Euler equations on semisimple Lie algebras

In the class of all smooth functions problem 3 is trivial, since on any symplectic manifold it is possible to construct a complete involutive family of functions. However, this collection (see [103]) is not interesting from the point of view of mechanics. To obtain an interesting problem it is necessary to impose additional conditions, for example, it may be supposed that M 2n is an algebraic, analytic, etc. manifold and it is natural to consider a narrower class of functions: algebraic, analytic, Bott functions, etc. Here there arises the problem of including a given Hamiltonian H defined on M 2n and of physical interest in a complete involutive family of functions on the manifold M =n. In connection with this there arises the problem of classifying finite-dimensional Lie algebras of functi6ns on a given symplectic manifold M 2n. For example, it is known that if M 2n is a compact symplectic manifold, then any finite-dimensional Lie algebra of functions on M 2n is reductive (a proof can be found, for example, in [67])i In solving these problems the greatest difficulty lies in the proof of completeness (i.e., in the proof of functional independence) of the collection of functions constructed. As a rule, involutive collections of functions can be constructed by general procedures. To verify independence of a given collection in each concrete case it is necessary to invent special methods. 2.

Euler Equations on Semisimple Lie Algebras

2.1. The Motion of a Multidimensional Solid Body with a Fixed Point and Its Analo u ~ on Semisimple Lie Algebras. Imbedding a Hamiltonian system in a Lie algebra guarantees a trivial family of first integrals - the functions constant on orbits of the coadjoint representation. As a rule, these rules do not suffice for complete integrability. If the Lie algebra is two-dimensional, then functions constant on an orbit completely solve the problem. The method of shift of invariants (see Sec. 4, Chap. 2) is used to construct a complete collection of integrals on semisimple Lie algebras. This general scheme is a development of an idea proposed in [53] for the case of the Lie algebra so(n). The fact that shifts of invariants are in involution was already noted in Sec. 4 of Chap. 2 (a rather simple fact). In this section we shall be concerned with functional independence of shifts of invariants. To prove this it is necessary to use in full scope the properties of semisimple Lie algebras, and this was first done in the works of Mishchenko and Fomenko; see [63, 65, 66]. The case so(4) was also studied in the works [59, 151]. The involutiveness of Mishchenko's integrals [59] was proved in [25]. Proposition 2.1.1. a) Suppose f is a function constant on orbits of the adjoint representation of a semisimple Lie group, i.e., f is an invariant. Then the complex-valued functions h~(X)=[(X+~a) are (for any X) integrals of the equation X=[X, ~bD(X)]; where ~ = ~ b D is an operator of the "complex" series. The function F(X)=<X, ~(X)> is also an integral. b) Suppose f6IG~, i.e., f is a function constant on orbits of the Lie algebra G u. Then the functions h~(X) =[(X+la) are (for any ~) integrals of the equation X = [X, ~(X)], where 9 is an operator of the "compact" series, X+%aEG~, %ER. The function F(X) =<X,~(X)> is also an integral. The next assertions for the classical operator ~ ( X ) = I X + X I [53] in investigating the Korteweg,de Vries equation.

on so(n) were obtained in

Proposition 2.1.2 (see [63, 65, 66]). LetfCIG=, i.e., the function f is constant on orbits of the Lie algebra G u. We consider functions qx(X), where ~6R, XEOncOu, which are restrictions of the functions h~(X)=/(X+Xa)to OncO~. Then the functions ql are integrals of the equation "X-----[X,~aoD(X)],where ~=~a~D is an operator of the "normal" series. The function F ( X ) ~ ( X , ~ ( X ) > is also an integral. The next theorem, which asserts complete integrability of the Euler equations on semisimple Lie algebras, was first proved by Mishchenko and Fomenko in the work [63] (see also [65, 66]). In the work [56] one may become familiar with explicit integration of Euler equations on semisimple Lie algebras using the ideas of Dubrovin [27-29, 54]. Here we shall not consider applications to the Euler equation of algebra-geometric methods of integrating nonlinear equations but refer the reader to the surveys [29, 30, 46, 73]. THEOREM 2.1.1 (see [63, 65, 66])~ a) Let G be a finite-dimensional, semisimpie Lie algebra, and let x = [x, ~bD(X)[] be Euler equations with an operator of the "complex" series. Then this system is completely integrable (in the Liouville sense) on orbits of general 2718

position. Let f be any invariant function on the Lie algebra G. Then all functions h%(X, a) ==f(X-k%a)" are integrals of the flow ~( for any %. Any two integrals h~(X, a) and g~(X, a) constructed on the basis of functions f, g6fG are in involution on the orbitso The Hamiltonian F(X)=<X, ~(X)> of the flow X also commutes with all integrals of the form h~(X, a). From the set of these integrals it is possible to choose a number of integrals equal to half the dimension of the orbits, which are functionally independent onorbits of general position. The integral F can be functionally expressed in terms of integrals of the form h%(X). b) Let G u be a compact real form of a semisimple Lie algebra, and le61~=[X, qg(X)] be a Hami!tonian system defined by an operator ~ of the "compact" series. Then the set of functions of the form f(X+%a), where foIG=, forms a complete commutative collection on orbits of general position in the Lie algebra G u. c) Let G n be a normal compact subalgebra in a complex semisimple Lie algebra G, and let ~?= [X, ~(X)] be a Hamiltonian system with an operator of the ~'normal" series. Then the set of functions f(X+)~a), where f6IG, forms a complete commutative collection of functions on orbits of general position. Thus, Euler equations on all orbits of general position in semisimple Lie algebras are completely integrable in the Liouvil!e sense. However, in reality a stronger assertion is true. It turns out that a complete involutive collection of algebraic functions (polynomials) exists on any semisimple orbit in a Lie algebra; in particular, this orbit need not be an orbit of general position. This result was first announced by Dao Ehong Tkhi in [24], but there were gaps in the proof of this theorem. These gaps were eliminated by A. V. Brailov. Let G be a semisimple Lie algebra of a Lie group ~. An orbit O of representation Ad of the group ~ in G is called semisimp!e if it consists of semisimple elements of the Lie algebra G. Let IG be the algebra of polynomials on G which are invariant relative to the adjoint representation. Let J1,...,Jr be a collection of homogeneous invariants generating the algebra IG, and let a be an arbitrary element in G. The coefficients of %J in the polynomial J~(%; X)=J~(X-~-, %a) we denote by J{7,G" These functions, as we already know, are integrals of the Euler equations. THEOREM 2.1.2. For any semisimple orbit G in the Lie algebra G and any element a of general position in G from the functions J[,a it is always possible to choose a number of functions equal to half the dimension of the orbit such that after restricting these functions to the orbit they become independent~ Moreover, the orbit need not be an orbit of general position. 2~176 Euler Equations on the_Lie Algebra so(4). In this case the state of the system is described by six variables l~=--lj~, i, ]=I ..... 4. We go over to the new variables

/t=~'(/2~'7-/x4)' /2"----7(lat~-[24), [3='~ (112~-[~), rot------~I (123-- l~4),

fLg2..~ ~1 (/31--/24),

F/~3~---.~1 ([12 --~34) 9

For them the Poisson brackets have the form

{lj,l~}=~jk~l~, {m i,m~}=~j~m~, {6, mk}=O. We consider a homogeneous

quadratic Hamiltonian of the most general form 1

" aiilfli-+-2

bJim~+

~ c~im~m i .

9

Such a Hamiltonian is characterized by three 3 x 3 matrices A = ( a i ) , B = (bij), and C = (cij) two of which (A and C) are symmetric and hence depend on 21 parameters. We remark that by using transformations of the group @ = S O (3)X S O (3),we can bring the matrices A and C to diagonal form so that we may assume with no loss of generality that H has the form H--=~

and depends on 15 parameters. A Hamiltonian system with a Hami!tonian of general form always possesses three integrals of the motion Iz = ~ 2 I~ = m ~, I~ = H. An orbit of the coadjoint representation has dimension 4 and is distinguished by the equations I~ = const, !~ = const. From a topological 2719

point of view the orbits are the product of two-dimensional spheres

~ = S = X S ~.

The following Theorem 2.2.1 holds which solves the questions of the existence of an additional quadratic integral of the motion. An answer to this question was given in a different but equivalent form in the works [7, 14] THEOREM 2.2.1 (see [14]). 1 ~ I - I = z2-

Let ! aj~lj+2

i,i=l b~jl~mj-~- z,;=l c~jm~ra; ,

whereby the eigenvalues of both the matrices A and C are distinct, and the matrix B is nondegenerate (detB ~ 0). Then for the existence of a fourth independent quadratic integral of the motion it is necessary that the following conditions be satisfied. a) The matrix B can be brought to diagonal form simultaneously with A and C, so that the matrices A, B, and C may be considered diagonal with elements aj, bj, and cj, respectively. b) The following conditions must hold:

b~ (a2 -- as) + O~ (a3 -- aO + b~(al-- ai) -'-,-(al - - a i ) ( a i - - a3)(a3-- a 3 = O. b~(c~ - - c ~j..: ~ ' b ~2(c~--c~)+O~(c~--c~)+(c~--ci)(c~--c~)(ca--c,)----O. An additional quadratic integral of the motion exists only in the Manakov case [53] and a Hamiltonian of the following form aj=a+~b71,

Bj=b;

-7~ Lb)o (blbzb3) _~,

~-.,- b ~ - b~) + ~c. We present another form of integrals of the motion taken from the work [7]. THEOREM 2.2.2.

The following three functions are in involution:

"61~ 14= [-( a~-a~ ),/2~1._{_(bl...~_.~2 b,-,% ),12 ],

i [a~

ai\lli,

, /b3--bi\l/2

-]z

L\ a t - - a 2

v

f6= [(~) I/2/2-'~\b~-' - -~}/~I--'2~I/2--/' ~-]'-~' 1/'2-[(al--a'~\l/21

' /bl--ba'I/2--

]1

We note that the functions l~, 12 and l~, I s , I~ are connected by a single linear relation, so that among them there are four independent functions. Therefore, the general form of a Hamiltonian admitting an additional integral of the motion is as follows: H = ~ I ~ + ~5T5 + 0~6T6. 2.3. Integrable Euler Equations Connected with Filtrations of Lie Algebras. We consider the classical Euler equations X = [X, C(X)], w h e r e X ~ G , G is a semisimple (real or complex) Lie algebra and C:G + G is a section operator. Suppose there is given a decomposition of G into a direct sum of linear subspaces G = L 0 + L I +... + L n, where all the Li are orthogonal relative to the Killing form and their commutators satisfy the condition ~[L~_, i~]~/~ for i > 0. It is clear that G k = L I + . . . + L k is a subalgebra. We therefore obtain a filtration of the Lie algebra: O = O ~ . . . ~ G 1 ~ O o = i o . Conversely, on the basis of each filtration such that the restriction of the Killing form to the subalgebra G i is nondegenerate we can recover the required decomposition of G into a direct sum of subspaces L~=G~[JG~-I • We denote by Pk the orthogonal projector onto the subspace L k. An arbitrary element I~G has the representation +1,,, l~=P~(1)~i~. We consider operators C:G + G of t h e form

I=lo+...

c ( o =ao(O +~.,h+ ... +~.&,

(1)

where a 0 is any symmetric operator in the Lie subalgebra L 0. Let f, g ~ I G be functions which are constant on orbits of the adjoint representation in the Lie algebra G. THEOREM 2.3.1 (see [8]). The Euler equation )( = [X, C(X)] with an operator C:G § G of the form (i) is equivalent to the equation 10= [10, a0(b0)] in the Lie subalgebra L 0 and to a chain of linear differential equations in the subspaces L k. The Euler equation X = [X, C(X)] has a collection of first integralsf(=m~+~l~), where +t~, m,=2, X'=lo+... +l,,, which depend on the spectral parameters ~z, ~, 0 ~ k ~ n The Poisson brackets of these integrals

m,~=/o+...

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{f(~mh+~lk), g(?m~+6lj)} ~=0, ~=i.

vanish in the following cases: a) k = j, b) k > j, ~ = i, $ = 0 or

Remark. The operation of "lifting invariants from subalgebras" was studied in the works of Trofimov [88, 91]. As a result of this operation integrals of the form f(m k) are obtained; that they are in involution was proved in [88] (see also Theorem 4.2.1 of Chap. 2). Generally speaking, this collection of integrals does not suffice for complete Liouville integrability of the Euler equations with operator c(x)=~0/0-~./. +~l~. 2.4. Inte~rability of Euler Equations for Singular Inertia Operators. Let a~G be an arbitrary semisimple element. For each such element a the operator ada'[al G]-+[al G] is invertible. Therefore, the skew-symmetric operator ads G]-+[a, G] is well defined. Let G a be the centralizer of the element a, and let D : O a ~ G a b e a symmetric operator. On the basis of the decomposition G~-[a, Gia-~Ga we define the operator ~alb,o by the formula

~a,b,D= \

0

Oj"

This definition o f t h e o p e r a t o r ~a,b,D, i s o b v i o u s l y an e x t e n s i o n o f t h e d e f i n i t i o n of the operators r for regular semisimple elements a The o p e r a t o r ~Pa,b,D i s c a l l e d s i n g u l a r if a is not a regular element in G. THEOREM 2.4.1 (A. V. Brailov). Let a be an arbitrary semisimple element in a semisimple Lie algebra G, let G a be its centralizer, let CentG a be the center of the Lie algebra G a, and let D:G a + G a be a symmetric operator invariant relative to all inner automorphisms of the Lie algebra G a. We consider the equation~(-~[X,~a,b.D(X)], XEG, on the Lie algebra G. For any invariant I of the Lie algebra G, for any number X, and for any element gGG a the functions J~(X)-~J(X~-%a) and Ig(X)-~- ( g, X ) are commutative integrals of the Euler equation X = [X,~a,~.D(X)]. From these integrals it is possible to choose a number of independent integrals equal to d i m G a ' - ~1-dim [a, O]. This theorem is an extension to the case of nonregular semisimple elements a of Theorem 2.1.1. Let G be a complex simple Lie algebra of rank r, and let H be the Cartan subalgebra; let R = R(G, H) be the system of roots; for any root ~E/~ the space G~-{XGG] [h, X]-~(h)X, ~@fY} is defined. For any root ~/~ the element HaE[G ~, G -a] is uniquely determined by the condition ~(H~)--2, H R is the real linear hull of the element -/-/=; (X~)~E~ is the Weyl basis of G modulo H. The real Lie algebras G O and G u are defined as follows: Go ~ HR + ~ G~-----V--1 H R q - ~

RX~,

(RX~+q-]/ - - 1 RX~),

where X~ = X r X-~. The s p a c e /-/R i s t h e s p l i t t i n g C a r t a n s u b a l g e b r a i n G0. On t h e L i e a l g e b r a GO t h e r e i s a c o l l e c t i o n o f i n v a r i a n t s 11 . . . . . I r o f d e g r e e s m l q - 1 , . . . . t o r T 1 , nh . . . . . ~ m r , w h e r e m l , . . . , m r a r e t h e e x p o n e n t s o f t h e Weyl g r o u p . The L i e a l g e b r a Gu i s t h e compact

real

form o f t h e c o m p l e x L i e a l g e b r a

K + L - the Cartan decomposition;

RX +, L ~ Z RXg--[- Hm

G.

Let

K-~Uof-lG u, s

K i s t h e maximal c o m p a c t s u b a l g e b r a

We h a v e Go = i n GO.

We h a v e t ( ~

aER

Let the element aEL, let G0~ be its centralizer in Go, let I(~I=K~Go~ be the centralizer of the element a in K, and let L==LNGo ~. Then [a, K] c L, [a, L]c/( and /(= [a, L] +1<% L=[a, ~ + L ~ Go=K=+[a, L]+L=+ [a, /(]. Let <', "> be the Killing form of the Lie algebra G O or any other bilinear, nongenerate, symmetric form invariant relative to all automorphisms of the Lie algebra G o . All the spaces participating in the decompositions of the spaces K, L, and G O are pairwise orthogonal, and the restrictions of the form <', "> to them are nondegenerate. We further identify the spaces participating in these decompositions with the spaces dual to them by means of the form <', "> or its restriction. The rank of an arbitrary reductive Lie algebra Q is the dimension of its Cartan subalgebra. The rank is denoted by rkQ.

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THEOREM 2.4.2 (A. V. Brailov). Let (G0,ffR)be a split simple Lie algebra; let G o = K + L be the Cartan decomposition indicated above; letias be an arbitrary element; let G~ be the centralizer of a in Go; let Cent G~ be the center of the Lie algebra G~; let the element bGLnCent O$; let D:Ka-+K a be a symmetric operator invariant relative to all inner automorphisms of the Lie algebra Ka; let Iz,...,l r be collection of homogeneous, algebraically independent invariants of the Lie algebra G O of degrees m I + l,...,m r + I, where ml,...,m r are the exponents of the Weyl group. Then for the operaterlqa,~,D, defined by the matrix ~ a , b, D --~i

relative to the decomposition A"~--[a, LI.-}-ffa , the Euler equation A~[X, ~a,b,D(X)], XEK, has the following integrals: the integrals I{,a(X), which are by definition the functional coefficients of ~2 in the polynomialP~,a(%, X)~-f~(X@%a), and the integrals[g(X)= (g, X>, where gGK a. All integrals of the formll[,w commute pairwise with each integral s From the integrals I{.~ and i it is possible to select a number of independent integrals equal to q + dimK a, where q is the number of integrals selected here of the form l{,a. For the number q we have the following expression: q = i/2(dimK/Ka + r k K - rkKa). The rank of the reductive Lie algebra K is equal to the number of odd exponents of the series ml,...,m r. Remark. In a manner similar to the way in which Theorem 2.4ol augments Theorem 2.1.1 of A. S. Mishchenko and A. T. Fomenko on the independence of integrals of the so-called "compact" series, Theorem 2.4.2 augments Theorem 2.1.1 on the independence of integrals of the "normal" series for the case of singular operators ~a,b,D3.

Euler Equations on Solvable Lie Algebras 3.1.

Euler Equations on Borel Subalgebras of Semisimple Lie Algebras.

Let G be a c o m -

plex simple Lie algebra, let H be its Cartan subalgebra, let G-----f-f@~G=be the Caftan de~0

composition, and let {hi, e~} be the Chevallier basi~.

We consider the Borel subalgebra

in

~>0

to it there corresponds the Lie group ~G. In the Weyl group W(G, H) of the Lie algebra G there exists an element w 0 of greatest length. A complete involutive family of functions on Borel subalgebras BG in simple Lie algebras was constructed in the works of Trofimov [88, 90]. THEOREM 3.1..1 (see [4]). Let d i be semiinvariants of the representation Ad* of the Lie g r o u p ~An. There exists an open, dense subset U c B A ~ * such that if a function f on BA~ depends in polynomial fashion on the functions ~i(X~-~a),i= 1..... n,%6R, a6U, then the system of equations x = {x,dfx} is completely integrable in the Liouville sense on orbits of general position of the representation Ad* of the Lie group ~Anl An analogous construction can be carried out for ~Sp(n)(see [88]). THEOREM 3.1.2 (see [88]). Let d i be semiinvariants of the representation Ad* of the Lie group ~Sp(n). There exists an open, dense subset~UcBSp(n) * such that if a function f on BSp(n)* depends functionally on the functions di(x+%a), i=l,..., n, %ER, a~U, then the system of equations x = {x, d[x} is completely integrable in the Liouville sense on orbits of general position of the representation Ad* of the group!~Sp(n). An explicit description of the semiinvariants for'~An can be found in the work of Arkhangel'skii [4] and for ~Sp(n) in the work of Trofimov [88]. In the work [90] a description is given of the semiinvariants and, in particular, the invariants for all Borel subalgebras BG in simple Lie algebras G. Let A~(X) be the lower-left-corner minor of order i of the matrix X, and let Oij(s) be the border of the minor As(X) corresponding to the element xij. THEOREM 3.1.3 (see [88]). Suppose a function f on BSO(n) ~ depends functionally on the semiinvariants of the representation Ad* of the Lie group ~SO(n) and, further, in the case n = 2k on the coordinates of the maximal Abelian subalgebra of BSO(n), i.e., on Yi,j+n (i + j < n), in the case BSO(4k + l) on the shifts Ak+l and the coordinates Yi,j+k (i + j < k) of

2722