergent vector field on D; the field being tangent to the boundary of D. As shown in [137], the simplest finite-dimensional analog of these equations is

M = [Q, M] - [H, J] I 1 _ Ell, H]

(4)

90


defined on the Lie algebra

so(n) of skew-symmetric matrices.

0 = (R9-,) * 4 is a right-translation to the unit of the group SO(n) of the velocity vector 4 e T;SO(n); J = AdY*-, j, j being flow density in the body; H = Ade h, h being the tension of magnetic field in the body, M being the kinetic moment in space. Let us give explicitly the embedding of the system (4) into the Lie algebra. Then, the system will be the Hamiltonian one on all orbits (the results of this section are given in [131]). Let G be any Lie algebra over the field K. Let us construct a new Lie algebra 0.,,(G), a, fi E K which contains G as a subalgebra. As a linear space, 52,,,(G) is the direct sum G ® G. Let us denote the elements of the first summand by a e G, of the second one by eb, b e G, i.e. any element of 52,,,(G) is of the form x + By, x, y c- G. Let us define on 0.,, (G) the product

[x1 + ey1,x2 + EYz] = [x1,x2] + $[YI,Y2] + E([Y1,x2] + [x1,Yz] + a[Y1,Y2]) Note that S28,0(G) is an extension of G since there is an epimorphism of Lie algebras f : S2,,o(G) -+ G, f (x + By) = x.

The linear space 52,.,(G) = G + eG over the field K, the product [x1 + CY1, x2 + eye] defined above, is a Lie algebra for any LEMMA 7.3

a,feK. The proof follows from the definition of the commutator [x1 + By, xZ + EYz]

If 4$ + a2 = 40 +, 2 = 0, then the Lie algebras 52,,,(G) and 0,,.8(G) are both isomorphic, being isomorphic to the standard LEMMA 7.4

extension 00,0(G) as well. Proof Let us define the isomorphism cp: i2=,#(G) -+ Q(),o(G) = G + SG, S2,,R(G) = G + eG, e2 = ae + $ (SZ = 0) by the formula T(x + By) _ (x + (a/2)y) + by. Evidently, [cpX, cpY] = p[X, Y].

In what follows, we shall denote the extension 0,,o(G) by S22(G). For the application to Hamilton mechanics we need the Lie algebra !00(G) which we denote by Q(G) = G + eG (e2 = 0). We can define the Lie algebra Q(G) in a different way. Let us consider the semi-simple sum G + G, where G acts on G by the coadjoint representation ad, the second summand being regarded as an Abelian Lie algebra. Then, the resulting direct sum coincides with Q(G).

91


Let us consider the Lie algebra S2(G) = G + eG, E2 = 0. We can assume (2(G)* = G* + (6G)* 3 (fl, f2) defining the isomorphism as follows

(fi,f2) -+ (.fi,f2)(x + y) = .fi(x) + f2(y),

x e G,

y e G.

Let us denote f e SZ(G)* by f = x* + Ey*; x*, y* E G*, By* e (eG)*.

We have denoted already the coadjoint representation in the Lie algebra G by a(x, y) = ad*(x), x e G*, y e G; a(x, y)(z) = <x, [y, z]>, z e G, <x, y> being the value of the covector x e G* on the vector y e G. PROPOSITION 7.7

Let x + By e S2(G) and x* + Ey* E S2(G)*. We state

that

a(x* + ey*,x + Ey) = a(x*,x) + a(y*,y) + ea(y*,x); x*, y* E G* and a(g, f) being calculated in the Lie algebra G. Proof Let x e G, x* E G*, y c- G. Then,

a(x*, x)(y) = x*([x, y]) = a(x*, x)(y) and

a(x*, x)(ey) = x*([x, By]) = 0 since x*IFG = 0. Let x c- G, x* e G*, y e G, then

a(Ex*, x)(y) = ex*([x, y]) = 0 since Ex*I G = 0,

a(Ex*, x)(ey) = ex*([x, By]) = x*([x, y]) = a(x*, x)(y) Let y e G, x* E G*, g c- G, then a(x*,Ey)(9) = x*([Ey,9]) = 0

since x*IEG = 0 and a(x*,Ey)(E9) = x*([Ey,E9]) = 0

because [Ey, Eg] = 0. Let y c- G, x* E G*, g c- G; then a(Ex*,Ey)(9) = ex*([Ey,9]) = x*([y, g]) = a(x*, y)(g)

and a(Ex*,ey)(E9) = x*([Ey,eg]) = 0.

92


Therefore, we finally obtain a(x* + ey*, x + ey) = a(x*, x) + a(ey*, x) + a(x*, ey) + a(sy*, ey)

= (a(x*, x), 0) + (0, a(y*, x)) + (0, 0) + (a(y*, y), 0)

= a(x*, x) + a(y*, y) + ea(y*, x) This completes the proof.

Let us consider the linear operator C: S2(G)* - S2(G) (we call it a sectional operator). Then, in the space S2(G)* we can give the system of non-linear differential equations a = a(a, C(a)), a e f2(G)*. This system appears when we integrate geodesic flows of the left-invariant metrics on

the Lie group 6. Let a = (X, Y) E G* + eG*; C(a) = (x, y) e G + eG. Then, from Proposition 7.7 we obtain the following corollary. PROPOSITION 7.8

The system of Euler equations a = a(a, C(a)),

a e O (G)* is of the form

X = a(X, x) + a(Y, y)

(5)

{ Y = a(Y, x). Our next aim is to construct the linear operators C: S2(G)* --,. S2(G) for which the Euler equations (5) are completely integrable in the Liouville sense on all orbits in general position of the coadjoint representation of the Lie group n((5) which is associated with the algebra S2(G). PROPOSITION 7.9

The Euler equations on fl(G)*, G being a semi-

simple algebra, coincide with the finite-dimensional approximations (4) of the equations of magnetic hydrodynamics in the case when G = so(n).

Proof Identifying G* = G with the use of the Cartan-Killing form, we evidently obtain a(y, x) = [x, y]. Then, the system of the equations of magnetic hydrodynamics (4) follows from the Euler equations (5).

Let G be a semi-simple complex Lie algebra, for which the nondegenerate symmetric scalar Cartan-Killing product B(X, Y), X, Y E G is given. We have the representation p = ad*: S2(G) -> End(Q(G)*). Let a e S2(G)*, then the mapping 0,: S2(G) - S2(G)*, fa(x) = ad* a, x e S2(G), a e S2(G)* is defined. In the case of the semi-simple Lie algebra

G we shall identify G* with G using the Cartan-Killing form B(X, Y): G* = G as we did before. It is easy to check that in this

93


isomorphism ad* y maps into the ordinary commutator [x, y]. In this case a = a1 + sae E fl(G)* = G + eG and we can assume aI, a2 E G. PROPOSITION 7.10

Let a c Sl(G)* bean element in general position, i.e.

the orbit of the coadjoint representation of the Lie group Q(6) which corresponds to a has maximal dimension. Therefore, we can take a from G. Then, a will be an element in general position in G. In this case,

Ker 0, = H + eH, H being a Cartan subalgebra in the semi-simple complex Lie algebra G.

Proof Let x = xI + x29 E Ker 0,. Then, a(a1 + eat, X1 + cX2) = 0

if and only if a(a2, x1) = 0, a(a1, x1) + a(a2, x2) = 0,

a, c- G* = G,

x, e G

(1 = 1, 2).

In the case of G we obtain the following condition [x1, a1] + [x2, a2] = 0, [x1, a2] = 0. Let us prove the first assertion of the proposition. We have dim OII(G)(x, y) = rk

CjXk Cit

C yk 0

)

If we take yin G (in general position) and assume e.g. x = 0, we obtain dim OO,G,(0, y) = 2 dim OG(Y) = 2 dim Oc

Since in a semi-simple Lie algebra the codimension of an orbit is equal to

the number of functionally independent invariants of the coadjoint representation (which is equivalent to the adjoint one because there exists a non-degenerate invariant scalar product-the Cartan-Killing form), (0, y) is an element in general position. This completes the proof

of the first assertion of the proposition (for the calculation of the invariants of the coadjoint representation of the Lie group fl(.T) see [132] and Theorem 13.1). Let us calculate Ker 0,. From the equation [x1, a2] = 0 it follows that x1 belongs to the Cartan subalgebra H which contains the element a2 (see [47]). Let aI = al.k + Y:#o al.ae,, x2 = X2.k + >4:o x2.:ea; G = H + Y,:o Ce, being the root expansion of the Lie algebra with respect to H and al.h, x2,,, E H, e, being the root vector which corresponds to the root a e H*. Then,

94


[x1, all + [x2, a2] = Y (aI.=a(x1)e: - x2,.xa(a2)e.2) = 0. a*O

We have a1,,a(x1) - x2,,a(a2) = 0, x,, a2 E H for any root a # 0. Since

a2 is in general position, a(a2) # 0 for any a # 0, a e A. Therefore, according to the first part of the proposition, a, can be taken as zero. Then, a,,, = 0 for any a # 0, and x2,, = 0 for any a # 0, i.e. x2 E H. Therefore, Ker 1,, 9 H + H. The inverse expression can be checked easily. Thus Ker 0. = H + eH. Let b = b, + e62 E Ker 0. = H + eH c S2(G), H be a Cartan subalgebra of G. Let b, be in general position. Then, PROPOSITION 7.11

Kerp(b)=H+eH eS2(G)* =G+eG (the isomorphism is given by the Cartan-Killing form).

Proof We need to find all x, + ex2 e S2(G)* such that a(x, + ex2, b, + e62) = 0. Calculating ad* for f2(G) we find that this is equivalent to

the system a(x2, b1) = 0; a(x b1) + a(x2, b2) = 0. By identifying G* = G using the Cartan-Killing form we obtain [b,,x1] + [b2,x2] = 0,

[b1,x2] = 0.

Since b1 is in general position, x2 belongs to the Cartan subalgebra H. Then, since b2, x2 e H, [b2, x2] = 0. Therefore, from the first equation we obtain [b1, x] = 0 from which it follows that x1 E H, i.e. x1 + ex2 EH + rH. This completes the proof. Let Ker p(b)1

PROPOSITION 7.12

0a: Ker 0;

a

be

as

in

Proposition

7.10.

Then,

For any b e Ker ¢, (b is as in Proposition 7.11) we have p(b)(Ker p(b)1) c Ker p(b)1. Here V' means orthogonal complement with respect to the direct sum of the Cartan-Killing forms on G: Ker 0, = V + e V and Ker p(b)1 = V + 6V, V = >,#0 Ce, being an orthogonal complement to the Cartan subalgebra H of G with respect to the Killing form.

Proof Let

a = a1 + eat e Ker p(b) = H + eH c S2(G)*. provided that Y-, x. e. + e Y. yae, a (Ker Oa)1,

Then,

a(a, + ea2,Ex,e, +eY_ yae,) _

x,a(a,)e, +

y,a(a2)e, + e 2

x,a(a2)e, c Ker p(b)1,


as

was

claimed.

Let

bl + eb2 a Ker 4 a = H + sH,

95

E, x,e, +

eY,y,e,EKerp(b)1. Then,

+eb2l

a/l

_

x,a(bi)e, + > y,a(b2)e, + e

y,a(b1)e, E Ker p(b)1 z

as was claimed. Here e, is the root vector which corresponds to the root a c- H*. REMARK Evidently, the mapping 0,: Ker 4 -+ Ker p(b)1 is an isomorphism of vector spaces.

With these preliminary considerations over, let us construct the sectional operators for the complex semi-simple Lie algebras. We shall use the general method given in Section 6. Let D: Ker p(b) -> Ker & be any linear operator. DEFINITION 7.1

Let us define the operator C: S2(G)* -- S2(G) by the

matrix

C = C(a, b, D) =

(&1 adb 0

0

1

D)

in accordance with the splittings S2(G) = Ker 0. + Ker 0a , 12(G)* _ Ker p(b) + Ker p(b)1 (see Figure 23). Considering what was said above, this definition is correct since all operators which are used here are well defined. As in the semi-simple case, we shall call the operators constructed operators of the "complex" series for the Lie algebra S2(G).

Let H be the Cartan subalgebra of the semi-simple complex Lie algebra G, a e H* be a non-zero root. We shall define h, e H using the

condition a(h) = B(h, h;) for any h e H, B(X, Y) being the Cartan-

Fig. 23.

96


Killing form of the Lie algebra G. Let Ho be the subspace in H which is generated by all vectors hz with rational coefficients. We shall consider the standard compact real form G. of the Lie algebra G (see [47]). We can take vectors ihz, e, + e_ i(e, - e_2) (i = as a real basis in G. Let us construct the sectional operators for 12(G,,). The operators

constructed we shall call operators of the "compact" series for the algebra Q(G).

Let a = i(al + ea2) E Ker 0, n S2(G.). Then, the operator ado for the Lie algebra f (G.) can be described as follows LEMMA 7.5

ex + e-, -+ a(a1)i(e, - e_,),

i(e, - e_) - -a(a1)(ea + e_2), e(e, + e_s) - a(a2)i(e, - e_,) + ea(a1)i(e, - e_a),

ei(e, - e_,) - -a(a2)(ea - e_,) - a(a1)e(ea + e_,).

The proof is obtained by direct calculation using Proposition 7.7. a = i(a1 + eat) E Ker p(b) n Then the LEMMA 7.6 Let operator ¢a for the Lie algebra fl(G,,) can be described as follows a(a1)i(e, - e_,) + ea(a2)i(e, - e_,),

e, + e_a

i(e, - e_,) -' -a(a1)(ex + e-2) - ea(a2)(e, + e-=), e(e, + e_,) - a(a2)i(e, - e_2),

ei(e, - e_,) - -a(a2)(e,, + e-,), Ker p(h) n f (G )* = Ker p(b) n The proof is obtained by direct calculation using Proposition 7.7. PROPOSITION 7.13

that a(a2)

Let a = i(a1 + sae) E Ker p(b) n S2(G.)* be such 0, b = i(b1 + eb2) e Ker 0, n S2(G,).

0 for any root a

Then the operator 0a 1 adb for the Lie algebra S2(G,,) can be described as follows e,

+ e_,

a(bI)

e ( e,

+ e_, ) ,

a(a2) i(e,

-

e-7!)

a( a

z) e

i( e,

- e_,

),

97


a(b2)

e(e,+e(o((a2) e_5)

a(b2)

(a(a2) -

a(b1)a(a1) a(az)

z

a(b1)a(a2)

a(a2)Z

E(es+e_,)+

a(b1)

a(az)

(e=+e ),

ei(e, - e-z + a(b1) Ilea - e-s a(a2)

The proof follows directly from Lemma 7.5 and Lemma 7.6.

Now we can construct the operator for

using the general

technique given in Section 6. Using the Cartan-Killing form we identify S2(G )* with S2(G,,) = G,, + eG,,. Let

(e,+e-,)+I Ri(e, - e_,) c- G. a

S

We have the splitting as a direct sum of linear subspaces

(Ho + eHo) + (V + n(Gu) = (Ho + eHo) + (V, + EVJ,

Ho =

Mhx. Q

In accordance with the direct decompositions

DEFINITION 7.2

(Ho + eHo) + (V + and

S2(G,,)* = (Ho + eHo) + (V,, +

we define C = C(a, b, D)

D C0

0

0a' adb J

D: Ho + eHo - Ho + eHo being any linear operator, a e S2(G,,)*, b c S2(Gu) being chosen as described above.

REMARK Remember that we can construct the "compact" series of sectional operators for the semi-simple Lie algebras as well. In the case of the semi-simple Lie algebras, the sectional operator C: G,* - G. is diagonal with respect to the canonical basis e, + e_2, i(e, - e_,). In our case, different from the semi-simple one, the operator 0a 1 c ad,*, is not

diagonal with respect to the canonical basis. Let us construct the sectional operators for 12(Gn), G. being a normal compact subalgebra in the semi-simple complex Lie algebra G. By definition, G. _

Loo l8(e, + LEMMA 7.7

e_2).

Let a e iHo + eiHo c f2(G,,)*, b e iHo + eiHo c O(G.).

98


Then the operators of the "compact" series C(a, b, D) preserve S)(G ). More precisely,

G+

G,, + eG -

C(a, b, D): f (G )*

The isomorphism S2(G )* = G. + eG is given by the direct sum of Cartan-Killing forms. The proof follows directly from the explicit form of the operator / 1 adb in Proposition 7.12.

Thus we have defined the operators C(a, b, D) = C: S2(G )* S2(G ). As in the semi-simple case, the elements a, b should not belong to i2(G.)*

and 12(G.) respectively. We shall call these operators operators of the "normal" series for the algebra S2(G). In conclusion, note that in the case of the Lie algebra G = so(3) from the construction of the sectional operators we obtain the following Euler equations. Let

1

0

x1

xz

-X1

0

-XZ

-X3

0

Y1

x3 J+e

-Y1

0

0

-Y2

-Y3

Yz

Y3 JES2(G)* = G* + eG*, 0

ez = 0. Then X1 = -klxzx3 + (k5 - k2)Y2Y3 + (k5 - k3)x2Y3 + (k6 - ka)x3Yz, X2 = k1x1x3 + (k2 - k5)Y1Y3 + (k3 - k6)x1Y3

+ (k4 - k6)x3Y1

,

X3=0, YI = -k1x3Y2 + (k6 - k3)Y2Y3, Y2 = k1x3Y1 + (k3 - k6)Y1Y3,

Y3=0. In this case, the kinetic energy H = <X, C(a, b, D)(X)> is of the form H = k1x3 + k5(Yi + yz) + k2Y3 + 2k6(x1Y1 + x2y2) + (k3 + k4)x3Y3 Using the diagonal form of the energy, we conclude that the quadratic form H always has at least two "minuses" and four "pluses." Also, note that the energy H coincides (except for one term) with the kinetic energy in the first case of Clebsch (which describes inertial motion of rigid body

in ideal fluid). More precisely, the kinetic energy constructed on 0(so(3))* is the limit (when b11 - 0) of the kinetic energy in the first case


99

of Clebsch (see notations in 7.3). Hence our case can be cited as an analog of the first case of Clebsch for non-positive-definite kinetic energy.

3

Sectional operators on symmetric spaces

S. CONSTRUCTION OF THE FORM F, AND THE FLOW XQ IN THE CASE OF A SYMMETRIC SPACE A. T. Fomenko's results given in this chapter can be found in [32], [34], [125]. Let Z3 = 6/t, be a homogeneous space. Then we can assume that the

Lie algebra G of the group 6 is split as a sum of linear subspaces: G = H + V, H = T, being the stationary subalgebra; V = TQ 13, i.e. it is identified with the space tangent to fly. The subalgebra H acts on V via the

adjoint action dp = ad, i.e. ad,. (X) = [H', X], H' e H, X e V. For the purpose of simplicity, we suggest that (5,J5 are semi-simple Lie groups; the space V is partitioned by the orbits 0 of the coadjoint action Ads.

Let T c V be a maximal commutative subalgebra of V (a Cartan subalgebra of V). Then V = T + T', T' being the orthogonal complement to Tin V (the sum of the linear subspaces). We assume that on G = H + V the Cartan-Killing form is used, so that its restriction to

V defines a non-degenerate scalar product. Let K c H be the annihilator of T, i.e. it is the subalgebra of H which consists of all the elements k e H such that [k, T] = 0 ([k, t] = 0 for any t e T). We assert that K is the annihilator of any element in general position t E T This annihilator is the same for all elements in general position in T Let H = K + K', K' being the orthogonal complement to K in H. Let us consider the subspace M = Ann K in V, i.e. m e M if and only if

[m, k] = 0 for any k e K. It is clear that M is the annihilator of an arbitrary element in general position in K. Evidently T c M. Let Z c M be an orthogonal complement to T in M. Then M = Ann K = T + Z (see Figure 24). Let R be an orthogonal complement to

Min V. Then V=T+Z+R=R+M. 100


101

Fig. 24.

For the purpose of simplicity, let us assume that 'I3 = 6/5 is a symmetric space. This means that on G there is defined an involutive automorphism a such that a- = + 1 on H and a = -1 on V (a2 = 1 on G). Hence [H, H] c V, [H, V] c V (these relationships are valid for any homogeneous space), [V, V] = H (the latter is valid in the symmetric case).

Let a e T be an element in general position in the subalgebra T. Then, the mapping ada: G - G allows us to identify the subspaces K' and T'.

For any a e Tin general position the mapping ada: H -> V defines an isomorphism between K' and T' (G, H being semi-simple Lie algebras, l3 = (5/55 a symmetric space). LEMMA 8.1

Proof Since K = Ann T = Ann(a) n H (a is the element in general position), H = K + K', because Im(ada) = ada K' c T' ada T' c K' c H because K = Ker(ada) n H. Conversely, T = Ker(ada) n V. It is clear that ada: T' - K' is a monomorphism since if a vector t' c- T' existed, such that ada(t') = 0, it would mean that t' e V commuted with the element in general position a e T, i.e. [t', T] = 0 and

hence T would not be maximal in V. Since dim H/K = dim K' = dim 0(X) = dim T' (O(X) is an orbit in general position in V), ada: K' - T' is a linear isomorphism. The lemma is proved. Since ada : K'--+ T' is an isomorphism, the inverse mapping ada 1: T' - K' is uniquely defined. The mapping ada: T' - K' behaves Similarly. It is clear that in general these mappings are different. Generally speaking, the compositions ad.: K' -+ K' and ad.: T' T'

102


are not the identity mappings. Let R = ada ' R, Z = ada' Z, then K' = 2 + R. We can define the subspaces Z and R' in an alternative way. K acts as the stationary subalgebra on the space tangent to fj/R which is isomorphic to T'. This action can be defined on the algebra H itself as

the coadjoint action K on the space K' which is orthogonal to K in H

and naturally identified with T'. Hence Ker(adK) n T' = Z is isomorphic with Ker(adk) n K'. Indeed Ker(adK) n K' coincides with Z = ada ' Z. The action of the mapping ada is illustrated in Figure 25. In

particular, ada Z = 2, ada Z = Z, ada R = R, ada R = R, as proved below. PROPOSITION 8.1

Let us consider the coadjoint action adK: H -+ H.

Then adK: K' -. K' and Ker(adK) n K' = Z. In particular, R is the orthogonal complement to 2 in K' and both spaces R and 2 are invariant under the action of adQ : K' - K'; a being an element in general position in T (see Figure 25). Proof Let k c- K be an arbitrary element in general position in K. Then

Ker(adK) n K' = Ker(adK) n K'. Similarly

Ker(adK) n T' = Ker(adK) n T'. Let a E T be an element in general position in T. Then, [a, k] = 0 and ada: K'-+ T' is an isomorphism. Let us prove that ada 'Z = Z c Ker(adK) n K'. If i c-,Z then [k, [a, z']] = 0 since [a, z'] E Z. From the Jacobi identity for the algebra G we have

[k, [a, -]] + [z, [k, all + [a, [z, k]] = 0,

Fig. 25.


103

i.e. ada [i, k] = 0 (since [k, a] = 0). Since [k, z] E K' and ad.: K' -, T is an isomorphism (see Lemma 8.1), [k, z] = 0 which was what we wished to prove. Conversely, let s c- Ker(adK) n K'. We need to prove that s r= Z = ada ' Z, i.e. ada s e Z, i.e. [k, ada s] = 0, [k, [a, s]] = 0. Since s c- K' and [k, s] = 0, from the Jacobi identity and because [k, a] = 0, we obtain

[k, [a, s]] + [s, [k, all + [a, [s, k]] = 0. Then [k, [a, s]] = 0 which completes the proof. The arguments for R and R reduce to studying the orthogonality relations. Let us prove that ada Z = Z = Ker(adK) n K'. If Z E Z, then [a, z] a K' and

[k, [a, z]] = - [z, [k, a]] - [a, [z, k]] = 0 since [k, a] = 0, [z, k] = 0, z E Z = Ker(adk) n T'. Thus, ada Z c Z. Since dim Z = dim Z, ada Z = Z. Therefore, ada R = It which completes the proof. REMARK The two mappings ada : Z - Z and ada ' : Z -+ 2 differ from each other by the transformation ad':;? - Z which is a non-degenerate linear self-mapping of Z. In order to construct the form F, we need to define the linear mapping C: V -. H. Let us construct the natural mapping C using the canonical properties of symmetric spaces, in particular, the relationship [V, V] c H. First we notice that we cannot construct the natural form on V by using the restriction of the standard Kirillov form on V Indeed, B(X, n]) = 0 for any X, , n E V since rl] E H and H is orthogonal to V. Of course, we could consider V which is a linear perturbation of the

plane V (produced e.g. by the parallel translation of the plane by the vector h e H, h j4 0). Then, we could restrict the Kirillov form on V and

obtain a non-trivial form on V (which is isomorphic to V = V + h). However, in general this form is not closed. Under such an approach we still need to consider the equations dFc = 0, X Q FC = 0. Since ads,: V - H, we need to construct the operator C: V - H using operators of the form ad,,, v e V. At the same time it is natural to choose as the operator C a transformation which would be as close as possible

to the identity in the case when e.g. 0 = (fj x fl/15 = fj. Since the mappings ad,,: V - H are not identical for the case fj = (.5 x fl/.5, the action of the operators ad, needs to be compensated by operators of the form ad-. We should combine the operators of both forms, i.e. ad and

104


ad -', because, as we mentioned above, the space V can be mapped in H

with the use of both ad and ad-' (see Lemma 8.1). All these considerations are given solely to clarify the general problem of identifying V with a subspace of H. Let us write down V and H as the sum of linear subspaces V = T + R + Z, H = K + + 2 (see Figure 25). We assume that this expansion is fixed. DEFINITION 8.1

Let us define the operator C: V - H by using the

following matrix C = ada. + ada ' ad,, + D: ada.

C=

0 0

0

0

ad, 'adb 0 0

,

D

a, a' c- T being arbitrary elements in general position in T, b E K being an

element in K (not necessarily one in general position), D: T K being any linear operator, . ada' adb: R ada.: Z - Z. Since Z = Ker(adK) n V, Ker(adb: V - V) 2 Z (if b is in general position in K, then Ker(adb: V - V) = Z) and adb: V --p R; ada' : R -p R (see Figure

25). We call the operators C: V -+ H sectional operators for the representation p : H -+ End(V).

REMARK The sectional operator C: V -+ H in Definition

8.1 is

constructed according to the general method given in Section 6. Let us define the form F, on the space V (and on the orbits 0 = O(X) (-- V) by the formula Fc(X, , r7) = B(CX, q]), the DEFINITION 8.2

operator C being as defined above C = C(a, b, a', D); X,, E V; B(X, Y) being the scalar product. It is clear that the 2-form Fc is skew-symmetric on V (with respect to ry). Following the general method, we would have to consider the operation [c', rl] instead of rl] ( = [X, s'], n = [X, q']). However, since we consider the semi-simple case only, the forms B(CX, rj) and B(CX, [ ', q']) are equivalent (as in the Kirillov case for the semi-simple algebra). DEFINITION 8.3

Let us define the vector field XQ on the space V by the

formula XQ = [X, QX], the operator Q being defined above: Q =

Q(d,b,d',D);Q:R+Z+T.i +2+K;Q =ada. +ada'adb+D. It is clear that the field XQ is tangent to the orbits 0 of the coadjoint action of Sj on V


LEMMA 8.2

105

Let f be a smooth function on V and the vector field

s grad f e V be such that F,(s grad f, Y) = Y(f)for any field Y e V, Y(f) being the derivative of the function f along the field Y. Then, grad f =

[CX, s grad f].

Proof We have Y(f) = B(Y, grad f ). Hence B(Y, grad f) = B(CX, [s grad f, Y]) = B([CX, s grad f], Y) ;

B(grad f - [CX, s grad f], Y) = 0. Since Y e V is arbitrary we obtain [CX, s grad f] - grad f = 0 which completes the proof. If the form Fc is non-degenerate at the point X (on V or on O(X)), then the field s grad f is uniquely defined by the equality Fc(s grad f, Y) = Y(f ).

9. THE CASE OF THE GROUP , =!6 = (, x -5)/Sj (SYMMETRIC SPACES OF TYPE II) Let us consider the semi-simple group .5 as a symmetric space. Then, as

we know (see e.g. [48]), this symmetric space can be written as (S5 x 6)/Sa, the involution a: S x Sa -+ .5 x Sa being defined by a(x, y) = (y, x). The corresponding expression in the Lie algebra H + H is as follows : if V = (X, - X), X E H; H = (X, X), X E H (both H and its

image in G = H (D H being denoted by the same letter), a V = - V,

aH=H. The following equalities hold: Z = Z = 0, H = K + R, V = T + R (a Cartan decomposition of the algebra H).

LEMMA 9.1

Proof Here T = {(t, -t)}, t e T', T' being a Cartan subalgebra in H and K = {(t, t)}, t c- T', so that clearly T and K are isomorphic to F. Since V and H correspond to the same group, the coadjoint action adH on V is of the form ad(h,h)(X, - X) = ([h, X] - [h, X]) e V, i.e. it is identical with the action ad,, X. Hence, to find Z it is sufficient to find the centralizer T' in H. Because we consider the semi-simple case, we have

Z = 0, i.e. the centralizer coincides with T. As a consequence of the semi-simplicity the orthogonal complement R to T (and R to K) is generated by the root subspaces of the algebra H. This means that the decompositions V = T + R and H = K + R are isomorphic to the

106


Cartan decompositions (linear for V). Note that although 3 is diffeomorphic to !5, v is not embedded in 6 = , x , as a subgroup. We

can assume that H = {(X,X)} and V = {(X, -X)} are identified by using the natural mapping a: (X,X) = (X, -X). In particular, the orbits 0 c V coincide with the orbits of the standard coadjoint action of

b on H. The form Fc(X ; , q), C(a, a, 0, E) = C (i.e. a = b, a' = 0, D = E being the identity operator) is identical with the Kirillov form on the algebra H (which is isomorphic as a linear space to V). In particular, it is non-degenerate and closed (and invariant) on the orbits 0 of the coadjoint action of Sa on V. PROPOSITION 9.1

Proof Since V = T + R, X = rzX + Y, nX a T, Y e R and

CX =DirX +ad,'adbY=xX +Y=XeH. Thus, Fc(X, , q) = B(X,

q]). In the semi-simple case this form

actually coincides with the Kirillov form, up to a linear transformation at each point. The forms B(X, q]) and B(X, q']) are invariant on V with respect to Ads. Therefore, it is sufficient to compare them at just one point in the general position X0 in V; = adXo ', q = adXo q'; B(X0, [adXo ', adXo q']) and B(X D, [c', q']) differ from each other by the non-degenerate linear transformation adXo which maps the tangent space TXOO into itself. Hence the form Fc in general is not closed on V (in this example, Fc is closed on the orbits only). It is in this example ('B = t') that we can see the

role of the operator ad, ' adb which enables us to identify R and R in a natural way: the action adb is compensated by the action ada ' from which we can obtain the identity operator E assuming a = b. Using only

one operator adb (or ada) we would not obtain the operator E as a particular case of the sectional operator C (because on the space T the set of the roots is a redundant basis and hence, the system of equations

a(t) = 1, a runs over all roots, generally speaking, does not have solutions on T). Let us consider the field XQ = [X,QX], Q = Q(a,b,O,D):

T+R-,.K+.R, D:T-K, ada'adb:R-,I. By using the identification of V and H, we find that the field XQ on V (on H) is given as follows . = [X, cpa,b,D(X)], the operator cpa,b,D: V - H (i.e. H -+ H) being identical with the multi-dimensional rigid body operator

107


introduced in [92]. In our case, D: T - T, ada 1 adb: T' - T, T' being the subspace generated by the eigenvectors of the operators ad,, t e T, H = T + T' being the Cartan decomposition. As follows from [89], all flows of the form cpa,b,o (a, b e T are elements of the general position in T), being Hamiltonian ones with respect to the form Fc (Kirillov form) on 0, are completely integrable on the orbits of the general position in H. Thus, the important case of completely integrable "rigid-body" like systems is one of the examples of the pair Fc, XQ.

10. THE CASE OF TYPES I, III, IV SYMMETRIC SPACES 10.1. Symmetric spaces of maximal rank

Let us consider the space 6/Sj with rank (i.e. dim T) equal to the rank of ($

(i.e. the dimension of the Cartan subalgebra of G). Example:

V = SU(n)/SO(n); the embedding of SO(n) in SU(n) being the standard one. If dim T = rk G, then T V is a Cartan subalgebra of G (not only of V), i.e. K = 0 (it is impossible to extend T by including T in a larger commutative subalgebra since the Cartan algebra is maximal). Hence

Z + T = V, i.e. R = R = 0 (the annihilator K = 0 in V coincides with V). The following lemma holds.

If the space' = fui/.5 is of maximal rank, then the 2-form Fc on V is generated by the curvature tensor of the symmetric space v: 4B(a', R(X, ), r1) = Fc(X, , r1), R being the curvature tensor, a' e T being a fixed vector. LEMMA 10.1

Proof Since K=R=R=0, b=0, X=nX+X', nXeT, X'eZ; CX = C(nnX + X') = DxX + ada. X' (a' e T). Since K = 0, D = 0, i.e. CX = ada. X',

Fc(X; , i) = B(CX,

B([a', X],

B(a', [X, [x,11]]) = 4B(a', R(X, )U),

R being the curvature tensor (see e.g. [48]).

Since K = 0, 0 = .5, i.e. the orbits of maximal dimension in V are diffeomorphic to the group .5. Thus, the 2-form Fc on 0 =

is obtained

by the restriction of the 2-form B(a', R(X, )q) to .5 c V, the field

108


R(X, )q being tangent to the orbits 0 = S5. The flow 9. = [X, QX] is of the form [X, [X, a]] = adX(a),

a E T.

On G = su(n) consider the involutive automorphism a(X) = X, X e G. Then H = so(n) and V consists of all symmetric imaginary matrices of order n, with trace equal to zero. The diagonal matrices in V constitute the maximal Abelian subspace T. In this case, K = 0, M = V and Z is the orthogonal complement to Tin V (i.e. R = 0). Let T J be the n x n matrix on n, with the only non-zero element being equal to one

and occupying the location (ij) (i is the row, j the column), Ii, j I = - p 0. Then, if A > 0, the solutions are given by the functions l CA

arctg (

if A < 0, by the functions 1

Jcx)=t,

IxI J .

arcth(-

The image in the phase space is given by T = 2(i)2 - Zy2; x (C2X, do = + 22x2 + µl ; U=

-f

Jl

o

z = y,

J(x) = 2c2x3 + 2cAx (see Figure 28).

REMARK A. V. Brailov suggested the following interpretation of some forms of the type F, in the case of the symmetric space Sn-1 in V. Let us consider the vector b e K c H and the parallel translation of the plane V along vector b. We obtain the plane Vb = b + V; here the restriction of the Kirillov form is not trivial. Further, we obtain the form wb which is closely linked with the forms of the type Fc (at least, in the real case on Sn-1).

U00

u

'\ O

A> O

u

N Fig. 28.

4

Methods of construction of functions in involution on orbits of coadjoint representation of Lie groups

11. METHOD OF ARGUMENT TRANSLATION 11.1. Translations of invariants of coadjoint representation

The most efficient method of construction of functions in involution on the orbits of the coadjoint representation of Lie groups is the method of argument translation. Originally, this method was used in the theory of the Korteweg-de Vries equation (see [74], [28], [92]). It is possible to

construct the set of functions fa(x) = f (x + )a), A e R, a e V for the function f (x) on a linear space V. Generally speaking, this process does not give new functions. For example, let us consider on the plane R '(x, y) expressed as a function of f(x,y) = ex. Then fa(x, y) = e"+2a = ea1ex,

i.e. fa(x,y) is expressed functionally by the initial function f(x, y). Taking a polynomial in general position as f, we obtain (translating fa(x) with a fixed) N new functions (N = deg f is the degree of the polynomial f): '(X) {{

f (X

=

N

+ Aa) _

pi(x, a)A'

i-0 In this case, general position to all appearances implies that repeated factors are absent in the expansion of f in irreducible factors. For example, the following statement holds: LEMMA 11.1

Let f(x, y) = axe + bxy + cy2. Let us consider the

translation of the function f : 136


(f)(x) =

d

137

f(x + via). A=O

Then, f and (f) are functionally dependent if and only if

0=b2-4ac=0. The proof follows from the explicit expression of the Jacobi matrix J

of the functions f = axe + bxy + cy2; (f) = 2aAx + bBx + bAy + 2cBy, the translation vector being of the form a = (A, B) 0 0. 2ax + by

J _ - (2aA

bx + 2cy

+ bB bA + 2cB

The determinant J is equal to zero if and only if (4ac - b2)B = 0 and (b2 - 4ac)A = 0 which gives the proof of the statement. A more general example (when the shifts are functionally independent) is given by invariants of the coadjoint representation of semi-simple Lie groups (see Chapter 5). The following theorem validates

the use of argument translation in the theory of Hamiltonian systems.

THEOREM 11.1 (see [88], [89], [90], [91]) Let F and G be two functions on the space K* which is dual to the Lie algebra K. These functions are constant on the orbits 0(t) of the coadjoint representation (invariants of coadjoint representation of the Lie group M- which corresponds to the Lie algebra K), a e K* being a fixed covector, A, u e 68 being arbitrary fixed numbers. We claim that the functions

FA(t) = F(t + Au) and 6,(t) = G(t + pa) are in involution with respect to the standard symplectic Kirillov structure on all the orbits 0(t) (for any A, It e R).

Proof Since s grad, F = adjF (t), we need to prove that means the value of the linear functional x on the vector y. Indeed,

{F, G} = -w(s grad F, s grad G) = - o (adap(t), ad jc (t))

_

Let A96 j and t=a(t+via) + fl(t+pa), fl =.1/(,1-µ),a=u/(µ- A).

138


Then

- (t + µa), di 2(t)> = 0

because the functions F and G are constant on the orbits Ad*. if A = µ, then the result is zero because of the continuity. This construction allows us to obtain a complete involutive set of functions for a wide class of the Lie algebras including the semi-simple Lie algebras. The idea of translation of invariants at first appeared in [74] (S. V. Manakov) for the case Lie algebra so(n) and was then developed by A. T. Fomenko and A. S. Mishchenko in [89] in the case of arbitrary semi-simple algebras. THEOREM 11.2 (Bolsinov, A. V.)

Let G be the complex Lie algebra,

x E G* be the regular covector, a e G*. We denote the space {df (x + .la) I A E C, f E 1(G)} by SI. The equality dim SI = ?(dim G + ind G) holds if, and only if, the element a + Ax is regular for any complex number A E C.

COROLLARY The translation of the invariants of the coadjoint representation is the complete commutative set if, and only if, the inequality dim M511, > 1 holds, where M,;ng is the set of the singular points for coadjoint representation. This theorem was proved in 1986 and generalized many previous results. 11.2. Representations of Lie groups in the space of the functions on the orbits and corresponding involutive sets of functions

There is a general construction of argument translation for functions from the finite-dimensional representations. This construction includes all known methods of translation (see [126]). Let W be a subspace of the space of analytic functions A(G*) on the

space G* which is dual to the Lie algebra G and is such that (a)

dimW = -

-[(Xjf)e`]O. Therefore (Xi f )e' = - f (x)c*.

Let us prove claim (a). Provided that W < A(G*) is a finitedimensional subspace of A(G*) invariant with respect to Ad* and since, as a consequence, W is closed, we obtain d dt

f (AdEXPrc X) E W.

t=o

The validity of claim (b) follows from the equation

(-f c*) = [X;(f )]e` = c; f e` LEMMA 11.2 Let f, g be smooth functions on G*. Then, If, g} - 0 on all orbits Ad* if and only if c;jxk of/8x; 7g/8xj = 0; e; being a basis in G,

G being the structure tensor for G in the basis e,, e' being the adjoint basis in G*, x; being the coordinates in the basis e'.

Proof We have the expression s grad fx = add f(x) for the skew gradient. Therefore, If, g}x = co(s grad f , s grad gx) =

= = [ad* (x)](dfx) _ <x, [dgx, dfx])

ag of = ax; axe dgx

09

= ax.

(x)e',

k

ag of ax; axi

f

dfx = axj O

(x)et.

THEOREM 11.3 (Trofimov, V. V.) Let W be a finite-dimensional subspace of A(G*) invariant under the coadjoint representation of the

group 6 which corresponds to the Lie algebra G. Let fl, ... , f, be a

141


, h, E W such that

basis in W. Let us define in W the functions hl, . 1 < i, j < p,

cj*(dhi.s) = 0,

1 < k < s,

cj* = c*(h with respect to the basis fl, ... , f,. We assert that (a) {hi, h;} = 0, 1 < i, j < p on all orbits of the coadjoint representation of the Lie group 6; (b) the translates of h; are in involution on all orbits of

the coadjoint representation Ad*(6): {hi(x + .a), h;(x + µa)} _- 0,

1

=

a<x,

[dfxa(x), dgpa(x)]> +

a<x, [dfxa(x), d9-µa(x)]>

+ a<x,

a<x, [df-xa(x),dg-,,,.(x)1>

{fxa, gµa}(x) +( 4{f. ,, 4,1W W) + 4{J-ia,gp:}(x) + 4{ f_xa,g-pa}(X) = 0. The above expression follows from Theorem 11.1. Since f, g, 7,4 are invariants of the coadjoint representation Ad*((5).

12. METHODS OF CONSTRUCTION OF COMMUTATIVE SETS OF FUNCTIONS USING CHAINS OF SUBALGEBRAS

The results of this section were obtained by V. V. Trofimov [127]. Let e1, ... , e" be a basis of the Lie algebra G; e', ... , e" be the conjugate basis of G*; X "-- . , x" and x,,. . . , X. be the corresponding coordinates. Let H be a subalgebra of the Lie algebra G. We have the projection G* - H* which is defined by the restriction of linear functionals. In this case, the functions defined on H* can be lifted to G*.

Let f and g be functions on H* which are in involution on all orbits of Ad*(,). Let us consider the extension of these functions LEMMA 12.1


144

to G*. They remain in involution on all orbits of the coadjoint representation Ad* of the Lie group 6 which corresponds to the Lie algebra G.

Proof Let e1, ... , e, be a basis for H. Let us extend it to a basis for

el,...,e e,+l,.... e in

G.

Let i=l,...,s; a=s+1,...,n;

a = 1, ... , n. We shall use these notations throughout this section. In this case, the lifting of functions from H* to G* does not depend on the coordinates xa. Let us check the involutivity using Lemma 11.2. We have of ag CayX,

__ of ag

aXa aXb -

Cs(tXe

ax" OXp

+ C2LXC

= C1j X,

of ag + C12Xc

-of ag

Ox" ax;

ax1 axe

+ C xc

'

of ag tX1 aXj

ax; ag

= CXk of ag ax; ax,

= 0.

Considering ag/ax, = 0 and [e;, e;] E H, we obtain c j = 0. Finally, we obtain zero because we have the condition that { f, g} _ C Xk Of/ax; ag/ax; - 0 on all the orbits of Ad*(.5). LEMMA 12.2 Let G D H be a chain of subalgebras. If f is an invariant of the coadjoint representation Ad*(6), and g is the lift of a function on

H*, then f and g are in involution on all orbits of the coadjoint representation of the group (5 which corresponds to G. The proof follows from the equality Of ag k

' Xk aX a

X..

Of

k

09

- (c ij aza ) Xk

= 0.

If H (-_ G', G' being the derived algebra of the Lie algebra, f being a semi-invariant of Ad*(f), ag/ax8 = 0, then { f, g} - 0 on all the orbits of the coadjoint representation Ad*((5). LEMMA 12.3

Proof The function f is a semi-invariant of Ad* if and only if Ck;Xk Of/ax; = A; f. Therefore


of ag _ (° bc

CX°

aXb O

x-l

of 1 ag _

J aXh OXc

145

ag (Xf) ax,

a9 f=(A1 but ,; = 0, since the character on the derived algebra equals zero. H, such Thus, for each chain of subalgebras G H1 H2 we can DHs_1DH, GD G'=) H1=) H1:DH2DH2::) that construct functions in involution on G* using a large number of the functions in involution on HS . The latter functions may be obtained by

e.g. translation of invariants, with the subsequent addition of semiinvariants of the coadjoint representation of the group .51 which corresponds to the Lie algebra H;. We can add not only semi-invariants but also functions from some representations.

Let K be a subalgebra of G and V c A(G*) be a finitedimensional subspace of the space of analytic functions on G* invariant with respect to Ad,*. A function f c V defines covectors c* E G* when we choose a basis in V. If the extension of a function g e A(K*) is constant along these covectors, then it is in involution with the function f on all orbits of the coadjoint representation of the group t . LEMMA 12.4

The proof is similar to that of Lemma 12.3. REMARK

It is enough to use the procedure described in Lemma 12.4 for

a construction of a complete involutive set of functions on Bore] subalgebras in the semi-simple Lie algebras (see [126], [127]).

The simplest example of the construction described above can be obtained if Fl, is a maximal Abelian subalgebra in G'. We can take as an invariant of an Abelian subalgebra any element of the subalgebra if we

regard it as a function on dual space. In general, the number of invariants in the maximal Abelian subalgebra is not enough to construct

completely integrable systems. Therefore we need to use some subalgebras. LEMMA 12.5 If there is a chain of subalgebras G H1 H2 such that Hi H2, H2 = 0, [H1,H2] = 0, then any function G G' H1

f E A(H?) and elements H2 (regarded as functions on Hz) are in involution on all orbits of AdT(G*).

146


Proof As follows from the condition [H1,H2] = 0, the operators Xd do not involve derivatives with respect to the coordinates conjugate to the coordinates in H1 and the function f does not depend on the rest of the coordinates. Therefore,

of c X. of ax -° =c,,xa - 5, aXb

aXb axe

of x,-=Xdf =0. h

Finally, we have a theorem. THEOREM 12.1

Let V

S be a chain of subalgebras. If the functions f

and g on S* are in involution on all orbits of Ad(' i), 13 being the Lie group which corresponds to the Lie algebra S, then the functions .I and j are in

involution on all orbits of Adm; B being the Lie group which corresponds to the Lie algebra V; 7 = f - it, g = g o 7r, n: V* -+ S* being the restriction mapping. If f is an invariant of the coadjoint

representation of the group V and g is the lifting of g e A(S*), then 0 on all the orbits of Ad**. If the chain V S is such that V V S, f being a semi-invariant of Ad,*, and g being the lifting of g e A(S*), then {f, g} = 0 on all the orbits of the coadjoint representation Ad* of the group !B. REMARK The technique used in this section is a generalization of the construction of M. Vergne (see [134]). In [134] for a construction of global symplectic coordinates on the orbits of maximal dimension of the representation Ad* in a nilpotent Lie algebra chains of ideals were used

such that G1 c G2 c

c: G; dim G,/G; -I = 1. Invariants of the representation Ad* of the ideals G; (i = 1,... , dim G) were lifted to G*

by means of the natural projection G* -+ G* which provided the coordinates needed.

In conclusion, we give two statements in which the operations of argument translation and function lifting are used simultaneously. Let a e G* be an element of the dual space of the Lie algebra G; let G° = {X e G: (ad X)a* = 0} be the annihilator of a with respect to the coadjoint representation of G on G*. LEMMA 12.6

Let a e G* be an element of the space dual to the Lie

algebra G; G° be the annihilator of a; f e 1(G) be the invariant of G; g be a smooth function on (Ga)*. Then the function fo(x) = f (x + a) and g

(regarded as a function on G*) are in involution on all orbits of the


coadjoint representation of the group (

(t

147

corresponds to the Lie

algebra G) on the space G*.

Proof We have { fa, g}(x) = <x, [dfa(x), dg(x)] >

= -

= - = 0 because dg(x) e G*. The lemma is proved. LEMMA 12.7

Let a- be an involutive automorphism of the Lie algebra

G, G = Go Q G1 and G* = Go Q Gi being the corresponding decompositions of G and G* (see Section 11); a e Gi, f e I(G), g be a smooth function on (Go)* which is the space dual to the annihilator of a in G. Then, the function (the restriction to G* of the translate of the invariant f by the element a) and the function g (regarded as a function on G**) are in involution on G*.

Proof We have { fa,g}(x) = <x, [d3(x),dg(x)]>

=

z<x, [dfa(x) + d]` a(x), dg(x)] >

= -i

= J = 0, because dg(x) E G. The lemma is proved.

13. METHOD OF TENSOR EXTENSIONS OF LIE ALGEBRAS 13.1. Basic definitions and results

The results of this section are given in the work by V. V. Trofimov [129]. Let us consider an arbitrary r-dimensional Lie algebra G' over the field k. We denote the ring of polynomials in the variables xl , ... , x over the

Let fI,...,f,Ek[x1,...,x,,] be such that field k by f,(0) _ = f,(0) = 0. Let 06 be the quotient-ring IK = k[x1, ... , xa] (f1., , f ); (fi, f,) being the ideal in k [xl , . . . , xa] generated by the be the natural VS polynomials f1,... , f,; let n: k[xl, ... ,


148

projection. We denote the image of x( under the mapping n by e,. Then, 6G is multiplicatively generated by the elements el,... , e" e K over the field k. Let us extend the ring of scalars of the Lie algebra G to 1K, i.e. consider the Lie algebra G'K = G'(& 6S over the ring K. The field k is contained in the ring K. Therefore, we can consider the Lie algebra GN over the field k. We denote this Lie algebra by Of (G), f = (fl, ... , f,) being a polynomial mapping f : R" -+ R' such that f (0) = 0. We claim that Of(G) is an extension of the Lie algebra G. It is enough to construct an epimorphism of Lie algebras g: Of (G) -+ G. Note that any element y e Of(G) can be represented as Y=

e" (at.....2,)

'

'

en"xal.....a" ;

xal.....a" E G.

Let us set g(y) = x0,....0. Evidently this is an epimorphism of Lie algebras. The Lie algebra Of (G) is a split extension of G. Let us explain this. Since G is embedded in Of (G) as a subalgebra, Of (G) is the semi-

direct sum of G and some Lie algebra M according to some representation p : G -' Der(M), Der(M) being the Lie algebra of derivations of M. In this case, M is the ideal in G ® 1K generated by the subspaces e1 G,... , E. G. Then Of (G) = G + M is a direct sum of linear spaces. The semi-direct sum of G and M is mapped into the Lie algebra

of derivations of M via the natural homomorphism x -+ ad. on G. Let us consider an arbitrary polynomial mapping K = 118[x]/(x2) in the construction described above. Then, we obtain the Lie algebra O (G) for which the algorithm for calculating the invariants REMARK

of the coadjoint representation (given the invariants of the coadjoint representation of the Lie algebra G) is given in [132].

Let us consider an arbitrary polynomial representation

REMARK

f:

ll

-' l omitting the condition f (O) = 0. Then, we obtain a Lie

algebra Of (G) which contains G as a subalgebra. Let us consider the Lie algebras Of (G) where

f = (zlm,..., x.,

We denote Of (G) by 52,,,E

)E(k[xl,...,zn])n

m"(G).

be a basis of the Lie algebra G. Then the vectors e "e;; 1 a; + m, - a; = m;, i.e. after multiplication by ell E'^ each new summand gives zero (i.e. has eai with S; > mi which is Tern) X(01 ... Enêt)F(z) = E1, ... En^C OF x(y1, ... , yn)kEl 1 -Y1 ... En^ aZj OSy9Sm, 1 n/2 we choose

the bases conjugate with respect to fi to the ones already chosen in A012) - ;. As a result we obtain el, ... , CN, a homogeneous basis of the algebra A self-conjugate with respect to fi; fl(e*i, ej) = 5 ,, * being a permutation. Let e1,.. . , e,,, be a basis in G*; x1, ... , x,,, be the corresponding coordinates. It is natural to regard the linear functions on G* as the elements of the Lie algebra G. Conversely, the elements xi ®e, of the Lie algebra GA = G ® A (regarded as a Lie algebra over k) can be regarded

as linear coordinate functions on G. Let x = xi ®Ej be coordinates on

G. For a polynomial function P(x) on G* co

Y>

P(xl,... ,

Pi1,...,i,xi, ... Xik = P1x1

(12)

k =0 (i,,...,ik)

with 1 q, then (Eq, a E,) = 0. Hence, if (Eq, 61) # 0 then ax*'/ax*' = 0. Therefore, when s > q the matrix M,q(x) consists of zeros. When s = q, we obtain aPjg)(x) 041,

= (Eq, EJ)PI

ax*' ax*8 *9

ax; *1 .. x;*1 ax*s x9,., t,r

t

1 = P!(y)

., -

xt*

a yi

Let us consider the matrix M,q;j(x) constructed with the matrices M,q(x).

This matrix is block-triangular, the diagonal being occupied by nondegenerate matrices aP,(y)/ay;. Therefore, it is non-degenerate and hence, the polynomials P(,) __P(" are independent at the point x. Proof of Theorem 13.7 Let P(x) be an invariant. It is equivalent to {P,x,} = 0, 1 ai(W)F}

a;(W){F", F;}

.

The following equation follows from the fact that (H, H) satisfies the Mcondition Y'"'X(n)WI(p.9)_'XT.X(q)wn+i-s-.(p.e)E'XT_

{F",F;} =

(4)

p.g

Hence we have the expression for {F", F.}. Clearly {F, F; } = 0, since otherwise I a;(W){F", F;}

0.

We will now consider the general case. Equation (4) follows from the M-

property. We have {F",Fp} = X'W"E*;{x;,x;}W*EXT - X'WpE*;{x;,xj}Wj*EXT. (5)

Let us show that the last two summands in (5) cancel each other. For this, it is enough to prove that the matrix N with (ij)-th element given by E;j{x;, commutes with W-or, which is equivalent, N(h) commutes with W;, for any h e H. Note that {h, N} = [N, Wh]. If {x;, xj} e H, then {h, N}(h) = {B([h, M], h) = 0. Thus, it follows that [N, Wh](h) = 0, which we wanted. Since {F", Fp} = Fp} - {Fp, F"}), z({F",

fXT =Z(XW"E*;{x',xj}W*EXT - XWpE*;{x;,xj}W*EXT) = 0.

We may transform the second summand in (5) similarly. Thus, the theorem is proved. Let us consider the H-module V + V assuming that V and V' are isomorphic. The vectors (x1, ... , x", x'1, ... , x',) = Y form a basis for V + V. The actions of Hon the basis 2n-tuple Y and the basis 2n-tuple Y' = ( A x , ,. .. , Ax", x1, ... , x") are isomorphic. ThL.efore we can construct H-invariants for the module V + V. If the representation of H


in V is given by the matrix of 1-forms W, then in V + V corresponding matrix W is of the form

(O

W).

171 the

Then,

y'W'gyT = AXWÈXT + X'WÈXT. Using this example, it is easy to obtain a corollary. COROLLARY

Linear combinations of similar canonical H-invariants of

the form FP = CCiJkX(`)WpEJX(k),

aijk E R

are in involution.

15. CONTRACTIONS OF LIE ALGEBRAS

The results of this section were obtained by A. V. Brailov (see [197]). 15.1. Restriction theorem

Let (M, co) be a symplectic manifold, H be a smooth function on M; s grad H be the vector field dual to the differential dH with respect to Co.

Also, we shall consider complex integrals of the system s grad H assuming that in our case the Poisson bracket is linearly extended to smooth complex functions. Let E be a group which acts on M by symplectic diffeomorphisms, H be a function which is constant on the orbits of E. Then, we shall say that

the group E is a group of symmetries of the Hamiltonian system (M, co, H).

Let E be a compact group of symmetries of the Hamiltonian system (M, co, H), a be a 1-invariant algebra of integrals, a, be a E-fixed subalgebra, N be a manifold of fixed points, a be the set of restrictions of the integrals in a to N. Then (a) PROPOSITION 15.1 (Restriction theorem)

(N, Co) is a symplectic manifold, Co = wl N; (b) a is closed with respect to

the Poisson bracket { f g}N; the restriction mapping is an epimorphism of Lie algebras (RE, { f, g} N) -. (R, {k, e} N); (c) ! is the algebra of integrals of the Hamiltonian system (N, Co, H); H being the restriction of

HtoN.

172


Let (M, co) be a symplectic vector space; p: E - End(W) be a completely reducible symplectic representation of the group E, Wo be a fixed subspace. Then, wl wo is a non-degenerate form. LEMMA 15.1

Proof Let i; E Wo and l be its orthogonal complement. Since

E Wo,

' is invariant under E. Hence, there exists an invariant onedimensional subspace R such that W = l O Rq. From the definition of n it follows that # 0. Given that l is 1-invariant, we deduce 7EWp. LEMMA 15.2

If F e aE, then s grad,,, F = s grads F.

Proof Let q e N. Because F is invariant, we obtain sgrad,,, F(q) e TqN. Therefore, to obtain the proof of the lemma it is enough to check the equality w(s grads F, X) = w(s gradN F, X)

(1)

for any vector X on M. Both sides of (1) are equal to X(F) which proves the lemma.

Proof of Proposition 15.1 Let q e N be a fixed point. Then, the correspondence a -+ dqa: TIN - TN is a symplectic representation of E on the space T. N. Since, under the hypothesis of the proposition, E is compact, this representation is completely reducible. Applying Lemma 15.1, we obtain assertion (a) of the proposition. From the definition of the Poisson bracket and equality (1) it follows that the restriction to aE is a homomorphism and the image of aE is closed in under the Poisson

bracket. We obtain the epimorphism given that the restriction of the function F to N coincides with the composition of restriction and averaging over the group E:

Foadp(a), which evidently belongs to a, Thus we have proved assertion (b) of Proposition 15.1. Let us prove assertion (c). Indeed, let F E a be the integral (M, w, H). Then {H, F},,, = 0. Applying (b) we obtain {H, F}N = 0. Assertion (c) is proved which completes the proof of Proposition 15.1. Let us give a simple example of a symplectic group action. PROPOSITION 15.2

Let E be a compact group which acts by

173


automorphisms on the Lie algebra G, G,. be a fixed subalgebra, x e G* c G*, OG(x) be the orbit of the representation Ad* of the Lie group (which corresponds to G) containing the point x; let OG.(x) be the orbit of G.. Then, (a) E acts by symplectic diffeomorphisms on OG(x); (b) OG,,(x) is open in the manifold of E-fixed points in the orbit 0,(x); (c) if co, co,, are the Kirillov forms of the Lie algebras G, G. respectively, then co = C0I G..

First let us prove two lemmas. Let 1,0,(x) be as in Proposition 15.2; ac, Q*: G* -+G* be the linear mapping conjugate to a. Then a*(OG(x)) - OG(x) LEMMA 15.3

Proof Let x e G*. Let us define the neighborhood Vx, in 0,(x') Vs. _ {Exp(Ad9*)(x'), g e G}.

(2)

Applying a* to (2), we obtain Q*(Vx,) = {Exp(ad*

g E G}.

(3)

Hence, Q*(Vz) C OG(x')

(4)

Thus the lemma is proved in the local case. To prove it in the general case, it is enough to notice that the orbit OG(x) is a connected set and OG(X) n a* -' (OG(o *(x)) is both open and closed as follows from (4). LEMMA 15.4

Let x e G*, , n e TxOG(x), co being the Kirillov form.

Then wx( , n) =

Q*n)

Proof By definition of the vectors and q, there are vectors ', n' E G such that i; = ads (x),

n = ad* (x),

then, by the definition of the Kirillov form,

(5)

<x, [g', n']>.

Applying a* to (5), we obtain

Q* = ad*-i.(a*x),

a*n = ad;-1,,'(Q*x)

(6)

Using (6))x it is easy to calculate the Kirillov form at the point u*x: Y a- ln]> = = cox(S' w a*(x)(c*S, Q*n) =

= GA if 2', 2" 0. Therefore, ind(G,i) = const(2) if 2 0. Let us prove that if 2 = 0, then the index can only increase. Indeed, the index is equal to the rank of the matrix l c (A)xk 11, c;`;(2) being the structure tensor of the Lie algebra GA; xk being the coordinates of a

covector in general position. If 2 varies slightly, then the rank of the matrix Ilci;(A)xk11 can only increase. Therefore, the index of GA can only

decrease. Hence, ind Go > ind Gx,

Proof of Theorem 15.1

A -A 0.

From Lemma 15.6 we obtain


177

ind(G°) < dim Ann(t*) = dim T = rank(G) = ind(G); t e T being an element in general position in a Cartan subalgebra T On the other hand, from Lemma 15.7 it follows that ind(G°) >, ind(G1) _ ind(G). Therefore ind(G°) = ind(G). This completes the proof. Let G = H ® V be as before a 712-graded Lie algebra. Similarly to the

above definition of the contraction of the commutator [x, y] -, [x, y]0 we can define an analog of the contraction for certain functions F on G*.

Let us suppose that the expansion of F in exterior powers on V terminates DEFINITION 15.4

F°(xH) + F'(xH, xV) + ... + F"(xH, xV);

Fk(xH,xv) being a k-form xV for fixed xH. Let us define FA(x) = 2n12F(xH, 2-'I2xv). Note that FA(x) is defined for A = 0: F0(x) = F"(xH, xV) as well.

Let G = H Q V be a 7L2-graded Lie algebra, F, F' be functions on G* such that (a) {F, F'} = 0; (b)F,1 and Fx are defined (see Definition 15.4; e.g. if F and F' are polynomials). Then, we assert that {F,1, F'} = 0 in particular, {F", F" 1 0 = 0; F", F'm being the terms of highest degree of the expansion into homogeneous components on V. PROPOSITION 15.3

1129v is a homoProof The correspondence gH + gV - gH + _ morphism of Lie algebras G - G,, if A 0. Under this F maps into

2"JZFA. Therefore IF,, F' }z = 0 if A : 0. This equation holds for A = 0 as well because both the commutator and the functions depend on A continuously. Proposition 15.3 can be used to construct involutive sets of functions

on G. Let us show how we can use the proposition to determine the invariants of the contracted Lie algebra G0. Let G = H Q V be a 7L2-graded Lie algebra, F be a polynomial on G*, invariant under the coadjoint representation; let F0 be the contraction of F (see Definition 15.4), G0 be the contraction of G. Then F is an invariant of the coadjoint representation of G0. PROPOSITION 15.4

Proof Let u E G0, then u can be regarded as a linear function on G. Evidently, here the contraction u - u0 is equal to uV if u V # 0 and equal to uH in the opposite case. We can express the fact that F is Ginvariant as follows

178


{F, uH} - 0

{F, u,,} -0,

uHEH,

u,,E V.

(15)

From the remark made above it follows that {F0, uH} - 0, {F0, u,,}0 - 0. These equations imply that the contraction FO is Goinvariant. EXAMPLE The algebra so(4) is defined by the equations [eij, efk] = eik, i, j, k taking values from 1 to 4; e,j = - ej,. The subalgebra H = so(3) is

generated by the vectors ell, e13, e23; let V be the linear subspace (e14, e24, e34). It is easy to check that the decomposition so(4) = H $ V gives a 1L2-grading, the corresponding contraction Go being a semidirect sum so(3) $ i83 (in other words, the Lie algebra of the group of

motions of the Euclidean space R3). We know that the functions F= xi, and F' = x12x34 - x13x24 + x23x14 are invariants of so(4). Contracting F and F' we obtain the invariants of the Lie algebra

so (3)$l :Fo=x14+x24+x34,Fo= F'.

5

Complete integrability of Hamiltonian systems on orbits of Lie algebras

16. COMPLETE INTEGRABILITY OF THE EQUATIONS OF MOTION OF A MULTI-DIMENSIONAL RIGID BODY WITH A FIXED POINT IN THE ABSENCE OF GRAVITY 16.1. Integrals of Euler equations on semi-simple Lie algebras

The Hamiltonian systems given in Section 7 allow embedding in Lie algebras and, in addition, are completely Liouville-integrable (in the commutative sense); the semi-simple case is given in [89, 90, 92]. In particular, for these cases we obtain positive answer to Hypothesis (a) (see Section 5) because we have complete commutative sets of functions

on the orbits in general position for the semi-simple and compact Lie algebras. The integrals of these Hamiltonian systems are very simple. To construct them, it is enough to know the invariants of the Lie algebra, i.e. a set of functions which are constant on orbits in general position. In outline, the construction of the integrals can be described as follows. Let f be some invariant of the algebra which is a function on G* (or on G for the compact and semi-simple case). Let a E G* be a covector in general position. Let us translate the argument of the function f (x), i.e. consider the function f (x + Aa), A e C or R. Since in our cases the functions f are

polynomials, we can expand the function f (x + Aa) in powers of the formal variable A and obtain an expansion of the form f (x + Aa) = Y-k Pk(x, a)Ak. Note that in all the above-mentioned cases it is these resulting polynomials Pk(x, a) (or, which is the same, the functions

f (x + Aa)) which form complete commutative sets of functions (integrals). We call this technique of the construction of integrals the

method of argument translation. It is a development of the idea suggested in [74] for the case of the algebra so(n). We already know that 179

180


translates of invariants are in involution. Therefore here we shall mainly prove the functional independence of the translates of invariants.

The method of argument translation gives a positive answer to Hypothesis (a) for many non-compact Lie algebras as well. We can obtain complete commutative sets of functions on orbits in general position by applying this method not only to the invariants of an algebra (sometimes those are not enough for obtaining the sets needed) but also to so-called semi-invariants (i.e. functions which are multiplied by the character of the representation under the (co)adjoint action of the group

on the orbit). Naturally, invariants are examples of semi-invariants because the former are fixed points of the action on the space of functions. Examples of semi-invariants can be found in [10], [127], [126]. Let us consider construction of commutative sets of integrals on the orbits in general position in semi-simple Lie algebras. We shall prove that these sets are complete in other sections. We shall mainly consider the complex semi-simple Lie algebras together with the Euler equations of the form i = [x, px]; the operators cp = TP.,D define the Hamiltonians of the complex series (see Section 7). Let us consider the coadjoint action

of the complex semi-simple Lie group (! on the corresponding Lie algebra G (we assume G = G*). The group partitions the algebra G into orbits. We set

Adgx=gxg-1,

gE6.

LEMMA 16.1 Any smooth function f (x), x c G which is invariant under the coadjoint action of the group (i.e. it is constant on the orbits) is an integral of the Euler equation i = [x, p x]; gyp: G - G being an arbitrary self-conjugate operator.

The proof follows directly from Tx0 = {[x, y] }, they vector running over the whole algebra G. Note that in the complex case there exist the elements which do not belong to the orbit 0(t), t e H; H being a fixed Cartan subalgebra. Let us consider the set of all complex vectors grad f(x), f e IG, IG standing for the ring of invariant polynomials on the algebra G. Let H(X) be a subspace of G which consists of all elements commuting with x. If x e Reg G, then H(x) is a Cartan subalgebra, and in a semi-simple

algebra any two Cartan subalgebras are conjugate. In particular, if x c- Reg G, then H(x) = g0H(a, b)go 1 for some go e 6, H(a, b) being the

Cartan subalgebra containing a, b. Evidently, H(x) belongs to the subspace generated by grad f (x), f e IG and if x e Reg G, then H(x) =


181

{grad f(x), f eIG}. This follows from the fact that the Killing form is non-degenerate and the space H(x) is orthogonal to the orbit's tangent space (see Figure 29). LEMMA 16.2

A smooth function f is constant on the orbits of an

algebra if and only if [x, grad f (x)] = 0 for any x c- G. We denote by grad f (x) the value of the field grad fat the point x c- G.

Proof Remembering that Tx0 = {[x,f]},

running over the Lie

algebra G, we obtain (grad f(x), [x, f]) = 0 for any

because [x, f] f (x) = 0. Since the operator adx is skew-symmetric, = 0 and because the Killing form is non-degenerate

this means that [grad f (x), x] = 0. The converse assertion can be checked similarly. REMARK The equation [x, grad f (x)] = 0 is obtained from c;ìxk of/8x; = 0 (see 2.3) by using the isomorphism G* = G given by the

Cartan-Killing form. PROPOSITION 16.1

Let f E IG, i.e. the function is invariant and

constant on the orbits. Then the complex functions h,A(x) = f (x + .la) are (for any ),) integrals of the equation z = [x, q bp(x)], W.,, being the

operator of the complex series. The function F(x) = <x, cpx> is an integral as well.

Proof Let us check the identity 0 = (d/dT)hx(x), T being the parameter along the trajectories of the flow z. This is equivalent to checking that = 0. We have

Fig. 29.

182


=

= - A

= - A = < [grad f (x + ;a), x + 2 a], cpx>

- 2 = 0

taking into account Lemma 16.2 and the identity [b, t + ,.a] = 0. Thus (d/dt)h,A(x) = 0 along z. Further d F(x) dt

(px> + <x, cpz> = 2 = 0

because

cp

is

symmetric and the operator ad is skew-symmetric. Let us consider the model example sl(n, C). Evidently, the standard symmetric polynomials on the eigenvalues of the matrix x are integrals constant on the orbits of the algebra. We manipulate the equation as follows (x + 2a)' = [x + 2a, apx + ).b]. Indeed, expanding the

commutator and making trivial calculations, we obtain the initial

equation z = [x, px]. We have used the fact that [a, b] = 0 and [x, b] + [a, px] = 0 because of the definition of cp,bp. Thus the equation has not changed but we notice a new series of integrals: the symmetric polynomials of the eigenvalues of the matrix x + via. These integrals can be presented in two ways: (1) Given the expansion of the polynomial det(x + Aa -,uE) = E,,O P,,,A",uO in powers of 2 and u, all polynomials

P,O(x, a) are integrals of the equation; (2) Using the functions Sk = trac(x + Aa)" and their expansions in powers of A: Sk = Ya Qak'(x, a).l°. The link between the Newton polynomials and the symmetric polynomials a' also interrelates the Qakl and P.


183

Let us construct the integrals of the compact series. Let Gu be a compact form of the Lie algebra G. Let X E G,,, a, b c- H,,, x - ,la c- G if .l

is real. Let us consider the action of 6 on G.. Unlike the complex

case, the union of the orbits defined by a Cartan subalgebra H. = H (a, b) coincides with G.. Let tp : G - G,, be an operator of the complex series.

Any smooth function f invariant under the coadjoint action of (5 (i.e. constant on the orbits) is an integral of the Euler [x, cpx], 'p: G. - G. being an arbitrary self-adjoint equation LEMMA 16.3

operator. The proof is trivial. Let IG. be a ring of invariant polynomials on G. Let us present explicitly the multiplicative structure of the ring 1G.. Let N be the normalizer of H. in G.; then, N/S = 0 is the Weyl group. Let

t E H., then the orbit 0(t) is orthogonal to the algebra H.. This orbit returns to H. intersecting H in a finite number of points which are the images of the element t under the action of the Weyl group. The ring IG,,

is identical with a ring of polynomials in H which are invariant with respect to the action of the Weyl group. This ring can be described

simply: if 6. is connected then the ring IG is a free algebra on r generators where r = the rank of G. among which it is possible to choose

homogeneous, algebraically independent polynomials Pk1'

, Pk,

where k. = deg Pk. For simple Lie algebras of number k., the degrees of the polynomials Pk; are as follows: A,:2,3,4..... n,n+ 1;

B,,:2,4,6,...,2n; C,,:2,4,6,...,2n; D,, : 2, 4, 6,

... , 2n - 2,n;

G2:2,6;

F,:2,6,8,12; E6: 2, 5, 6, 8, 9, 12; E7: 2, 6, 8, 10, 12, 14, 18;

E8:2,8, 12, 14, 18, 20, 24, 30.

The polynomials Pk, can be given explicitly. Consider the linear representation of the algebra G. of minimal dimension with matrices of

184


the size (m x m), let A,,.

.

.

,

A,,, be the weights of the representation, i.e.

the linear functionals in H. corresponding to the eigenvectors of the operators in H. on the representation space. The coordinates A1,... , A, in H. can be linearly dependent. The polynomials Pk have the form: n+1

An: I A;'; j=1

Bn: Y j=1 n

Cn: Yj=1 n

D. : > j=1

k;=2,4,6,...,2n-2; P;,=A1A2...An A;'. It is clear If G. is a particularly simple algebra then Pk, 1 that all the rings IG are subrings of the ring of symmetric polynomials S(A...... A,,,). All of the indicated functions are of the form Trace xki = A;'; except for the series D. in which a further polynomial D-et x is added. PROPOSITION 16.2

Let f c IG,,, i.e. the function is constant on the

orbits of the algebra Gn. Then the functions hx(x) = f (x + )a) are (for any A) integrals of the equation x = [x, cpx] where cp is the operator of the compact series x + .la E G,,, a. E R. The function F(x) = <x, (px> is also an integral. The proof proceeds in the same way as the proof of Proposition 16.1. See [90] for details.

Now consider the integrals of the normal series. Consider the embedding of G. to Gn. The operators cpab: G. --> G. are generated by the vectors a, b c H,,, in particular a, b 0 G. and hence x + .la 0 G,,, if

xEG.,AcD. Let f e IG,,, i.e. the function f is constant on the orbits of the algebra Gn. Consider the functions gjx) where A E f8, PROPOSITION 16.3


185

x c- G. c G. which are the restriction of the functions h1(x) = f (x + Aa)

to G. c G. Then the functions gz are integrals of the equation z = [x, (pabx] where gpob is the operator of the normal series. The function F(x) = <x, cpx> is also an integral.

16.2. Examples for Lie algebras so(3) and so(4)

For a clear illustration of this consider some examples of the series of integrals constructed above for very simple Lie algebras. In particular, we find that there are well-known classical integrals contained among these integrals. Let G. = so(3), we can consider so(3) as su(2), by making

use of the well-known isomorphism. Let us include the algebra su(2) into the compact real form G. of the algebra G = sl(n, C); then su(2) coincides with the stationary points of the involution ox = It is

clear that G. decomposes into the sum of three one-dimensional subspaces generated by vectors of the following form:

E- = i(E2 - E-s), Eo

E+ = Ex + E-_, where E0 E H. = iHo,

Eo=(i

Oil,

E+=I-0 O),

E__(0 O).

The operator of the compact series q : su(2)l- su(2) acts in the following way: a(b)

coE+ =

a(a)

a(b)

E- = a(a) E_

E+

,

coEo = ) 0Eo,

0 is an arbitrary real number, b = A+a, A.+ j4 0, a(a) : 0, i.e. A+ = a(b)/a(a). Finally, cpE+ = ),+E+, cpE_ _ A+E_, cpEo = )LoEo and differs from zero. For cp in general position A+ A0. In the case where G. = su(2) the operators cp of the compact series form a twoparameter family (A+, 2o). If where A0

x=

(-

iz

x+iy

x + iyl

E su(2) -iz )

then

V(x) _

i)lo

+(x + iy)

2+(-x + iy)

-iAoz

)

186


It is clear that <x, i> = 0, i.e. the velocity vector x is tangent to an orbit of the adjoint action of SU(2) on su(2). The orbits are two-dimensional spheres centered on the point 0 together with the point 0 itself. All the orbits apart from the point 0 are orbits in general position. Let us fix an arbitrary orbit in general position, then the integral trajectories of the flow x on the sphere coincide with the trajectories of the points of the

sphere during its rotation around the axis E0, see Figure 30. The following functions should be the integrals of the flow: Trace(x + Aa)k. We have

i(z+qA)

x+iy

-x+iy -i(z+qA))' where a = qEo, q 96 0. Hence S1 = 0, S2 = -2(x2 + y2 + z2 + 2zqA + q2 .2). The integrals are the coefficients of each power of A,, i.e. Q1(x,a) = x2 + y2 + z2,

Q2(x,a) = zq,

Q3(x,a) = q2,

i.e. in reality the functions z = const, x2 + y2 = const. The integral trajectories are the intersection of the spheres with the planes z = const. We have obtained one of the simplest classical cases of the movement of a solid: the integral Q1(x,a) = Q1(x) is the integral of kinetic moment,

the integral z = const is equivalent to the integral of energy in the common case in which the invariants 11,12, I3 are connected by the relationship 11 = 12 (ellipsoid of rotation). Let us consider once more the algebra so(3) and let us write it now in the usual form, i.e. let us study the integrals of the normal series for so(3).

Let G = sl(3, C), G. = su(3), G. = so(3), ax = _T, TX = z, G is the manifold of stationary points of the involutions a and T. The subalgebra

G. = so(3) is generated by the three vectors E,j = E,, + E,

Fig. 30.

187


0

E12=

1

0

- 0 0.

E13=

1

0

0

-

0

0

0

0

0 0,

1

0

1

0

Let a, b e H. = (diagonal purely imaginary matrices 3 x 3 with zero trace). Then the operators cpab: G. - G take the form: gE12 =

bl - b2 E12, a1 - a2

pE13 =

bl - b3

E13

pE23 =

a1 - a3

b2 - b3 E23 a2 - a3

The manifold {cpab} for the normal series forms a three-parameter family

compared with the two-parameter one for the compact series. Not one of the compact operators is normal. Let us put Aij = (bi - bj)/(ai - aj) then x - vP(213 - )23)E12 + ay(A23 - A12)E13 + a$(212 - )13)E23

where x = aE12 + YE13 + yE23. It is clear that <x,. > = 0, i.e. the vectors z are tangent to the spheres with centers at point 0. Let us recall that for so(3) = su(2) the Killing form coincides with a Euclidean scalar product. We have

fi .a1

x+2a=

a

fl

-a

i2a2

y

-fl

-y

iAa3

The integrals are defined by the functions Trace(x + ).a)", 1 < k < 3.

Fig. 31.

188


Computation gives P(x) = a2 + Q2 + y2, Q(x, a) = a2(a, + a2) + #2(a, + a3) + y2(a2 + a3). These integrals coincide with the classical

P = M2-the integral of kinetic moment and Q = E-energy. The integral trajectories are depicted in diagram 31. As opposed to the previous case, this depicts the level-energy ellipsoid E = const and its intersection with the spheres M2 = const. The Euler equation is fully integrable for any a, b e H. The flow .x of the compact series is obtained by passage to the limit from the flow of the normal series. In this classical case the Liouville tori permit a differential-geometric description. This can be confirmed as follows: Consider the curve of intersection of an ellipsoid and a sphere Ax2 + B y2 + Cz2 = 1

I x2+y2+z2=a2 then it can be proved that at all points on this curve 4

(ABC)3/4

PIP2 PI + P2

ABCa2 - BC - CA - AB -

const,

where p, , P2 are the principal radii of curvature of the ellipsoid. This can be verified by direct calculation. It would be interesting to obtain a similar description of Liouville tori in higher dimensions also. For our next example let us analyse the flows of the normal series for

G,, = so(4) c G c su(4) c G = sl(4, C).

The algebra so(4) can be represented in G as the skew-symmetric matrices spanned by the vectors in standard form E;3 = E, + E_,. Let us write x c so(4) in the form x = aEI2 + PEI3 + yEI4 + SE23 + pE24 + EE34 where all the coefficients are real. We recall that the rank of so(4) = 2

and the four dimensional manifolds S2 x S2 are orbits in general position. Let a, b e H4 c su(4) then = a b, (panx

b2

EI2 + fi b, - b3

a, - a2 +S

b2 - b3

a2 - a3

a, - a3 E23 + p

+ y b,

EI3

b2 - b4

a2 - a4

b4 EI4 a, -- a4

E24 + E

b3 - b4

a3 - a4

E34.


189

For every pair a, b in general position we obtain a flow .z on S2 X S2. The integrals will be the functions Trace(x + 2a)", 1 S k < 4 where

x+2a=

Aa 1

a

fi

-at

.ia 2

6

-Ii -6 Aa3 -y

-p

-E

y

E

Aa

Computations give four integrals: h1 = Trace x2, h2 = Trace x4, h3 = Trace x2a, h4 = 2 Trace x2a2 + trace xaxa. The integrals h1 and h2 are constant on the orbits and have the form h1 = a2 + #2 + y2 + 62 + p2 + E2 , h2 = h; + 4(Jibyp - a&ye + aple) - 2(a2E2 + $2p2 + y262).

In fact h2 is the square of the second degree integral q (after computing

the function hi from h2) where q = aE - fip + O. Thus, the two quadratic integrals h1 and q are the generators of the ring I so(4), i.e. any polynomial which is constant on the orbits can be decomposed in terms

of h1 and q. It is easy to prove that h1 and q are independent. The equations h1 = p, q = t where p, t are constants define orbits in general position. These integrals, in particular q, were discussed in [65]. The integrals h3 and h4 are no longer constant on the orbits and have the form :

h3 = a2(a1 + a2) + #2(a1 + a3) + y2(a1 + a4) + 62(a2 + a3) + p2(a2 + a4) + E2(a3 + a4),

h4 = a2(a2 + ata2 + a2) + $2(ai + a1a3 + a3) + y2(al + ala4 + a4) + O2(a2 + a2a3 + a3) + p2(a2 + a2a4 + a4) + E2(a3 + a3a4 + a4). It is easy to prove that the integrals hl, q, h3, h4 are functionally

independent and that the integrals h3 and h4 are in involution on all orbits. 16.3. Cases of complete integrability of Euler's equations on semi-simple Lie algebras

Here we shall give a short sketch of the proof while the technical details

can be found in [89] and [90].

190


THEOREM 16.1 (A. T. Fomenko, A. S. Mishchenko)

(1) Let G be a complex semi-simple Lie algebra and let z = [x, rp.,,(x)] be Euler's equations with an operator of the complex series. Then this system is fully integrable (by Liouville) on orbits in general position. Let f be any

invariant function in the algebra. Then all the functions hx(x, a) = f (x + .la) are integrals of the flow x for any Z. Any two integrals hA(x, a)

and gµ(x, a) arising from the functions f, g E IG are in involution in orbits. The Hamiltonian F = <x, Tx> of the flow x also commutes with all integrals of the form hx(x, a). From the set of these integrals one can choose integrals, functionally independent on orbits in general position equal in number to half the dimension of the orbit. The integral F may be expressed as a function of integrals of the form hx(x, a). (2) Let

G. be a compact real form of a semi-simple Lie algebra and let z = [x, Tx] be a Hamiltonian system determined by a compact series

operator gyp. Then the set of the functions of the form f (x + .la) where f EIG forms a complete commutative set on orbits in general

position in the Lie algebra G, (3) Let G be a compact normal subalgebra in the compact algebra G. and let z = [x, cpx] be a Hamiltonian system of the normal series. Then the set of functions of the form f (x + %a) where f e I G,, forms a complete commutative set of functions on orbits in general position.

The involutivity of the translation of the invariants stems from the results of Section 11. Let us reproduce this proof for the case of a semisimple Lie algebra. We can calculate in its explicit form s grad f for any smooth function f of G, expressing s grad f in terms of grad f. LEMMA 16.4

Any smooth function f of G satisfies the identity

s grad f (x) _ [grad f (x), x].

Proof Let

be a vector in TO, then w(s grad f,

f f (x) _

; on determining the form co we obtain w(s grad f, ) _ = [x, y], and hence _ <s grad f, y> where <s grad f, y>, i.e. = <s grad f, y>. As this identity is true

for any y then s grad f = [grad f, x]. If F = <x, (px> then cpx = grad f (x) and hence - )i = [qpx, x] _ s grad f which proves that is Hamiltonian. Thus, if f and g are two functions of G then { f, g} _ . We have proved the following assertion. Any smooth functions f and g on G fulfil the identity If, g} = (x, [grad f, grad g] >.

LEMMA 16.5


LEMMA 16.6

191

Let f and g be smooth functions on G which are constant

on each orbit. Then [grad f, grad g] = 0. Proof Let x c- Reg G initially. As f and g are constants on each orbit their gradients are orthogonal to the orbit, i.e. they both lie in H(x) and, consequently, commute. As regular elements are everywhere dense the lemma is proved. This lemma can also be obtained from the results of Proposition 2.3. PROPOSITION 16.4 Let f and g be smooth functions of G which are constant on each orbit. Let us consider the functions hA(x, a) =

f (x + .1a), dx(x, a) = g(x + µa) where a c- H(a, b). Then the integrals hx and dN commute. Moreover, {F, hA, } = 0 for any f c IG.

Proof Let us recall that the functions hx and dµ by Proposition 16.1 are integrals of the flow [x, cpx]. By Lemma 16.5 it is sufficient to prove that <x, [grad hz, grad d'] > = 0,

<x, [grad f (x + 1a), grad g(x + µa)]) = 0.

Let us put Y = x + %a then x= Y - .1a, x + pa = Y + va where v = p - .1. Let us suppose initially that v

0.

z = = - . As f e I G then [Y, grad f (Y )] = 0 by Lemma 16.2. As g e I G then by the same lemma

[Y + va, grad g(Y + va)] = 0. Hence [Y, grad g(Y + va)]

v[a, grad g(Y + va)].

Substituting in z we obtain z= - < [Y, grad g (Y + va )] , grad f (Y) V

_ = 0 V

as f EIG. The assertion is proved for A 0- p. If A = p then

192


0 = <x, grad f (x + .la), grad g(x + .la)] > by Lemma 16.6. It remains to be proved that IF, hx} = 0, i.e. to compute L = <x, [(px, grad f (x + tia)] >

as grad F(x) = (px. Let us put Y = x + )a then L = _ - ) - ) + X1.2 = 0

as in the first term [Y, grad f (Y)] = 0, similarly in the second, in the fourth cpa = Da e H(a, b), i.e. [a, 9a] = 0, in the third [a, pY] = adb Y = [b, Y] and again [Y, grad f(Y)] = 0. The involutivity of the integrals of the compact series can be proved in the same way. PROPOSITION 16.5

Let f; g e IG., let us put h,(x, a) = f (x + da),

dµ(x, a) = g(x + µa) where a, b E H (a, b). Then the integrals h,, and d,, commute and IF, h2 } = 0 for any f e IG,,.

The same line of reasoning gives the proof of the involutivity of the

integrals of the normal series. Let us move on to the proof of the completeness of the commutative sets of functions which have been presented. This final part of the proof is technically more subtle and, therefore, we shall restrict ourselves to setting out only the plan of the constructions. Let G be a complex semi-simple algebra, x e Reg G. Let fl , ... _f, e I G

be the complete set of invariants of the algebra. At the point x there arises a set of complex vectors grad h2,k where hA,k(x, a) = fk(x + .la). Let V(x, a) be the subspace of G generated by vectors grad hx.k(x, a). We

must find a lower bound for dim V(x, a). Let us consider the p'2` where qk + 1 = deg fk. Let fk increase with increasing degree. Let N + 1 = q, + 1= deg f, be the highest degree among the generators fk. All the polynomials hx,k can be considered as polynomials of degree N + 1 and of these certain coefficients in high degrees in A are equal to zero. We have grad h.,,k = El,, o U'1' where Uk(x, a) = grad pk(x, a) are polynomials of degree i in x and a. It is clear that UZk(a) does not depend on x as plk is linear in x. The vectors UZk generate the Cartan subalgebra H independent of the choice of x. It is decomposition h.,,k

clear that V(x,a) is generated by the vectors U.


193

For each k the following recurrence relationships on the vectors Uk are satisfied: LEMMA 16.7

[ Uk , x] = 0

[Uk,x] + [Uko,a] = 0

[Uk,x] + [Uk I,a] = 0

[Uk,x] + [U'-', a] = 0 [Uk,a] = 0. Proof By Lemma 16.2 [x, grad fk(x)] = 0. Applying this identity to the functions hx.k we obtain [x + Aa, grad hA,k(x, a)] = 0, i.e. N

x+1a, I U.l' =0. =o

Our assertion follows from this. As H(x) is a Cartan subalgebra, it is possible to construct the root decomposition G relative to H(x) and to select a Weyl basis. LEMMA 16.8

If a EH(x) O+ V(x) then

grad hk,k(x,a)EH(x)Q V+(x), i.e. Uk EH(x) $ V+(x).

Further, let us consider that a e H(x) $ V+(x). Let us consider the simple roots al, ... , a, then every positive root can be given in the form

a = Yi-l m;a; where m; > 0 and m; are whole numbers. The whole number k = k(a) = Y = 1 m; is called the order or the height of the root a. We shall denote the subspace of V+(x) generated by the vectors xa for which k(a) = k by Vk+(x). Then, it is evident that V '(x) = VI+ $ . $ V , + and V I+ is generated by x,,, ... , xa,, i.e. by simple roots. Let us specify a choice of a a c- H(x) $ V + (x). Let a E VI+ and a = Y; = I v; xa, where all v; A 0, 1 < i < r. Then [ Vk+, a] c Vk+1, [H(x), a] c VI+ LEMMA 16.9

Let x, a be chosen as shown above. Then VI+ = [H(x), a]

and ada: H(x) - VI+ is an isomorphism.

194


LEMMA 16.10

[l

,

Let x, a be the vectors shown above. Then V k' i =

a] i.a. ado: 1 k+ --* V +I is an epimorphism.

The following relations: U,° E H(x), Y+ Q ... (D Vi+ are valid for 1 < k , dim H(x) ® V '(x) = 2(dim G + rank G), where V(x) is a complex subspace generated by all the vectors grad hz,k

for all points x e G from an open subset everywhere dense in the Lie algebra G. In order to complete the proof it is enough to note that the elements x and a have been used symmetrically in all the previous statements, as

f

a polynomial of degree q. The proof of the completeness of the commutative sets, constructed above, in the cases of compact and normal series can be obtained from the mentioned scheme, taking into account the involutions defining those two series. For details see [89], [90]. 17. CASES OF COMPLETE INTEGRABILITY OF THE EQUATIONS OF INERTIAL MOTION OF A MULTI-DIMENSIONAL RIGID BODY IN AN IDEAL FLUID

Let us consider the embedding of the type of system mentioned in the

title, in the non-compact Lie algebra of the group of motions of Euclidean space (this has been done in the authors' works [123], [124],

[125]). It turns out that in this case too the method of argument translation makes it possible to construct a complete commutative set of integrals on the orbits in general position.

Let f be an invariant of the coadjoint representation of the group of motions of the Euclidean space Vi". Then the functions LEMMA 17.1


195

f(x + Aa) for any real A are the integrals of the Euler equations z = adQx x, where Q(a, b, D) is the family of sectional operators Q: E(n)* -+ E(n), constructed above.

Proof It is sufficient to check the equality (a(x, Qx)>, df (x + Aa)> = 0 where <x, > is the value of the functional x on the vector . Obviously: A =

_ - - A.

As f is invariant, the first term is zero. Using the definition of the sectional operator Q(a, b, D) we obtain

-AA =

1 ad*, x1,a),df(x +)ia)> + ,

where x1 E K*I, x2 E K*. The second term is zero, as Dx2 E K, a E K*. The first term is equal to . As x2, b c- Ann(a), we have = = 0

because f is invariant. This concludes the proof of the lemma. THEOREM 17.1 (A. T. Fomenko, V. V. Trofimov) (1) The differential equations x = adQx x where Q = Q(a, b, D) on E(n)* is completely integrable on the orbits in general position. (2) Let f be an invariant

function on E(n)*. Then the functions hx(x) = fix + Aa) are motion integrals any number A. Any two integrals hx and gu are in involution on all the orbits of the representation Ad* of the Lie group '(n), while the

number of independent integrals of this type is equal to half the dimension of the orbit in general position. Also, if 0 is the maximal dimension of an orbit in general position of the coadjoint representation, then codim 0 = [(n + 1)/2]. Proof The fact that the given functions are integrals has been checked in Lemma 17.1. Their involutivity was, in fact, shown in Chapter 11. The

only thing to check now is that the shifts of the invariants f (x + .la) comprise a complete commutative set on the orbits in general position.

The statement codim 0 = [(n + 1)/2] can be checked by standard means (see 2.3). We shall give the complete set of invariants of the algebra. For this, we write E(n)* in the matrix form:

196


0

E(n)* =

I

SO(n) 0

\yl...y.

0

Let us call the minor of a matrix x at the intersection of rows i1,...

, i,

and the columns 11, where 1 < i1 < ... < is ,j: M'1 1 < jl < < js 5 n. Then the functions

n,

t

are the invariants of the algebra. The functions with even numbers are

equal to zero and the functions with odd numbers represent the complete set of invariants, as can be found from direct calculations. Let (f) be the full set of polynomial invariants, then N,

f (x + Aa) _

pis(x, a)A', s=0

df e E(n)** = E(n). Suppose df (x + ;.a) =

ui,(x, a)A', s=0

where ui, a E(n). LEMMA 17.2

The following recurrence relations hold: a(x, uio) = 0 a(x, ui1) + a(a, uio) = 0;

a(x, uiN) + a(a, ui,N,-1) = 0 a(a, ui,N) = 0.

Let n = 2s + 1. Consider the complexifcation CE(n). The Lie algebra so(n, C) is simple. Let

so(n,(C)=H®YG; ®EG; i;1

i31

where the subspaces Gt are spanned the root vectors ea with the weight

197


of the root a equal to ± i and H = graded subspace of E(n).

Q

E(n) + = (H (D Ce") $ (Gi $ B1)

I CE 2k + l,zk+ z Consider the

Q... $ (G; $ B3) $ E Gk k3s+1

= Iao QH where Bs+1 -j = C(ez j_ I + ie2j) c C", j = 1, ... , s. The subspace E(n) +

with this grading may be considered to lie in E(n)*. We choose x, a e CE(n)* in the following way : x E K* in general position, a e G; + B1 such that all the components in decomposition upon the root basis in G' and the component of the base vector e" _ z + ie" _ I E C" are nontrivial. LEMMA 17.3

Let a E Gi ED B1 c E(n)* be the element mentioned E(n) we have a(a, Hj) c Hi+1 c E(n)* for i > 0.

above. Then, for H;

LEMMA 17.4 Let x, a c- E(n)* be chosen as indicated above, then the mapping H. - Hi,, c CE(n)* defined by y - a(a, y), y c- Hi c CE(n) is an epimorphism. LEMMA 17.5

The relations (a) ujo E Ho, (b) ujk E Hk hold for any j.

LEMMA 17.6

The vectors ujk generate the entire subspace Hk. We

conclude that the dimension of the subspace, generated by df(x + .la) is at least dim E(n)' = sz + 2s + 1; but for complete integrability we need

codim 0 + z(dim E(n)* - codim 0) = s2 + 2s + 1 functionally independent integrals on G*. The theorem, therefore, has been proved in the case n = 2s + 1. The case n = 2s could be examined in a similar way. We shall not dwell here on the technical details.

198


18. THE CASE OF COMPLETE INTEGRABILITY OF THE EQUATIONS OF INERTIAL MOTION OF A MULTIDIMENSIONAL RIGID BODY IN AN INCOMPRESSIBLE, IDEALLY CONDUCTIVE FLUID 18.1. Complete integrability of the Euler equations on extensions fl(G) of semi-simple Lie algebras

The results given here have been obtained by V. V. Trofimov. Consider the embedding of the equations of magnetic hydrodynamics into the Lie

algebra Q(so(n)) described in 7.4. It turns out that in this case the method of tensor extensions of Lie algebras makes it possible to construct a complete commutative set of integrals on the orbits in general position. We shall study first the Euler equations on KI(G)* with the "complex" series' sectional operators, constructed earlier.

Let G be a complex semi-simple Lie algebra and R(G) the set of functions on G* representing shifts of invariants F of the coadjoint representation G, i.e. it consists of the functions h(x) = F(x + .la), A E C, a E G* fixed covector. Applying algorithm (91) from Theorem 13.1 to the functions h(x) ER(G), we can construct functions h(y), h(x, y) on f )(G)* and obtain a set of functions R(f2(G)) on space fl(G)*. We remind the

reader that the construction of Section 13, applied to the Lie algebra Q(G) = G + EG, EZ = 0 enables us to construct functions F1(x, y), ... , F,(x, y) E Coo(S1(G*)), with F;(x, y) = (8F;(y)/8yj)xj (x, coordinates in G*

and y; in EG*), using functions F1(x), ... , F,(x) E C'(G)*. If F; is in involution on all the orbits of the coadjoint representation of the Lie group (! associated with Lie algebra G, then the functions (y), F,(y), P, (x, y)..... F,(x, y) are in involution on all the orbits F1(y), of the coadjoint representation of the Lie group D (O) associated with Lie algebra K )(G). If F; are functionally independent on fl(G)*, then Fl(y), Fj(x, y) are functionally independent on f)(G)* too. (See Section 13, Theorem 13.2).

Let a function h be functionally dependent on the family of functions a(f (G)), then the Euler equations z = a(x, dhx), x e Q(G)* are a completely integrable Hamiltonian system on all the THEOREM 18.1

orbits in general position of the coadjoint representation Ad* of the Lie group KI(T)) associated with Q (G).


199

Proof The involutivity of the given functions follows from Theorem 13.2. Let FI (x), ... , FN(x) be a complete set of involutive functions on G* (constructed above), then FI (y), ... , FN(y), FI (x, y), ... , FN(x, y) are functionally independent. In order to achieve complete integrability it is necessary to have s integrals, where s = Z(dim S2(G) + ind S2(G)) = Z(2 dim G + 2 ind G)

= 2['I(dim G + ind G)] = 2N according to Theorem 13.3. This concludes the proof. THEOREM 18.2

Let G be a complex semi-simple Lie algebra, x =

a(x, C(a, b, D)(x)) the Euler equations on KI(G)* for x e O(G)* with the complex series operator, then this system is completely integrable in the

Liouville sense on all the orbits in general position of the coadjoint representation of the Lie group 0((1i), associated with fl(G). Or, more precisely, let F(x) be any smooth function on i2(G)*, invariant with respect to the coadjoint representation of the group f)(65 ), then all the functions F(x + .la), A e C are first integrals of the Euler equations for any A e C. Any two of those integrals F(x + 2a), H(x + pa), A, p c- C are

in involution on all the orbits with respect to the Kirillov form. It is possible to choose from the given set of integrals a set of functionally independent integrals equal in number to half the dimension of a general

position orbit of the coadjoint representation of the Lie group S2(6).

Proof The involutivity of the shifts of the invariants is a well-known fact (see Section 11). We shall prove that the shifts of the invariants are first integrals of the Euler equations . = a(x, C(a, b, D)(x)), x e S2(G)*. To do that it is enough to check that = 0, where <J, x> is the value of the functional x on the vector f We have:

= (dF(x + .la), a(x + .la, C(x))> - = - .

The first term is equal to zero, as F is an invariant, thus

2,

as DheKer4a, where x=x'+h, x'eV+eV, heH+eH, H is a

200


Cartan subalgebra in G and V is an orthogonal complement with respect

to the Cartan-Killing form, therefore

A

A = 0. We used here the facts that h, 2a e Ker p(b) and that, as F is an invariant, the expression a(x + Aa, dF(x + Ad)) = 0 is zero. We have, thus, proved that the shifts of the invariants are integrals of the Euler equations. We

shall prove now the functional independence of the shifts of the invariants. Let F1(x + Ad), ... , FN(x + Ad) be a complete set of functions in involution for G. Then, using these, a complete set of functions in involution on (2(G)* can be constructed: OF, (y + Ad) Ad),

.

x;, ...

ClYr

(y

x; . (1)

Let us examine now the invariants Ad* for S2(G): F1(Y), ... , FN(Y),

ay,Y)

x;, ... , ay Y) x;

(2)

we should take shifts of these invariants along a vector a where a is taken

from eG*, then after shifting functions (2) along the vector a, we,

obviously, obtain functions (1) and the latter, as we know, are functionally independent. The theorem has been proved in full. In the same way as for the complex operators the following theorems can be proved. THEOREM 18.3

Let function h be a function which is functionally

dependent on the family of functions R(S2(G,,)), than the Euler equations

x = a(x, dhs), x E )(G )* are a completely integrable Hamiltonian

system on all the orbits

in

general position of the coadjoint

representation of the Lie group f2(6.), associated with S2(Gu). Let G,, be a compact form of a complex semi-simple Lie with f2(Gu). Then this system compact series operators C(a, b, D): S2(G )* is completely integrable in the Liouville sense on all the orbits in general THEOREM 18.4

algebra, z = a(x, C(a, b, D)(x)) the Euler equations on

position of the coadjoint representation Ad* of the Lie group S2(0 ),

201


associated with the Lie algebra SZ(GJ. Or, more precisely, let F(x) be any then smooth function on SQ(G.)* invariant with respect to

all the functions F(x + Aa) are integrals of the Euler equation for any 2 E R. Any two such integrals F(x + Aa) and H(x + µa) are in involution on all the orbits with respect to the Kirillov forms. From the mentioned set of integrals one can choose functionally independent integrals equal

in number to half the dimension of an orbit in general position of We shall examine now the construction of integrals of the Euler equation with the "normal" operator series. Let f (x) be a function, invariant with respect to the Consider functions hs(x) = f (x + 2a)I Then the functions h2 are, for any A E U8, first integrals of the Euler equation z = a(x, C(a, b)(x)) on where C(a, h) is a "normal" series operator, a e iHo, h e iHo + EiH'. PROPOSITION 18.1

coadjoint representation of the Lie group

Proof Suppose that we have a differential df(x + Aa) in G,,, then df I G.(x + .la) is an orthogonal projection df (x + Aa) on G In the case

G + eG we obtain df (x + a ,a) = vl + Eve, vi e G (i = 1, 2) and df I n(,.)(x + Aa) = n(vI) + eir(v2)

where n

is

an orthogonal

projection on G. with respect to the Cartan-Killing form. Then = (df I II(G.)(x + .la), [Cx, x])

= (7C(vI) + En(v2), [Cx, x])

= (v1 + eve, [Cx,x]); as [Cx, x] e G it is possible, therefore, to add any term to the first factor, orthogonal to G. We have thus: = (df(x + via), [Cx, x])

= = 0,

as has been proved already in the case of the "compact" series operators. Proposition 18.1 has been proved. PROPOSITION 18.2

Any two integrals of Proposition 18.1 "normal"

series type are in involution on all the orbits Ad* of the Lie group with respect to the Kirillov form. Moreover, the number of functionally

independent integrals, given by Proposition 18.1 is equal to half the

202


dimension of an orbit in general position of the coadjoint representation Ad* of the Lie group associated with Lie algebra S2(G.).

Proof We reduce all cases to that of a semi-simple "normal" series of operators. We have G. c G. Let f (x) be an invariant of G,,, then on G.

the complete set of functions in involution (see [89], [90]) can be obtained as the restriction to G. of functions f(x + .la). Applying to these functions the (21) algorithm, one finds a complete set of functions in involution on These functions have the form fl Y + ).a)/G (as

the operations of restriction and substitution are commutative) and W G. (Y + .la)/?yj)x;. Let us check that those functions are identical to the functions mentioned in the proposition. According to Theorem 13.1 the S2(Gu) invariants have the form f (y) and (Of (y)/8y;)x;. It is clear that the shifts of these functions along a suitable a after restriction to S2(G.) will give the necessary results. Thus, the algorithm (91) leads to the necessary set of functions, which proves the proposition.

The results are summarized in the following theorem.

Let G be a normal compact subalgebra in a complex semi-simple Lie algebra G; z = a(x, C(a, b)(c)). The Euler equations on with the operators of the "normal" series C(a, b):

THEOREM 18.5

S2(G,,)* --

are a completely integrable system in the Liouville

sense on all the orbits in general position of the coadjoint representation Or, more of the Lie group S2(6j, associated with the Lie algebra f precisely, let F(x) be any smooth function on Q (G.)*, invariant with respect to Ad*(S2((6.)), then all the functions F(x + Aa)I ((;,) are integrals of the Euler equations for any a. E R. Any two such integrals F(x + .la), H(x + µa) are in involution on all the orbits Ad* One can choose

from the indicated set of integrals a set of functionally independent integrals equal in number to half the dimension of an orbit in general position of the representation Ad* In a way similar to the complex case the following theorem can be proved.

Let function h be functionally dependent on the functions of the family, then the Euler equations )i= a(x, dhx), x e f2(G.)* are a completely integrable Hamiltonian THEOREM 18.6

system on all orbits in general position of the coadjoint representation Ad* of the Lie group C)(6 ), associated with Lie algebra S2(G ).

203


18.2. Complete integrability of a geodesic flow on T*f2(6)

Let G be a complex semi-simple Lie algebra, with compact form, G. and

G a normal compact subalgebra in G. Let f2(6), f2(6 ), f2((%) be Lie groups corresponding to the Lie algebras f2(G), f2(Gu), f2(G ). Consider

the cotangent bundles T*f)(6),

Let quadratic

forms (so(n) ® 1fi", so(n - 1))

is practicable for other Lie algebras as well, e.g. (su(n + 1), su(n - 1)) - (su(n) (D C", su(n - 1)) (u(n + 1), su(n - 1)) -, (u(n) (D C", su(n - 1))

(sl(n + 1, li), sl(n - 1, R) - (sl(n, li) (@ i", sl(n - 1, R)).

For all right-hand pairs we can construct entire involutive sets of functions. Let us formulate this in a way similar to Theorem 19.1. THEOREM 19.3 Let 0: G -+ G. be the contraction of the algebra G. If we take (so(n) ® R", so(n - 1)), (su(n) ® C", su(n - 1)),

(u(n) (9 C", su(n - 1)), or (sl(n, R) ® li", sl(n - 1, 118)) as a pair (GB, H),

then the Hamiltonian systems are given by z = s grad B* V, the Hamiltonian

V = E (a;,(X(`)W", X1 )kk + fl;;P(X11)W(P , XU)

being

constructed for the pairs of the algebras (so(n + 1), so(n - 1)), (su(n + 1), su(n - 1)), (u(n + 1), su(n - 1)), (sl(n + 1, l8), sl(n - 1, li)).

We can prove this theorem following the standard technique given above and using as well the dimension of the orbits in general position of coadjoint representation of semi-direct sums, in connection with this see [118].

19.3. Functional independence of integrals

Note that the functions in the set ¢ = {F;, i = 1, ... , rk W, f e (H*)*, 1 < i < dim H}, where the Ft are similar functions are functionally

independent on the space G* dual to the algebra G. Functional independence can be proved in two ways: either (a) by showing that the

invariants defining the orbits are functionally independent of the functions in the set 0; or (b) if this cannot be done, for obvious reasons, by proving that the skew gradients of the set of functions 0 are linearly independent almost everywhere. The semi-simple case

In 19.2 examples of Hamiltonian systems on semi-simple algebras are


213

considered. As is known, in this case the orbits of the (co)adjoint representation of the group Exp G are given by traces of powers of the (co)adjoint representation of Lie algebra G. It is enough, then, to show functional independence within the set 0, when the F; are canonical H-

invariants. It is clear that then the similar linear combinations of invariants included in the set 0 as the functions Fi will comprise a functionally independent set. THEOREM 19.4

The following sets of functions 0 are functionally

independent on orbits in general

position of the

coadjoint

representation of the corresponding Lie algebras: (1) so(2n): 0 = {elements so(n - 2) viewed as linear forms on so(n - 2)* and either (a) XW2pX`, or (b) X'W2pX", (c) X'W2pX" or (d) 0 5 p , 1. Taking into account show below that the the projection of the E su(m)9 we obtain that d,c that onto T, generates T +. As dim T + = i dim 0, the differentials theorem is proved. R" ,q Esu(m)7+_1. Let

generate Tq

Let us prove in addition that the differentials (q >, 1). As

E su(m)q 1, m-q dz pa.9 =

CJI(a)Un

C, 9(a)

0.

j=1

It is enough therefore to show that the matrix Vq, l' = U7 is nonsingular, where n(1) < n(2) < < n(mq) is a maximal sequence of q (it follows from the definition of the indices such that xn(j) < sequence n(j) that mq = dim Ti). LEMMA 20.1 Let y1, ... , yN be indeterminates, for 1 <j, p < N, j + p < N, let W?(y) be the homogeneous component of i in the formal series Wp: W"(y) = (1 - y)-' - (1 - yj+p)-1. Let WP be the

(N - p) x (N - p) matrix of these polynomials 0

i < N - p - 1,

1<j =2=0 as dyf eGu.

It is known that dxI4 = d,lx (for any set of functions a, by definition dx a = {linear span dx f, f e tR}). Let 0 be an orbit passing through x, as follows from Theorem 20.1 it is possible to select an element a in such a way that dim xT(dXI4) = i dim 0 and therefore

dim nT(d,I,) _' dim 0 ='(dim G. - dim G;) too. Therefore we can choose from the set Ix functions fk + i , , fk +s where s ='(dim G,, - dim Gx), so that TT(da f) (i = k + 1, ... , k + s) generate T +. As T + c T = Ts01 Gv and d, f E Gv (i = 1, ... , k) all the functions fi , ... , fk +, are also independent. Their number is equal to Therefore this is a completely involutive set on G. Statement (i) of the theorem is proved. (ii) Wodd = O G2n+1 Wed = W o G+ d.

For

any

subspace

WcG

let

It has been proved in Theorem 20.1 that a vector a e su(m) can be chosen

from so(n) such that the projection of the differential dxla on T is nT(dxl,) = T+. As dxl, = dalx nT(dalx) = T. As dalx = nT

(dalx) = nTpdd ° nc,(dalx) = nc. ° nTo(daIx) = n,q(T+dd) =

Todd

219


LEMMA 20.4 We claim that !(dim G' + rg

dim(Gx)o+da

Proof As Gx = so(k1) ®... (D so (k,), GU = s(u(kI) (G4)

odd

®... (D u(kt)),

= (su(kl) ®... ® Su(kt))

dl

it is enough to show that z(dim so(k) + rg so(k)) = dim(su(k))'d but this was proved in [90]. The lemma is proved. It is possible to construct on Gx an involutive set

consisting

of k = !(dim G' + rg Q independent functions. In the same way as in (i) we shall extend the functions f to G. taking them to be constant along vectors orthogonal to G. As in (i) we have {Ix, j } = 0. And indeed,

{f,}(y) = =

_ -A<x, [!(d, - dyf-z),dyfi]> _ = 0 as

dyjEG'. It has been shown above that it is possible to select from set 1

functions fk+I,

.

, fk+s

in such a way that fk+s}) = T+odd.

fk+, are independent. Their dd. But, as was number is k + s = dim T+dd + dim(Gx)odd = noted above, dim(G) °+dd = !(dim G + rank G.). Therefore the set 11, ,fk+S} form a ,fk+s together with the set Ix u {jk+I,

As T+dd 1 G all the functions A

,

completely involutive set. The theorem is proved. 20.2. Completely involutive sets of functions on affine Lie algebras

Let G be a Lie algebra and let G = H ® V where H is a subalgebra and V is a commutative ideal. Let p = ad,, H be the adjoint representation of H on V. G in fact is the split extension of Lie algebra H determined by the representation p. Such Lie algebras are called affine Lie algebras.

220


For any representation p of a Lie algebra H in a vector space V the number ind p = {the codimension of an orbit in general position} (i.e. of an orbit of the action of the group , corresponding to the Lie algebra H)

is called the index of the representation. The index of the coadjoint representation ind G = ind ad* is called the index of the Lie algebra G. Let G be a Lie algebra, W c G a vector subspace, x e G* an element of the space dual to G. We define the vector subspace Wx = Ann(W, x) =

{g a W, adg x = 0} c W. If W is a subalgebra in G, then Wx is a subalgebra too. We shall need, when calculating the index of an affine Lie algebra. Let G be an affine Lie algebra which is the split extension of Lie algebra H determined by a representation p of the

THEOREM 20.3 (see [118])

algebra H on V. Then for an element x e G* in general position the equality ind G = ind Hx + ind p* holds, where p* is the representation of H on V*, dual to p. Proof Let x e G*, x = xH + xv, xH e H*, xv e V be an element such that the following conditions are satisfied: (a) ind p* = dim H dim Hx; (b) ind Hxv = inf find HY, y e V*); (c) dim Ann(Hxv, xIH=,,) =

ind Hxv. All such elements x constitute a non-empty Zariski-open

set in G*. Thus, the general position elements in G* satisfy the conditions (a)-(c). Therefore, in order to prove the theorem it is enough

to check that the equation dim Gx = ind Hx + ind p* follows from (a)-(c). Let g = gH + gv a Gx, then ada (xH) + ad* (xv) = 0

(2)

ad* (xv) = 0.

(3)

It follows from (3) that gH a Hx°. Consider the restriction of equation (2)

to H'v: = - = 0;

Hxv> = <xH, ad* (Hxv)) = <XHI H=v, ad9H(Hxv)>

= 0and gk E (Pk(Gx) _ ((P')k(Ann(Hx'', x')) O V,

gk = gH + gv, g' E (9')k(Ann(Hxv, x')), g,, E V. We want to prove that


223

nH((p(g)) = cp'(gH) a consequence of which is the equation (5) fork + 1. According to the definition of the operators cp and (p' we obtain pp(gk) = {gk+1, (ad

(p'(gH) =

g+1)* x = (ad gk)* a} ,

{gH+l, (adH gH+1)* x' = (adH gk)* a'}.

Let gk+' E cp(gk), which means that the following equations hold:

(adgH+l)*xH + (adgV+l)*xv = (adgH)*a,

(6)

(ad gH+')* xv = 0.

(7)

We obtain from

(7) gH+ 1 E Hxv, from which it follows that ((ad gH+ )* xH) I H=v = (ad gk,+' )* x'. As for gH, gH E H"' so that ((ad gH)* a)I H=v = (ad gk )* a'. In addition, ((ad gV+')* xv)I H=v = 0,

therefore restricting elements of G*, which appears in (6) to Hxv, we get gk+ 1 E gp(gk) if (ad gk,+' )* x' = (ad g1)* a'. Thus, then nH(gk+1) = gH+1 E rp(g'). On the other hand, let gH+' a (p'(gk), then it is possible to choose gk + 1 E (p(gk) in such a way that 7CH(gk + 1) = gH+' And indeed, it follows from gH+I e gp'(gk) that

((ad g')* a - (ad

gH+ 1)*

xH) I H=v = 0.

Therefore (Lemma 20.5) there is an element gkv+' such that (ad g ,+ l) * x v = (ad gH)* a - (ad gH+' )* xH . k+1 and n1(gk+1)=g'+'. Thus E co(gk) + nH((p(gk)) = 4'(gH) and equation (5) has been proved for k + 1. The

Hence

gk+1 =

gH+1

lemma is proved.

We obtain from Lemma 20.7 dxI,(G*) + V = dx,Ia,((Hxv)*) O V,

therefore

dim(d., I, + V) = dim dx,I', + dim V = '(dim Hxv + ind Hxv) + dim V

='(ind Hxv + dim G + dim Hxv + dim V - dim H). Note now that ind p* = dim V - dim H/Hxv, therefore

dim(dxI, + V) ='(ind Hxv + dim G + ind p*) ='(ind G + dim G)

224


in accordance with Theorem 20.3. That means that I. + V is a completely involutive set on G*. The theorem is proved. COROLLARY

Let G be the extension of a compact Lie algebra H

determined by a representation p: H -p gl(V), Then for an element a e H* in general position the set 1,(G*) u V is a completely involutive set.

Proof For any y E V* the subalgebra HY is reductive. Therefore, Ia.(H'')

is a completely involutive set for an element a' c- (HY)* in general position. That polynomial invariants exist follows from the fact that [G, G] = G. REMARK The stabilizers Hy of elements yin general position have been studied, for example, in the work [31]. This makes it possible to build

completely involutive sets on extensions of some non-compact semisimple Lie algebras too. The possibilities of using Theorem 20.4 are not restricted to extensions of semi-simple Lie algebras only. It gives the existence of completely involutive sets on some Lie algebras with noncommutative radicals too. 21. COMPLETELY INVOLUTIVE SETS OF FUNCTIONS ON EXTENSIONS OF ABELIAN LIE ALGEBRAS 21.1. The main construction

In this section we give the results of Le Ngok Tyeuyen. We will use here the method of constructing involutive sets given in Section 13. Let % be a connected Lie group, G its Lie algebra and G* the dual space G. We shall

use, for the sake of simplicity, the notations Ad* f = g x f, g e (fi, fEG*, ad* f = x f, E G, fEG*. The number r = ind G = dim G* - sup fEG* dim 0(f) is called the index of Lie algebra, where 0(f) is the orbit of the coadjoint representation passing through f E G*,

r= ind G = inf dim G(f) where G(f) = { E G ad* f =c x f =01. The point f e G* is called a point in general position if dim 0(f) = dim G* - r or, equivalently, dim G(f) = r. Let the Lie algebra G be

decomposable as the direct sum of an ideal Go and an Abelian subalgebra H: G = Go + H, let 6o and .5 be the connected Lie subgroups of (fi corresponding to Go and H. We obviously have (for a

given decomposition) that G* is isomorphic to the subspace G*


225

of G*, Go = { f E G* I f I H ° 0} and H* is isomorphic to H* _ {h E G* I hlGo = 0} c G*. We can therefore consider G0* and H* as subspaces of G*. The representations Ad*:60 - GL(G0*) and ad*: Go End(Go*) are defined. We introduce the notations Ada f =

g Of if g e 60, f c -Go* and ad* f = ®f if E Go, f c G0*. Thus, if

fEG**cG*, ge60c6, EGo CG then x fEG*, ®fEGo*, g x f E G*, g® f c- Go and, generally, x f and ® f as well as g x f and g ® f do not coincide in G*. Let n0 be the projection of G* onto Go along H* (n1 the projection of G* onto H* along Go), then we obtain the following simple relations

no( x f) = Of, EGo, LEMMA 21.1

na(g x f) = g Ox .f,

.fEGo,

gE60.

(1)

Let h e H*, fEG*, then O(f + h) = 0(f) + h, i.e. the

orbit of the coadjoint representation passing through the point f + h can be obtained by a translation of the orbit passing through f along the vector h.

Proof We have g x (f +h)= g x f+ g x h, g E 6, f E G*, h E H. As H is a commutative subalgebra and Go an ideal in G, g x h = h for all

g c 6, i.e. 0(h) = {h}. It follows from that fact that g x (f + h) _ g x f + h which was to be proved. COROLLARY 1

Let the space H* in G* be obtained by translating space

H* along the vector f, i.e. f' E H* if and only if no(f') = f. Then 0(f) n H* is a subgroup with respect to addition in H* . Indeed, if f + h; E Off ), h; E H*, i = 1, 2 then it follows from the

lemma that f + hI + h2 and f - hl belong to 0(f). COROLLARY 2 It follows from the lemma that f c- G* is a point in general position if and only if n0(f) is a point in general position for G*. Because the set of all points in general position for any Lie algebra is open and everywhere dense, it is not difficult to deduce, using this fact, that there is an open and everywhere dense set Wo in G*o such that for any f e Wo, f is, at the same time, a point in general position both in G* and G*o. Let f c- W0 c Go, i.e. f is a point in general position both for G* and

Go, and let 00(f) be an orbit of the coadjoint representation of 60 on Go, i.e. 00(f) = {6 ® f I g e 60} = 60 ® f. Then the tangent space to ®f I E G0} = Go ®f. 00(f) at the point f is the space Tf 0(f) On the strength of equation (1) we have

226


00(f) = no(u(f)),

(2)

where

u(f)={gx Go ®f = Tf00(f) = 7C0({c x f,

E GO})

LEMMA 21.2 Let f c- W0 c G**, h e H*. Then if x f = f + th. for all the t e E8: (Exp

7ro(Go x f).

x f = h, e G0 then

Proof If h = 0, then our assertion follows from the definition; we can thus assume h 0.{ In addition, it is enough to prove the lemma for the case when dim H = 1, as the general case can be reduced to this by supposing G* = Gi + R h (i.e. by considering a new decomposition of G as the direct sum of an ideal G1 and a one-dimensional subalgebra). Let T(t) = Exp(tf) x f be a curve in G*. We have to prove that y(t) = f + th. It follows from the condition x f = h, e G0 and equation (2) that ® f = ic0(h) = 0, therefore (Exp t t a 60. But n0[(Ex0 x f ] _ (Exp ®f, therefore y(t) _ (Exp x f E H f. The latter means that y(t) = f + a(t)h, a(t) E R. Let t1, t2 e ER, then

y(t1 + t2) = [Exp(t1 + t2) ] x f = (Exp t1) x [(Exp i2) x f] = (Exp t1) x If + a(t2)h] = f + a(t1)h + a(t2)h because

(Exp t1) x h = h. Therefore a(t1 + t2) = a(r1) + a(t2). 0 = h or i(t)j, 1, therefore a(t) = t, which was to be

Moreover y(t)1

proved. COROLLARY

Let f c- W 0 c Go, then 60 x f

{ f+ (G0 x f n H*)}

where (j0 is the Lie group corresponding to G0. There is a subspace Vo c H* such that Vo = G0 x f n H*, f e W1 where W1 is open and everywhere dense in G. In particular, Vo c O(f) n H f for any f E W1. LEMMA 21.3

Proof We shall prove the lemma using the method of induction. Let G

be decomposable as a direct sum of an ideal G0 and an Abelian subalgebra H, dim H = 1: G = G0 + H. Consider the restriction of the coadjoint representation on G* to 60, then G* is partitioned into the orbits of this action60 x f, f e G*. Similarly to Corollary 2 of Lemma

21.2 we have: f E G* is in general position (60 x f has maximal dimension) if and only if 7r0(f) is in general position for this action. This means that there is in Go an open everywhere dense set W' of points in


227

general position for the action of 60 on G*. Let Wl = Wo n W'. It is obvious that W1 is open and everywhere dense in Go also. Let us take any two vectors fl, fl e W, then fl, f2 are in general position both for G* and G*, i.e. dim(G0 x fl) = dim(G0 (9 f2) and dim(G0 x fl) = dim(G0 x f2). Suppose that G0 x f1 n H* A 0 thus G1 x fl n H* = H* as dim H* = 1. According to (2) we have: implying n0(Go x fl) = G0 ® f dim G0 0 f1 = dim G0 x f - 1, therefore

dim G0 ®f2 = dim G0 ®f1 = dim G0 x f1 - 1 = dim G0 x f2 - 1, but G0 ® f2 = n0(G0 x f2), thus G0 x f2 n H* = G0 x fl n H* = H*, which means that there is a subspace Vo c H* (here either Vo = 0, or Vo =H*) such that G0x fnH*= l' for all the feW1. Suppose we have proved the lemma for the case dim H = m. Consider the decomposition G = G0 + H, G0 an ideal, H an Abelian subalgebra, dim H = m + 1 . Let us choose in G a basis ell e2, ... , ejo,

e0+1..... ejo+m+I for H. We denote the conjugate basis by G* as eI , ejo+m+I where e'(ej) = 5 , i, j = I.... , j0 + m + 1. Consider G1 = Go + <ejo+I, .. , ejo+m) = G0 + H1 where <ejo+1, ...., e,,+,> is the subspace of H with basis ejo+I, ... , ejo+m It is obvious that G1 is a Lie algebra which can be decomposed as the direct sum of an ideal G0

and an Abelian subalgebra H1, dim H1 = m. In accordance with the

inductive hypothesis, there is V0* in H* = <ejo+I, .... ejo+m> such that of/the G0 x (1) f n Hi = V0* where i; x (1) f denotes the action vector e Go on the vector f c- Wl c Go under the representation of G1 in Gi, where Wl is an open and everywhere dense set in G. According to the construction, we have a decomposition

G=G1+<en>=G1+Re,, =G1+H2, where G1 is an ideal in G, H2 an Abelian subalgebra, dim H2 = 1. In the same way as when dim H = 1, we find in Go an open and everywhere dense subset W' such that dim(G0 ® fl) = dim(Go ®f2),

dim(G0 x fl) = dim(G0 x f2) for all the fl, f2 E W', but G0 ®f = n0(G0 x fl), i=1,2, therefore dim(G0 x f1 nH*)=dim(G0 x f2 r) H*) for all fl, f2 e W c G*0. Write W1 = W' n W1, then W1 is the open and everywhere dense set in G. Let TII be a projection of G* on Gi along H2, then from relation (2) we obtain :

228


Go x(1)f = nt(Go x f),

(Go x(1)f)nHip=n1[(Go x f)nH*],

i= 1,2, f1,f2EW1.

We have the following equations:

dim[(Go x fl) n H*] = dim[(Go x f2) n H*]

[(Go x(1)f) nHi] = n1[(Go x f) r) H*],

i = 1, 2;

dim[Go x (1) fl) n Hi] = dim[(Go x (1) f2 n Hi] = dim Vo for any f1, f2 E W1. But H* = Hi + H2* and dim HZ = 1, therefore it follows from these equations that there is a subspace Vo of H* such that Go x f n H* = Vo for all f E W1, which was what we had to prove. We introduce the notation k = dim Vo , where Vo = Go x f n H*, f e W1. It follows from Lemma 21.3 and Corollary 1 of Lemma 21.1 that for f e W1 there is a basis e1, ... , ego, ego+1, . . . , e,, of G such that Go = <e1, e2, ... , e,0>, H = <eio+1, ... , e.) and REMARK

k

60 x f n H* = f + > Re* +1 + i=1

s(f)

7Le*+k+s

s=1

where k = dim V* does not depend on f and Ek= 1 Re*+ 1 = V*. Let, as before, G be expanded as the direct sum of an ideal Go and an Abelian

subalgebra H: G = Go + H, let 6, 60,.5 be the connected Lie groups corresponding to these Lie algebras. Consider the restriction of the coadjoint representation 6 to G0*, so that 6 acts naturally on G0*. We denote by 6 ® f for f E Go the orbit of the point f under this action.

Obviously, 60 ®f contains the orbit 0,(f)=000f of the coadjoint representation of 60 on G. According to equation (2) we have 6 (& f = no(6 ® f). There is in G*0 an open and everywhere dense set W2 of points in general position for the action of (5 on G. We denote its intersection with W1 by W: W = W1 n W2. Then W is also an open and

everywhere dense set in G. In a way similar to Lemma 21.3, the following lemma can be proved. LEMMA 21.4

There is in the space H* a subspace V*(V*

Vo ),

dim V* = k + I such that for all f E W we have:

6x fnH*=V*.

(3)

It follows, obviously, from Lemma 21.4 that V* may be written as a direct sum of two subspaces V* = Vo + Vl*, where Vo is as constructed

229


in Lemma 21.3, dim V1* = 1.

Let G be any Lie algebra, n = dim G, r = ind G. Let m smooth functions be defined on G*:F1(f),...,Fm(f), F.(f)eC°°(G*), 1 < i < m. We remind the reader that such a set of functions is called a completely involutive set for G, if the functions F1, . . , F. are mutually in involution on all orbits, i.e. (Fi(f ), F1(f )} - 0, 1 (2jo+r+2-n)=z(2jo+r+2s-n) and in this case the inequality (4) is proved also. Suppose that the inequality (4) has been proved for all s < so (so 3 2) and let us prove it now for s = so. Let so = k0 + l0; consider the two cases: (a) k0 0, (b)

k0=0,10=s0

0.

The first case. We choose for G a basis e,, e2, ... , ejo, ejo+, , ... , e Go = <e1, ... , ejo>. Let

such that G = Go + H, H = (ejo+I, ... ,

H'=(eo+i, -e.>, e,*eGox f r) H*. Write G°(el,...,ejo,ejo+ H, = G = G1 + H1 then G1 may be decomposed as a direct sum of an ideal Go and an Abelian subalgebra H2 = G1=Go+Hz. Let G1x(1) f be the coadjoint representation of Lie group 6, in G'. In accordance with equation (2): Go x (l) f* = 7rG,(G0 x f), where ire, is the projection of G* on G' along H' and Go x(1) f nHz = n6l(Go x f n H*). The construction is such

that Hi =

c Go x f n H*, therefore

dim(Go x (1) f n HZ) = dim(G0 x fnH*) - 1 = k0 - 1.


231

Similarly, from the fact that G1 x (1) f = it,,(G1 x f) it follows that dim[(G1 x (1) f) n H*] = 1o + ko - 1 = so - 1. Therefore, according to the inductive assumption, we have Mo Z(2j0 + 2(s0 - 1) + r1 - n1), n1 = dim G1, r1 = ind G1 but -dim(G1 x(1) f) 3 -dim(G x f) + 1, therefore nl = n - 1;

r1 = nl - dim G1 x(1) f > r. Thus, Mo%i(2jo+2(so- 1)+r-n)+Z.

i(2j0 + 2(so - 1) + r - n) are integers we Having noticed that M0 and obtain Mo ?(2j0 + 2s + r - n). Likewise, for the second case, when ko = 0, l0 = so: Hi = <e*> = G x f n H*, dim(G0 x(l) f n HZ) _ k0 = 0 and dim(G1 x (1) f n HZ) = 10 - 1 = so - 1. The inequality (4) is proved. COROLLARY

Let G be a Lie algebra of the "radical" type, i.e. G may

be decomposed as the direct sum of a nilpotent ideal Go and an Abelian subalgebra H : G = G0 + H, then there is a complete commutative set of functions on G.

Proof It has been shown in [134] that for any nilpotent Lie algebra

there is a completely involutive set of functions F1(f ), ... , Fm(f ) (m = '(n0 + r0)); the corollary follows from this assertion and from our theorem.

Let G be any Lie algebra of dimension n, r = ind G, and let I1, ... J, be a complete set of invariants for G (i.e. I. is invariant under the coadjoint representation). It is obvious that for any F(f) E C`°(G*):

{F(f),1;(f } - 0, i = 1, ... , r. This means that if a set of smooth functions F,(f ), ... , F(f) on G* is complete and commutative, then the set F1,... , Fm, Ii,. .. ,1, is commutative. From Theorem 21.1 immediately follows the theorem:

Let the Lie algebra G be decomposable as the direct sum of an ideal Go and an Abelian subalgebra H: G = Go + H and let I1(f),. . . , I,(f) be a complete set of invariants for G(r = ind G). Let F1(f ), ... , Fr(f) be a completely involutive set of smooth functions on THEOREM 21.2

G. Then the set F1(f ), ... , Fm(f ), 11(f),. .. , I,(f) is a completely involutive set on G, where F1(f ), ... , Fm(f) are the liftings of functions F1 , ... , Fm to G*. COROLLARY 1

Let G be decomposable as the direct sum of a nilpotent

232


ideal and an Abelian subalgebra : G = Go + H. If there is a complete set of invariants for G, then a completely involutive set of functions on it exists.

Let G = Go + H, Go a nilpotent ideal, H an Abelian subalgebra and r = ind G = 0, then a completely involutive set of

COROLLARY 2

functions on G exists. COROLLARY 3

Let BG be a Borel subalgebra in a semi-simple complex

Lie algebra G. It is obvious that BG = Go + H is the direct sum of a

nilpotent ideal Go and an Abelian subalgebra H. It follows from Corollary 2 that there always is a completely involutive set of functions on G for Borel subalgebras BG.

Let G be the semisimple complex Lie algebra, BG = Q+; Rh; ED Y,,,, Re. be a Bore] subalgebra in G, coo be the element of the group Weyl of the maximal height (see [11]). If 0 be the orbit of the maximal dimension of the representation, coadjoint then codim 0 = i card A, where A = {a EA I (-coo)a a}; A be the set of the simple roots of the Lie THEOREM 21.3 (Trofimov, V. V., [126], [127]).

algebra G.

Remark The completely involutive set of functions for Borel subalgebras of semi-simple Lie algebras is given explicitly in [126], [127]. The set of functions constructed here differs from the set of functions given in those papers. 21.2. Lie algebras of triangular matrices

Let F be a smooth function on the dual space Lie algebra G. We recall that F is called semi-invariant if F(AdB f) = X(g)F(f) for any g E 05, f E G* where X(g) is a character of the group 05 and Ad* is the coadjoint representation of the group (5 in G*. Recall also the main properties of

semi-invariants from [10]. If F is a semi-invariant for G, then s grad F(f) = -F(f) dX for any f e G*, dX E G* (see Section 11). Therefore, any function 0(f) is in involution with a semi-invariant F(f) if and only if = 0 for all f c- G* (doff) E G). Let the semiinvariants F(f) and t(f) of the algebra G be in involution

{F(f ), 0(f)) = 0, f e G*, then for any h c- G* and any 2, p e R the functions Fz,,,(f) = F(f + Ah), 0,,,ti(f) = Off + ph) involution {FA,h(f ), 4,,,h(f )}

0.

are also in

233


Let {el , ... , en } be a basis in G; let G - be the structure tensor of the Lie algebra G with respect to this basis. Denote by (f1 ,f2, ... , fn) the system

of -coordinates in G* relative to

(e1, e2, . . , e") where e'(ej) = S;, i , j = 1 , ... , n (el, ... , e") is the conjugate basis in G*. Then for any .

vector f c- G* we have dim O(f) = rk Ckj fk II , where 0(f) is the orbit of the coadjoint representation passing through f, and f = Y_h- 1 fie' gives f with respect to the basis (e1, e2, ... , e"). Let M(n, l) be the space of all matrices with n rows and n columns. Define the matrices Toj, by Tjo = (olio bjj.), i, j = 1, 2, ... , n. Then the T j, i, j = 1, 2, ... , n form a basis for the space M(n, 68):

M(n, R) _ Y7 J_1 RTij. Let T. be the space of all upper-triangular matrices T. = [_1,i,j," RTij. T. is a Lie algebra. Using the scalar product (x, y) = T,(x, y), x, y c M(n, l) it is possible to identify the space T,* with the space of all lower-triangular matrices Y_ 1 <j [(n + 1)/2] = m}. It follows from (7) that these polynomials are mutually in involution on L*, that they are independent at all points in general position and that there are Z(dim L + codim O(xo)) of them. But ?(dim L + codim O(x0)) , 2(dim L + ind L).

Thus the theorem is proved.

22. INTEGRABILITY OF EULER'S EQUATIONS ON SINGULAR ORBITS OF SENII-SIMPLE LIE ALGEBRAS

This section is devoted to some more precise results and extensions to those of Section 16. These results are due to A. V. Brailov (see [199], [201] ).

22.1. Integrability of Euler's equations on orbits 0 intersecting the set

tHR, teC

Let G be a semi-simple Lie algebra. An element a E G is called semisimple if the endomorphism ad° is semi-simple. In this case G = [G, G] Q G°, where G° is the centralizer of a in G. The restriction ad°I Ic,c] is invertible. Thus the mapping ada 1: [a, G] - [a, G] is well


237

defined. Let b e G°, then (adb)([a, G]) [a, G] and therefore the mapping ada 1 adb: [a, G] - [a, G] is defined. Let D: G° - G' be an operator symmetric with respect to a non-singular Ad-invariant form Q. We define operators PabD by the matrix

/,

_

`YabD -

ada 1 adb

0

0

D)

(see Section 7) with respect to the decomposition G = [a, G] ® Ga. PROPOSITION 22.1

The decomposition G = [a, G] @ G° is orthogonal.

PROPOSITION 22.2

Let Q be a non-singular symmetric bilinear

invariant on the Lie algebra G, a e G being a semi-simple element. Then the restriction of Q to the centralizer G° of a is also a non-singular form. PROPOSITION 22.3

Suppose that the hypotheses of Proposition 22.2

are satisfied, and that b E G° and D : G° -> G° is an operator symmetric with respect to Q, then the operator cpahD: G -> G is symmetric with respect to Q also. Let G be a Lie algebra, a e G a semi-simple element, Q an invariant symmetric bilinear form on G, G° the centralizer of a, Cent G° its center, D: Ga --+ G° an operator symmetric with respect to Q, Q the restriction of Q to G°, and f1(Y),... , f,(Y) integrals (in involution) of the equation of motion Y = [Y, D(Y )], Y E Ga. Then (a) if b c- Ga, Euler's equations of motion x = [x, PabD(x)], x c- G have integrals J(x + .la) f (Y), where J e I(G), A e R; (b) if b E Cent G°, the functions where Y is the projection of x onto G° along [a, G], are integrals of the LEMMA 22.1

equation of motion; (c) all the above-mentioned integrals of the equation of motion x = [x, (pabD(x)], x c G are in involution with respect to the Kirillov bracket transferred to the Lie algebra G using Q. Let a be an involutive automorphism of the Lie algebra G, G* the space dual to G, a* the automorphism of G and G* arising from a and a* (see Section 11). Let <X, Y> be a non-singular Aut(G)-invariant symmetric bilinear form on G. Then <X, Y> is invariant, in particular with respect to a and, using <X, Y> to identify G with G*, G** is identified with Go, Gi

with G1, where G1 is the orthogonal complement of Go. LEMMA 22.2

Let G be a Lie algebra, <X, Y> a non-singular symmetric

bilinear form on G invariant under Aut(G), a an involutive automorphism of G, G = Go p G1 the decomposition of G arising from

238


a; let f , g be invariants of G, a E GI, A, µ e ll ; let fa(x) = f (x + .la), g,,,a(x) = g(x + µa) be their translates, fx a, g a the restrictions of fl,a, g,,,a to Go, Ga the centralizer of a, Cent Ga its center, Go = Ga n Go,

D : Ga - G a, an operator symmetric with respect to the restriction <X, Y> of (X, Y> to Go, f1(Y),... , fk(Y) integrals in involution (under the Kirillov form transferred to Go by <X, Y>) of the equation of motion

3' = [Y, D(Y)], Y E G. Then we claim that (a) if b e Ga n GI, then functions J(x + Aa), J E 1(G), 2 E l are integrals of the equations of motion z = [x, (pabD(x)], x c Go; (b) if b c- Ga n G1, the functions f(x) = f,(Y), where Y is the projection of x onto G a along [a, G1] are also integrals of x = [x, (pabD(x)], x e Go; (c) all the above-mentioned integrals of the equation x = [x, (pabD(x)] : J(x + Aa) and f(x) (for b c (Cent(Ga) n G1)) are in involution on Go under the Poisson bracket, which corresponds to the restriction of <X, Y> to Go.

The proof of these statements is standard and we omit it. We note a further property of the center of the centralizer Ga. Let G be a real or complex semi-simple Lie algebra, a c G semi-simple element, Ga the centralizer of a, Cent Ga its center, <X, Y> LEMMA 22.3

an invariant non-singular symmetric bilinear form which we use to

identify G with G*, fl, ... , fk generators of I(G), the algebra of invariants. Then the differentials dfl(a), ... , dfk(a) generate Cent Ga.

Proof (1) The complex case. Let H be a Cartan subalgebra of G. Then, as is well known, the restriction mapping j: 1(G) - S(H) where S(H) is the algebra of polynomial functions on H, is an embedding j(I(G)) = S(H)W where S(H)W is the subalgebra of S(H) comprising the elements invariant under the Weyl group W. Let b c- Cent Ga, Wa the stabilizer of a in W, Wb the stabilizer of b in W. We have then Wa c Wb. Let {a1, a2, ... , aa} be an orbit of a under the Weyl group. We choose a positive function g on H in such a way that dg(a) = b and dg(a;) = 0 for a;

a. Let g = n ZW E W g - co. Then d4(a) = b and g c- S(H) W. Therefore,

f = j -1(g) is an invariant of G such that df(a) = b. Thus we have shown

that if fl, , f are generators of the invariants of the algebra G, f = P(f1, ... , fk) for a suitable polynomial P. Therefore, . . .

b = E;`= I OP/8f d f (a), which was what we had to prove.

(2) The real case. Let G be a real Lie algebra. Consider the complexification G, of G. Then G is a real form of Gc; let a be the

conjugation on G. Let r = rk G and fl, ... , f, be generators of


algebra

I(G).

Let

239

9,+I=

( , / -l ) _'(fi - J ° Q),

92, = (.,/- l)-'(f2 - f, ° u), where the line

denotes complex conjugation. Then g, , ... , 92r c- I (G,) and all of these are real on G, therefore their restrictions to G, which we also denote by 91, ... 1 92,, are invariants of G. Let g be an invariant of G. Let gc be the complex extension of g to Gc. Then gc is an invariant of Gc and gc = f2) for a suitable polynomial P. As f, = g, + 9r+I,

f, = 9, + 1/ -192x, gc and g are polynomials in g,,... 1 92,. Let G' be the centralizer of a in Gc and Cent Go its center. Then Cent GaC _ Cent G° + /_-1 Cent G° and any element b e Cent G° is a linear combinations of differentials df, (a), ... , df,(a) with complex coefficients

by dint of (1), therefore, b is a linear combination of differentials dg, (a), ... , dg2r(a) with real coefficients which was what we had to prove.

Let G be a semi-simple Lie algebra over the field k = l or C; H a splitting Cartan subalgebra of G; R = R(G, H) a root system of the split Lie algebra (G, H). For any root a c- R let G° = {x c- G: [h, x] = a(h)x for all the h e H}; the dimension of each of the vector spaces G', [G', G-'] is equal to one. For any root a c- R the space [G', G -'] is contained in H an element H° E [G1, G-'] is uniquely defined by the condition a(H2) = 2. We define the real subspace HR in H in the following way HR = LER RH°. Note that in case of k = R we have H;, = H. Let (G, H) be a split semi-simple Lie algebra over the field k = l or C; 0 an orbit of G, intersecting the set tHR where t c- k; let a c G be a semi-simple element, G° its centralizer, b c- G°, Q a nonTHEOREM 22.1

singular invariant symmetric bilinear form on G, D: G° -* G° a symmetric operator with respect to Q. Then Euler.s equation of motion X = [X, (pObp(X)] ,

XE0

(1)

has integrals in involution J(x + Ia), J E I(G), 2 E R, from which it is possible to choose independent functions on the orbit 0 equal in number to half of its dimension for any general position element a in G. We need, further, the following result, due to B. Kostant (see [26]). LEMMA 22.4

Let G be a semi-simple Lie algebra with rank r, H a

splitting Cartan subalgebra, R = R(G, H) the root system, B a basis of R, h an element of H such that a(h) = 2 for any at c- B. Suppose h = E°ER a2H°. For any root a e B denote by b,, and c° scalars such that

240


b,c, = a, and let x, a G", x _, e G where [x x _,] = H8, U = Y_,EB b,x, v = Y_8EB cax_s, S = ku + kh + kv. We claim that (a) [h, u] = 2u, [h, v] = -2v, [u, v] = h, with dim G" = dim G" = r; (b) consider G as an S-module under the adjoint representation. Let G = Al Q . p A. be some decomposition of this module as a direct sum of simple S-modules of dimensions v1 + 1, ... , v" + 1, where v1 < < v". Then n = r; (c) let J1, ... , J, be homogeneous algebraically independent generators of the algebra of invariants 1(G) of degrees

m1 + 1, ... , m, + 1, where m1 < < m,. Then vi = 2m, for any 1 < i 5 r; (d) differentials of functions J1, . . , J, are linearly independent at any point in the set u + G For the element h of this lemma all the eigenvalues of the .

endomorphism ad,, are even. For an integer n let G" be an eigenspace of ad,, for the eigenvalue 2n. This subspace is called the n-th diagonal of the Lie algebra G (with respect to basis B). We have [G`, G']

G'+'

(2)

Let R+(B) (and, respectively R_(B)) be the set of positive (negative) roots in the basis B. Let a e B; the height of the root a in basis B is the number lal = Y-eEB mfl, where mp are integers such that a = E,,, ma - P. From the definition of the diagonals of the Lie algebra G it follows that for any integer n : 0 we have G" = " G8. For any element x of the Cartan subalgebra H of G and basis B of the root system R we define the following subsets of R:

R°(x) = {a a R: a(x) = 0} ,

B°(x)=BnR°(x), R° (x, B) = R + (B) n R°(x),

R'(x) = R - R°(x), B'(x) = B n R'(x), R' (x, B) = R+(B) n R'(x).

Let C = {x e HR: a(x) 3 0 for all a E B}, the closure of positive Weyl

chamber, t e k and x e tC. Then any root a in R° = R° (x, B) is an integer linear combination of roots in B° = B°(x) R'+ = R'+ (x, B) we have an embedding

(R'++B)nRuR'+.

and for (3)

Let (G, H) be a split Lie algebra over k, R the root system, B a basis of R, t E k, C the closure of the positive Weyl chamber, x e tC, 0 LEMMA 22.5

an orbit in G passing through x, T = Ts0 the tangent space, T" = T n G" the intersection of T with G", the n-th diagonal of Lie


241

algebra G. Then: (a) T = 0"E, T"; (b) ads: T" -+ T" is an isomorphism; (c) (ada)(T") c Tn+1 for a = LE, xa and elements xa as in Lemma 22.4.

Proof (a) The equality T = ED. c, T" follows from the fact that T = [x, G], x e H = G° and from formula (2). (b) ads(T") c T" as a consequence of formula (2). As the endomorphism ads is semi-simple, ads: T -+ T is an isomorphism. It follows from this that ads: T" -+ T" is also an isomorphism. (c) This follows from formula (3). The lemma is proved.

Let J1, ... , J, be algebraically independent generators of the algebra of invariants I(G), m1 + 1, ... , m, + 1 their degrees, m1 < . < m,. The numbers m1, . . , m, are called the indices of Lie algebra G. Let a be an .

element of G. We define polynomial functions J;,a (i = 1, ... , r;

j = 0,...,mt + 1): m" 1

Jt(x + .la) = Y A'J,a(x).

(4)

1=0

As J1, ... , Jr are invariants

[x + 2a, grad Jt(x + 2a)] = 0.

(5)

We obtain from (4) and (5) m;+1

Y t'([x, uf] + [a, u; -']) = 0,

(6)

where u = grad Jj,a(x) (i = 1 , ... , r; j = 1, ... , mt + 1), u; 1 = 0 (1 < i < r). As J"'Q+ 1(x) does not depend on x, u;"; + 1 = 0 and we obtain,

as a result, the following chain of equalities (see [89], [90]):

[x, u°] = 0 [x, us] + [a, u,°] = 0 (7)

[x,

[a, um; -1 ]

=0

[a, u""] = 0. LEMMA 22.6 Let (G, H) be a split semi-simple Lie algebra, R its root system, B a basis for R, G' the first diagonal of the Lie algebra G, x E H, a e G1; J1,.. . , J, homogeneous algebraically independent generators of the algebra of invariants 1(G) of degrees m1 + 1,. . ,m, + 1, where .

242


,O we define Fp Rys , FO=O, F°_ = HR, FP = Ft p... Q Fr . I et J...... J, be homogeneous algebraically independent generators of the algebra of invariants of G° of degrees m, + 1, ... , m, + 1, m, LEMMA 22.10

+_+_=

m2; functions J j,,, (1 < i < r and 0 < j ` mi) are defined by the expansion J,(a + Ax) = Y, , AjJ{x(a) for any integer p > 0; let VP = Vp(a, x) be the linear span of the gradients grad J ,x(a) such that

I 0. LEMMA 22.13

Let Go and G. be Lie algebras as in Theorem 22.4,

r = rk G0, rk G be the rank of let m1, ... , m, be the indices of the Lie algebra Go. Then the rank of G is equal to the number of odd indices among m1, ... , m,. Proof It is enough to prove the lemma for simple Lie algebras Go only. Suppose, first, that Go is a simple Lie algebra with the root system of type A1, B, (r 3 2), C, (r > 2), D, (r is even and r 3 4), E.,, E8, F4 or G2.

Then all the indices m1, ... , m, are odd (see [11]). Therefore an automorphism of Hs equal to (-1) belongs to the Weyl group [11]. Therefore the canonical automorphism a: Go -. Go equal to (- 1) on HR and mapping xa into x for any root a is an inner automorphism. As G coincides with the set of fixed points of a, it follows that rk G = rk Go = r. As all the indices ml,. . . , m, are odd, for these Lie algebras Go the lemma is proved. Let us consider the remaining simple Lie algebras Go case by case.

The series of roots of A, (r 3 2). Indices: 1, 2, ... , r; G. = so(r + 1).

For even r the number of odd indices is equal to r/2 and rk so(r + 1) = r/2. For odd r the number of odd indices is equal to (r + 1)/2 and rk so(r + 1) = 2(r + 1). The series of roots of D, (r is odd, r 3 3). Indices: 1,3,5,.. . , 2r - 3 and r - 1. The number of odd indices is equal to r - 1,

G = so(r) p so(r) and rk G = r - 1. The series of roots E6. Indices: 1, 4, 5, 7, 8, 11; G. = sp(4) number of odd indices is equal to 4 and rk G. = 4. The lemma is proved.

The following lemma supersedes the argument in [90]. LEMMA 22.14

Under the hypotheses of Theorem 22.4, assume

(Go, HR) is a real split semi-simple Lie algebra. Let R = R(G0, HR) be the

root system, B its basis, R+ the system of positive roots, R°dd the set of


251

roots of odd height, Rodd = R n R°dd. For any finite set M let Card M be the number of its elements. Then

J(rk G. + dim G") = Card(R'd).

Proof We have G = Q+.,R, R(xa + x_8) and dim G. = Card R+. Therefore it is enough to prove that

rk G = 2 Card(R°dd) - dim G = 2 Card(R°+dd) - Card(R+)

= Card(R°dd) - Card(R+°)

_

m(2i + 1), i30

where R+ is the set of roots of even height and m(2i - 1) the number of indices of the Lie algebra G° equal to 2i + 1. This is exactly what was proved in Lemma 22.13. The lemma is proved. Again under the hypotheses of Theorem 22.4 we remind the reader that a is an element of HR, G0 its centralizer, G" = G" n G. Let R = R(G°, HR) be the root system of the split Lie algebra (G0, HR) and B a basis such that for any root a c- B we have a(a) > 0. Then LEMMA 22.15

i(rk GQ + dim

Card(R° r) Rod +d),

where R° is the set of roots a e R, equal to zero on a and R°+dd the set of roots at e R positive (with respect to B) and of odd height.

Proof As the element a is semi-simple in G° (see [26]) Ga is a reductive Lie algebra. Let Cent Go be its center, S = [Go, Go] is a semi-simple

ideal in G. Then Go = (Cent Ga) Q S. We have HR c Go and the normalizer of HR in Ga coincides with H. Therefore Cent G0 c HR and HS = HR n S is a split Cartan subalgebra in S. Let R = R(S, Hs) be the

root system of the split Lie algebra (S, Hs), R' the set of roots a e R

not equal to zero on a. We have Go = HR ® ($ ER° Ga) and Ga). Hence it follows that associating with each root S = Hs ® a c R° its restriction to Hs we obtain a one-to-one mapping of the set R°

onto R. Let B° = B n R° _ (a°, ... , ak), a e R°. Then for suitable integers n l , ... , nk we have a = D-1 ni a°. Let y c -A and y = a/Hs. Then y = Y+=1 ni fiiQ where fii = a°/Hs for every 1 < i < k. Therefore

B = (f1, . .

.

, Yk) is a

basis of R and the height of root a e R° with respect

252


to B is equal to the height of the root y = a/HS e R with respect to B. Let

°+, where S. = S n G. By Lemma 22.14 z(rk S + dim S,) = Card R°d R+d is the set of roots fi e R of odd height and positive with respect to B. It follows from these considerations that Card(R°+dd) = Card(R° n R°+dd).

Note that S. = QaERo R(xa + x _a) = G., where R° is the set of positive The roots a in R°. Therefore J(rk G.' + dim G.a) = Card(R° n lemma is proved. R°+dd).

Proof of Theorem 22.4 The functions specified in the theorem are integrals in involution for equation (13) as a consequence of Lemma 22.2. Let V(x, a) be the linear span of the gradients grad J(x + .la) where J E 1(G°), .? e R and the functions J(x + la) are viewed as functions of

the variable x E G°, let V(x, a) be the linear span of the gradients grad, J(x + Ia) where J e 1(G°), 1 e B and the functions J(x + la) are viewed as functions of the variable x e G. Then for any x c- G,: V(x, a) is the projection of V(x, a) onto G, along V. As a consequence of Lemma

22.9 V(x, a) = V(a, x). Let VT (a, x) be the projection of V(a, x) onto T = [a, G°] along Go and VT (a, x) = VT (x, a) the projection of V(x, a) = V(a, x) onto T n G. = [a, V] along G', F = Fp , where the space FD is defined in Lemma 22.10 and p is greater than all the indices of the Lie algebra G°. As a consequence of Lemma 22.12 we have VT (a, x) = T n F - for x = Y-1EB (xa + x -J, given the basis B such that

for any a e B the value a(a) > 0. Therefore, VT(x, a) = VT(a, x) _ T n F°dd, where F°dd = JaER.dd ICY: (for the definition of ya see Lemma 22.10, here R+d is the set of positive roots of odd height). We have, further, dim F°dd = Card(R' n Rod +d) where R' is the set of roots a e R such that a(a) # 0. As (fl_ .. , fk) is a completely involutive set on then k = Z(rk Ga + dim G"). Since grad g;(x) E G: (1 0. Let 7 1, ... , y, be numbers such that

254


al = ... = aqi = YI > aq,+1 = ' = a4, = Y2 > ... =aq.=Y >aqi_1+I =

Yl >...>Y3

Then so(n)° = so(P1) O ®so(P,) where pl =9j, P2=q2 - ql, ..., p, = q3 - R,- 1. Let the operator D: so(n)° _+ so(n)" be multiplication by y; on so(p;) (i = 1, ... , s). Then it is easily verified that A-l(S2) = W°bD(Q) Let G = sl(n, Vi), G = so(n) in Theorem 22.4. The functions tr M'` (2 < k 0. The functions y1,

are local coordinates on S+, while yn+l = ' +

, Y"

+ y,2. Let

z1,. .. , zn be the corresponding impulse variables on T*S+. Then , y") is a system of coordinates on T*S"+ and the (z1, , zn, Y l, standard symplectic structure is co=y"=1 dz, A dy,. The matrix X = I x. j II E so (n + 1) has a corresponding vector field on S" which in the

local coordinates yl, ... , yn equals n-1

(

"

a

a

)+

11

y E xr,ly;a-Y;Yja+ Y,E x,.n+lY"+1 aY, . ,=1 j i+1 i=1 Thus, P(z, y) = 11 P,j(z, y) II , where P,j(z, y) = z, yj - Zj Y, provided that i, j < n, and P,," + 1(z, Y) = zi Yn + 1 provided that 1 2 in case a = 0 and n > 3 in case a = 1. Then Ka = Ko x Tn-2, K1 = Ki x Tn-3 and Ki = K2 x S'. Let us consider the two-fold covering p: T" - K" and let Kn+1 = Kp1 is the cylinder of the map p. It is clear that 8K,+1 = T° We will describe five types of rearrangement of the torus T. (1) The torus T", implemented like the boundary of the dissipative complete torus D2 x Tn-1, contracts to its "axis," the torus Tn-1 (we will put T" -> T" - 1 -+ 0). (2) The two tori Ti" and T20--the boundaries of the cylinder T" x D1 moves in opposite directions and merge into one torus T" (i.e. 2Tn - Tn - 0). (3) The torus T"-the lower boundary or the oriented torus saddle N2 x T" -1 rises upward and, in accordance with the topology N2 x T"', splits into two tori Ti and Tz (i.e. T" -+ 2T").

272


(4) The torus T"-one of the boundaries A,, rises with respect to A. and is rearranged in its "middle," becoming once more a single (twice wound) torus (i.e. T" - T"). These rearrangements are parametrized by the nonzero elements a e H1 (T"-1,1L2) = 7L2-1 (5) Let us realize the torus T" as the boundary of Kr I . Let us deform T" in K" + 1 and collapse T" on K.". We obtain p: T" -+ K. We shall fix the values of the last n - 1 integrals

f2,. .. , f" and shall consider the resulting (n + 1)-dimensional surface Xn+1 Limiting in it f1 = H, we obtain the smooth function fin X"'. We will say that the rearrangement of the Liouville tori, which generate the non-singular fiber B. (assumed compact), is a rearrangement of the common location if, in the neighborhood of the rearrangement the torus T", the surface X" +I is compact, non-singular and the restriction f of the energy f1 = H on Xn+1 is a Morse function in the sense of Section 1 in this neighborhood. In terms of the diagram E, this means that the path y along which a moves, transversally intersects E at the point C, whose neighborhood in E is a smooth (n - 1)-dimensional submanifold in IR", and the last n - 1 integrals f2,. . ., f" are independent on Xn+1 in the neighborhood of the torus T". THEOREM 24.4 (A. T. Fomenko) (Theorem of the classification of the rearrangements of Liouville tori) (1) If dim E < n - 1, then all the non-singular fibers B. are diffeomorphic. (2) Suppose dim E = n - 1.

Suppose the non-degenerate Liouville torus T" moves along the common non-singular (n + 1)-dimensional surface of the level of the integrals f2,... , f", which is entrapped by the change in value of the energy integral f1 = H. This is equivalent to the fact that the point a = F(T") E li" moves along the path yin the direction of E. Suppose the torus T" undergoes rearrangement. This occurs when and only when T" meets the critical points N of the mapping of the moment F (i.e. the path y at the point C transversally pierces the (n - 1)-dimensional sheet E). Then all the possible types of rearrangement of the common location are

exhausted by the compositions of the above five canonical rearrangements 1, 2, 3, 4, 5. In case 1 (the rearrangement T" - T" -1 -+ 0) as the energy H increases the torus T" becomes a degenerate torus T" -1,

after which it disappears from the surface of the constant energy H = const (the limiting degeneration). In case 2 (the rearrangement 2T" -+ T" - 0) as the energy H increases the two tori T1 and Ti" merge

into one torus T", after which they disappear from the surface H = const. In case 3 (i.e. T" -+a 2T") as H increases the torus "penetrates" the critical energy level and splits into two tori T, and T2"


273

on the surface H = const. In case 4 (i.e. T" -a T") as H increases the torus T" "penetrates" the critical energy level and once more becomes the torus T" (a non-trivial transformation of a double coil). In case 5 the torus T" merge into the manifold K; and disappears from the surface H = const. Changing the direction of the motion of the torus T", we obtain five inverse processes: (1) the production of the torus T" from the torus T", (2) the trivial production of the two tori Tl and Tz from one torus T", (3) the non-trivial merging of the two tori Ti" and Tz into one torus T", (4) the non-trivial transformation of the torus T" into the torus T" (double coil), (5) the transformation of Ka into the torus T. The previously known rearrangements of two-dimensional tori in the

Kovalevskii case and in the Goryachev-Chaplygin case (see M. P. Kharlamov, A topological analysis of classical integrable systems in solid body dynamics, DAN SSSR, 273, no. 6 (1983), 1322-1325) are special cases (and compositions) of the rearrangements described in Theorem 4. When changing H, the torus T" drifts along the surface X"" of the level of the integrals f2,. . , f". It can happen that T" contracts to 1

-

the torus T" -1. These limiting degenerations emerge in mechanical

systems with dissipation. If we introduce small friction into the integrable system, we can assume, to a first approximation, that the energy dissipation is modelled using a decrease in the value H and causes, consequently, a slow evolution (drift) of the Liouville tori along

Xn+1. An answer to the question-What kind of topology is the topology of the surfaces Xn+1?-is given by the following theorem. THEOREM 24.5

Suppose M2 is a smooth symplectic manifold and the

system v = s grad H is integrable using the smooth independent f2,.. ., f ". Suppose Xn+1 is any fixed noncommuting integrals H = singular compact common surface of the level of the last n - 1 integrals. Suppose the restriction H on X"+1 is a Morse function. Then

X"+i

m(D2 x

T"-1) +

p(T" x D1) + q(N2 x

T"-1)

+ Y sa(A j) +

raK" 1,

a#o

i.e. a splice of boundary tori (using some diffeomorphisms) of the following "elementary bricks" is obtained: m dissipative complete tori, p

cylinders, q torus oriented saddles, s = aeo sa torus non-oriented saddles and r = ro + r1 non-oriented cylinders. The number m equals

274


the number of limiting degenerations of the system v in X"', in which H

reaches a local minimum or maximum. Theorem 24.3 follows from Theorem 24.5 when n = 2. All the above results also hold for Hamiltonian systems permitting "noncommutative

integration." In these cases the Hamiltonian H is included in the noncommutative Lie algebra of G functions on Men, such that the rank G + dim G = dim Men. Then the trajectories of the system move with respect to the tori T', where r = rank G. When proving the above results we use the following statements.

Suppose in the singular fiber B, there is exactly one critical saddle torus Tn-1. (1) Suppose the integral f is orientable in Xn+1 and a < c < b, where a and b are close to c. Then Cb = (f < b) is LEMMA 24.1

homotopically equivalent to C. = (f < a), to which the manifold P-1 x D1 is attracted with respect to the two non-intersecting tori T;,- I and T2a 1. (2) Suppose the integral f is non-orientable. Then Cb is homotopically equivalent to Ca, to the boundary B. of which, using the

torus T"', is attracted the n-dimensional manifold Y" which has the

a fibration y",, 'D' Tn- 1, which corresponds to the nonzero element aE71z-1 =H,(Tn-1,7L2). (3)

boundary P-1 and which

is

Further, each of the tori Tl a 1, T2, a 1, T" 1 always realizes one of the generatrices in the group of homologies Hn

(T.", Z) = 7L" -1. If any of

these (n - 1)-dimensional tori are attached to one and the same Liouville torus T", they do not intersect and they realize one and the same generatrix of the group of homologies H. -I (Ta , 71), and therefore

they are always isotopic in the torus T".

We will provide one more description of the three-dimensional surfaces Q of constant energy of the integrable (using an oriented Morse

integral) systems on W. Let us suppose that all critical manifolds of integral f are orientable. Suppose m is the number of stable periodic solutions of the system in Q, on which f reaches the minimum or maximum. Consider the two-dimensional connected closed compact orientable manifold Ma of the genus g, where g > 1 (i.e. a sphere with g handles) and take the product Mg x S'. We shall separate an arbitrary finite set of non-intersecting and self-non-intersecting smooth circles a;

in Ma , among which there are exactly m contractible circles (the remainder are non-contractible in MB ). In Mp X S2 the circles 01i determine the tori Ti' = at x S1. We will cut out Ma x S' with respect to all these tori, after which we will inversely identify these tori using


275

some diffeomorphisms. As a result we obtain a new three-dimensional manifold. It appears that the surface Q has precisely this form.

Find an explicit convenient corepresentation of the group ir1(Q), where Q' is the surface from Theorem 24.3. Give an explicit PROBLEM

classification of the surfaces of constant energy of the integrable systems

of arbitrary dimension. How can we make an upper estimate of the number of complete tori (i.e. stable periodic solutions) in Q', in terms of the topological invariants Q (homologies, homotopies) in the general

case. Discuss the complex analytical analog of the Morse theory of integrable systems constructed above. Does an integrable foliation to the two-dimensional (in a real sense) complex tori exist in the analytical manifold M4? Probably, we can obtain these obstacles in explicit form in examples of surfaces of the K3 type. 243. New topological invariant of integrable Hamiltonians

In this section we describe the topological invariant, which was introduced by A. T. Fomenko on the basis of his Morse-type theory of Bottian integrals. Let M4 be a symplectic manifold, v be a Hamiltonian system with

Hamiltonian H; v is completely integrable on the compact regular surface Q' = (H = const); f : Q - R is a second independent Bottian integral on Q. The critical submanifolds of f are isoenergetic

non-degenerate in Q. The Hamiltonian H will be called non-resonance if the set of Liouville's tori with irrational trajectories of v is dense in Q. The set f -'(a) is the set of tori in the case when a e R is regular.

THEOREM 24.6 (A. T. Fomenko) There exists a one-dimensional graph Z(Q, f), two-dimensional closed compact surface P(Q, f) and the

embedding h : Z(Q, f) -- P(Q, f ), which are naturally and uniquely defined by the integrable non-resonance Hamiltonian H with the Bottian integral f on Q. The triple (Z, P, h) does not depend on the choice of the second integral f This means that if f and f are two arbitrary Bottian integrals of a given system, the graphs Z and Z' are homeomorphic, the surfaces P and P' are homeomorphic, and the diagram

h:ZP h':Z'-P'

276


is commutative. Consequently, the graph Z(Q), surface P(Q) and the embedding h(Q): Z(Q) - P(Q) are the topological invariants of the integrable case (of Hamiltonian H) proper.

The triple Z(Q), P(Q), h(Q) allow us to classify the integrable Hamiltonians corresponding to their topological types. In particular, we can now demonstrate the visual difference between the invariant topological structure of the Kovalevskaya case, Goryachev-Chaplygin case and so on. The subdivison of the surface P(Q) into the sum of the domains is also the topological invariant of the Hamiltonian H and describes its topological complexity. The graph which is dual to the graph Z on the surface P, has the vertices of the

multiplicity no more than four. The collection of the graphs Z(Q), surfaces P(Q) and embeddings h(Q) is the total topological invariant (topological portrait) of integrable Hamiltonian H. Let us construct the graph Z(Q, f). If a is a non-critical regular value for f, then fQ is a union of a finite number of Liouville's tori. Let us represent these tori by the points in R3 lying on the level a. Changing the value of a (in the domain of regular values), we force the points to move along the vertical in R3. Consequently, we obtain some intervals, viz. the

part of the edges of our graph Z. Let us suppose that the axis R is oriented vertically in R3. If the value a becomes critical (we denote such values by c), the critical (singular) level of the integral f becomes more complicated. Let f, be a connected component of a critical level surface of the integral. We denote by N, the set of critical points of the integral f on f c.

Let us consider two cases: (a) N, = f, (b) Nc c f c. In Section 24.2 A. T. Fomenko gives the complete description of all cases and the topological structure of f,. (See [149], [150].) Let us consider case (a). Here only three types of critical sets are possible.

The "minimax circle" type. Here Nc = f and this set is homeomorphic to a circle S'. The integral f has a minimum or maximum on S'. The circle S' is the axis of the filled torus which foliated

into non-singular two-dimensional Liouville's tori. We represent this minimax circle by the black point (a vertex of the graph) with one edge (interval) entering the point or emerging from it. The "torus" type. Here N, = fc. This set is homeomorphic to a two-

dimensional critical torus. The integral f has a local minimum or maximum on this torus. The tubular neighborhood of this torus in Q is


277

homeomorphic to the direct product T2 x D1. We represent this minimax torus by the white point (the vertex of the graph). Two edges of

the graph Z enter this vertex or emerge from it. The "Klein bottle" type. Here N, = f . This set is homeomorphic to a two-dimensional Klein bottle K2. The integral f has the local minimum or maximum on this manifold. The tubular neighborhood of KZ is homeomorphic to the skew product of K2 and the interval D'. We represent this minimax Klein bottle by a white disc with a black point at the centre (the vertex of graph Z). One edge of the graph Z enters this vertex or emerges from it. Let us consider case (b). Here N, c f and N, # f. Here N, is a union of non-intersecting critical circles in f,. Each of these circles is a saddle circle for f. We shall call the corresponding connected component f a saddle component. Each saddle component f, is represented by a flat horizontal square in R' on the level c. Some edges of the graph Z enter the square from below (when a -+ c and a < c). Some other edges of graph Z emerge upwards from the square (when a > c). Finally we define some of graph A which consists of the regular edges described above. Some edges enter the vertices like the three types described above. Graph A is a subgraph in graph Z. Graph A was obtained from the union of the edges which are the traces of the points representing the regular Liouville tori. Let us define the graphs T. We consider a vector field w = grad f on Q. Let us call by separatrices the integral trajectories of w which enter the critical points on critical submanifolds (or emerge from them) and call

their union the separatrix diagram of a critical submanifold. Then we consider the local separatrix diagram of each saddle critical circle S'. Let us consider two regular values c - e and c + e which are close to c. They

define the regular Liouville tori above and below fc. The separatrix diagrams of critical circles meet these tori and intersect them along some smooth circles. These curves of intersection divide each torus into the sum of two-dimensional domains which will be referred to as regular. Each inner point of a regular domain belongs to the integral trajectory of the field w, which is not a separatrix. The trajectory goes upwards and leaves aside the critical circles on the level f . Then the trajectory meets

some torus on the upper non-singular level fc+,. We obtain a certain correspondence (homeomorphism) between regular domains on the

levels f _ and f,+,. Let us consider the orientable case, when all separatrix diagrams are orientable. Since each regular torus is

278


represented by a point on graph A, we can join the corresponding points

by arcs which represent the bundle of parallel integral trajectories. Consequently we obtain some of graph T . All edges of the graph T represent the trajectories of single regular domains of Liouville's tori. The tori break down into the sum of single pieces, then these pieces are transposed and joined into new tori again. Each upper torus is formed from the pieces of lower tori (and conversely). The ends of the edges of the graph T, are identified with some ends of the edges of the graph A. Graph T demonstrates the process of transformation of lower tori into

upper tori after their intersection with a saddle critical level of the integral.

Let us consider the non-orientable case when we have the critical circles with non-orientable separatrix diagrams. Let us consider all Liouville's tori which are in contact with the level surface f, with a nonorientable separatrix diagram of some critical circles on f. Let us mark by asterisks all regular domains on these tori which are in contact with non-orientable separatrix diagrams. We mark by asterisks the corresponding edges of the graph. Finally we double all edges of the graph (preserving the number of its vertices) and denote the resulting

graph as T. Finally, we define the graph Z as the union Z = A + E, T, where {c} are the critical values of f Let us construct the surface P(Q, f ). This surface is obtained as the union P(A) + E, P(T) (here {c} are the critical values off) where P(A) and P(T) are two-dimensional surfaces with boundary. Here P(A) = (A x S') + > D2 + Y U2 + Z S1 x V. Here A = Int A, > D2 denotes the non-intersecting 2-discs corresponding to the vertices of the graph A, which have a "minimax circle" type; 2]p2 denotes the non-intersecting MObius bands, corresponding to the vertices of the graph A, which have a "Klein bottle" type; Y S' x D1 denotes the non-intersecting cylinders, corresponding to the vertices of the graph A, which have a "torus type".

The corresponding boundary circles of A x S' are identified with the boundary circles of D2, t2, S' x D' by some homeomorphisms. Let us construct the surfaces P(T,). Let us consider the orientable case. Fomenko proves (see Section 24.2 and [149], [150]) that in this case the surface f is homeomorphic to direct product K, x S1, where K, is some

graph. The graph K, is constructed from several circles, which are tangent in some points. Such circles can be realized as a cycle on the


279

torus contained in f.. This cycle intersects with a critical circle on f only in one point. The surface f is obtained as a two-dimensional cell-complex by the union of several species of two-dimensional tori along some circles. The tori stick together along the critical circles realizing a non-trivial cycle on the tori. The critical circles do not intersect and they are homologous

in f,. They cut f into the sum of flat rings. Consequently, the circle y (non-homologous to zero) is uniquely defined on a critical level surface

f. We can choose the circle a which is a generator on the torus contained

in f. The circle a is complementary to y. We obtain the set of circles a which are tangent to one another at points on critical circles. Each circle a will be called oval. The ovals can be tangent to one another at several points. The graph K, is the union of all ovals. The surface P(TA) can be realized as "normal section" of a small neighborhood of a critical level surface f in Q. The intersection of P(T)

with f is the graph K,. To realize the surface P(T,) in Q, we must consider the small intervals on the integral trajectories of the field w = grad f, which intersect the graph K, This definition is correct in all non-critical points on K, Let us consider the vertices of the graph K, i.e. the critical points of the integral f on f . Then we consider the small squares orthogonal to the critical circles on f. The surface P(T) is the union of these squares and the bands, which are formed from the small intervals defined above. Finally, we identify the boundary circles of the surface P(A) with the boundary circles of the surfaces P(7 ). The graph K, is embedded in the surface P(T). We obtain some graph K as the union of all graphs K, and all boundary circles, described above.

THEOREM 24.7 (A. T. Fomenko) The graph Z(Q, f) is conjugate to

the graph K(Q,f) in the surface P(Q, f ). Consequently, the graph Z(Q, f) is embedded in the surface P(Q, f ). The surface P(Q, f) does not embed (in general case) in the surface Q. The construction of the triple Z, P, h is finished. Theorem 24.6 states that this triple does not depend from the choice of the Bottian integral f PROPOSITION

Let f and f be two arbitrary Bottian integrals of a

system v. Then the homeomorphism h : Z(Q, f) -+Z(Q, f') (see

Theorem 24.6) transforms the subgraphs TT into the subgraphs T,'. The asterisks of the graph Z(Q, f) are mapped into the asterisks of the graph

280


Z(Q, f'). The vertices of the types "minimax circle" and "Klein bottle" on the graph Z(Q, f) are mapped into the vertices of the same type on the graph Z(Q, f '). The vertices of the "torus type" on the graph Z(Q, f) may change their type and be mapped into the usual inner points of some edge on the graph Z(Q, f'). Conversely, some usual inner points of the edges on the graph Z(Q, f) can be mapped into the vertices of the "torus type" on the graph Z(Q, f'). This event corresponds (from the analytical point of view) to the operation f -+ f 2 (square of function) or, conversely, to the operation f (square root). If a non-resonance Hamiltonian H is fixed, we can consider all its nonsingular isoenergetic surfaces Q. This set consists (in concrete cases) usually of a finite number of triples (Z, P, h). We formulate the new definition based on Theorem 24.6.

DEFINITION We shall call the triple Z(Q), P(Q), h(Q) an invariant topological portrait of a non-resonance integrable Hamiltonian H on a fixed isoenergetic surface Q. The discrete set of all triples {Z, P, h} will be

called the total topological invariant portrait of the integrable Hamiltonian. We shall obtain the following corollary from Theorem 24.6. If two integrable systems have non-homeomorphic topological portraits, then there exists no transformation of coordinates which would realize the equivalence of these systems. So, the systems with non-homeomorphic

topological portraits are non-equivalent. On the other hand, nonequivalent systems with homeomorphic topological portraits do exist. Practically all the results listed above are also valid in the multidimensional case. These results will be described in a separate paper by Fomenko.

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Selective key to the notation used

l8-the set of all real numbers. C-the set of all complex numbers. 7L-the set of all integers. Q---the set of all rational numbers. l8"-n-dimensional real linear space. Cmt A (02 -the exterior product of differential forms co, and W2.

X`, X'-the matrix transpose of a matrix X. T; the elementary matrix: (Tj)Pq = (Sjp 5Jq).

t

E1 = T1 - T; the elementary skew-symmetric matrix. Ii, j I = Tj + T the elementary symmetric matrix, ad, (x) _ (ad )* x = a(x, ) where e G, x E G* is the coadjoint representation of the Lie algebra G in the space G* dual to the Lie algebra G. B(X, Y)--the Cartan-Killing form. <X, Y>-pairing between the space V and the space V* that is dual to it. C°°(M)-the space of all smooth functions on smooth manifold M.

H(M)-the space of all Hamiltonian vector fields. A(W)-the space of analytical functions on space W V(M)-the Lie algebra of all vector fields on a smooth manifold M under the commutator of vector fields. Exp-the exponential mapping Exp: G -+ 6 of the Lie algebra G into the Lie group 6. F(M)-the full commutative set of functions on a symplectic manifold M. Reg(G)-regular elements of Lie algebra G. Exp G-the Lie group corresponding to the Lie algebra G. W(M)-the space of the differential k-forms on the manifold M.

Index

Adjoint representation, 18 Affine Lie algebras, 219 Algebra with Poincarb duality, 162 Algebraic variety, 8 Argument translation, 137

Functions in involution, 30

Bifurcation diagram, 271 Bounded domain, 8

Index of the Lie algebra, 33 Integral, 12 Invariant, 27

Canonical H-invariants, 168 Cartan-Killing form, 20 Cartan subalgebra, 20 Case of Steklov, 85 Chain subalgebras, 144 Coadjoint representation, 18 Compact series, 75 Complete torus, 270 Complex semi-simple series, 71 Condition (FJ), 48 Configuration space, 5 f-connective vector fields, 52 Contraction of the Lie algebra, 174 Cylinders, 270 Dissipative complete torus, 271 Dynamic tensor, 56

Embedding of the dynamic system into a Lie algebra, 55 Equations of magnetic hydrodynamics, 89 Euler's equation, 55 First case of Clebsch, 84 Fubini-Studi metric, 7

Geodesic flow, 10

Hamiltonian field, 8

Kahler manifold, 7 Kirchhoff integrals, 83 Lagrange case, 210 Lie algebra, 17 Lie group, 17 Locally Hamiltonian vector field, 9

M-condition, 169 Morphism of symplectic manifolds, 53 Morse-type integral, 267 n-dimensional rigid body, 61 Non-oriented saddle, 270 Non-resonance Hamiltonian, 275 Normal nilpotent series, 75 Normal series, 75 Normal solvable series, 75

One-parameter subgroup, 17 Oriented saddle, 270 Poisson bracket, 11 293

294


Realization in a symplectic manifold, 54 Reduction of a Hamiltonian system, 37 Regular element, 20 Root, 21 Second case of Clebsch, 84 Sectional operator, 68 Sectional operators on symmetric spaces, 104 Semi-invariant, 27 Similar functions, 168

Simple root, 21 Skew-symmetric gradient, 2 S-representation, 142 Stable trajectory, 267 Submersion, 52 Symplectic atlas, 4 Symplectic coordinates, 3 Symplectic manifold, 1 Symplectic structure, 1

Toda chain, 62

Integrable systems on Lie algebras and symmetric spaces

Integrable Systems on Lie Algebras and Symmetric Spaces (Advanced Studies in Contemporary Mathematics)

Integrable Systems on Lie Algebras and Symmetric Spaces (Advanced Studies in Contemporary Mathematics, Vol 2)

Differential geometry, Lie groups, and symmetric spaces

Differential Geometry, Lie Groups and Symmetric Spaces

Lectures on Lie groups and Lie algebras

Lectures on Lie groups and Lie algebras

Lectures on Lie Groups and Lie Algebras

Lectures on Lie groups and Lie algebras

Infinite Dimensional Algebras and Quantum Integrable Systems

Notes on Lie Algebras

Notes on Lie Algebras

Infinite Dimensional Algebras and Quantum Integrable Systems

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Lectures on integrable systems

Geometric Analysis on Symmetric Spaces

Lie Theory: Harmonic Analysis on Symmetric Spaces General Plancherel Theorems

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Mirror Geometry of Lie Algebras, Lie Groups, and Homogeneous Spaces

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Lie Algebras and Lie Groups

Lie groups and Lie algebras

Lie Groups and Lie Algebras

Lie algebras, geometry, and Toda-type systems

Lie Algebras

Lie Algebras

Lie Algebras

Lie Theory. Unitary Representations and Compactifications of Symmetric Spaces

Euler equations on Lie algebras

Integrable systems on Lie algebras and symmetric spaces