Euler equations on nonsolvable Lie Algebras with nontrivial radical

(o:)

Poisson bracket on T~ goes over into the standard Poisson bracket on the

space

If Ht(F)=2tr(F+a) ~-, where F 6 A and

(y~)|

0

a-----

0

9

~

~Tn.

,~-1

then direct computation shows that H=H[ [. This realization makes it possible to construct a complete collection of first integrals of the Toda chain [83]. 4.

Euler Equations on Nonsolvable Lie Algebras with Nontrivial Radical

4.1. Cases of Complete Integrability of the Equations of Inertial Motion of a Multi-dimensional Solid Body in an Ideal Fluid. The system indicated in the title we have already imbedded in the Lie algebra E(n) of the group of motions of the space R n. It turns out that in this case the method of shift of argument makes it possible to construct a complete commutative coilection of integrals on orbits of general position. THEOREM 4.1.i (see [i01, 102]). a) The system of differential equations)~=adQ(x)(X), where Q = Q(a, b, D) is the section operator constructed earlier for E(n), is completely integrable on orbits of general position, b) Let f be an invariant function on E(n)*. Then the functions h~(X)=[(X-b%a) are integrals of the motion for any numbers ~. Any two integrals h A and g~ are in involution on all orbits of the representation Ad* of the Lie group ~(n), and the number of independent integrals of the form indicated is equal to half the dimension of an orbit of general position. Moreover, if ~ is an orbit of maximal dimension +l of the coadjoint representation, then codim~7= t[ n~-]. The integrals are described in the work of Trofimov and Fomenko [I01] or see, for example, the survey [103] or [107] (see also [108, 109, IIi, 130-132]). Complete integrability of the corresponding geodesic flow on the Lie group ~(n)is proved in [95]. The complete commutative collection of functions on the space E(n)* constructed in Theorem 4.1.i plays the role of the "compact" series of integrals for semisimple Lie algebras (see part 2.1 of the present chapter). An analogue of the "normal" series was constructed by A. V. Brailov. Let E(n) be the Lie algebra of the Lie group ~(I~) of motions of Euclidean space R ~. The standard basis of the Lie algebra E(n) consists of elements xij and yk, where xij is an infinitesimal rotation in the (i, j)-th plane, and Yk is the infinitesimal shift in the k-th coordinate, i < j = ]...... n, k = I,...,n. The linear coordinate functions on E(n)* corresponding to the elements xij, Yk we denote by xij and Yk" Let a I ..... a n be arbitrary numbers. We define matrices of dimension(n-~-i)xin+l):E~s=]I6ik6~]]l; here ~ik is the Kronecker n+l

symbol;

n--I

,

Y._---Eu--(n-.cl)-1~E/j;

Y~Eo+Eii(i.~j);

Al~a~Fu, i=l

7=I

xt---HxiiII, where x~i=--xT~,

+ A_~=anE~,.+~, A=A~-FA_t;

x~.n+a=xn+1, k = 0 for k = l .....~ - I ;

X-I---- YIE~.~+~; X=X~n-~' ~-i. i=I

Thus, X, X I, X_ I are matrix functions on the space E(n)*. THEOREM 4.1.2 (A. V. Brailov). Let at> ... >an-t, ~ln~AO, bl, b2..... bn be some numbers. the d~fferential equations i = ad~(x)x, xCE(n)* with the quadratic function n--2 n--1

n--I

2 H = ~'~, "~ b~--bi, x~i__ 2 ~--_lj=~+la~-aJ .=

Then

n--1

yiXin--~L

yi-i'~n Yn

are completely integrable in the Liouville sense on an orbit ~ of the representation Ad* of the group ~(n) in E(n)* for an orbit 0' of general position in E(n)*, A complete collection of commuting integrals is formed by the functional coefficients hk,s (k = 2,...,n + I; s = 0 ..... k) of Xs~-2 in the polynomial hk(~,~-!)=A~(XI~%A~t2(X_~-[-~A_~)), where 5k is the sum of all syrmmetric minors of k-th order. 4.2. Cases of Complete Integrability of th_e Equation of Inertial Motion of a Multidimensional Rigid Body in an Incompressible, Ideally Conducting Fluid. The method of tensor extensions makes it possible to construct complete, com~nutative collections of first integrals for finite-dimensional analogues of the equation of magnetohydrodynamics described above. Let G be a complex, semisimple Lie algebra, and let ~a(G) be the set of functions on G* which are shifts of the invariants F of the coadjoint representation Ad* of the Lie algebra G,

2726

i.e., ~a(G.) consists of functions of the form h~(X) = F(X§ %~C, where a~G* is a fixed covector. Using the procedure of tensor extensions, we shall construct a family of functions ~(~a(G)) on the space ~(G)* (see part 4.3 of Chap. 2). The following theorem was proved by V. V. Trofimov. THEOREM 4.2.1 (see [93, 97]). a) Suppose a function h depends functionally on the family * X ) form a completely integrable Hamiltonian sys~(~a(G)). Then the Euler equations 2~adah<x)( tem on all orbits of general position of the coadjoint representation Ad* of the Lie group ~(~), associated with the Lie algebra ~(G). b) Let G be a complex, semisimple Lie algebra, and let 2==ad~(a.~,o)(x)(X) be the Euler equations on ~(O)*, X6~(O)*, with an operator of the "complex" series. Then this system is completely integrable in the Liouville sense on all orbits of the coadjoint representation of the Lie group Q(~), associated with ~(G). More precisely, let F(x) be any smooth function on ~(G)* which is invariant relative to the coadjoint representation of the Lie group Q(~). Then all functions F(X+s %~C, are first integrals of the Euler equations for all ~6C. Any two such integrals F(X+Ea),H(X+pa),E, >GC, are in involution on all orbits relative to the Kirillov form. From this set of integrals it is possible to select a number of functionally independent integrals equal to half the dimension of an orbit of general position of the coadjoint representation of the Lie group Q(~) associated with the Lie algebra ~(G). An analogous theorem is valid for the compact real form of a semisimple Lie algebra and also for a normal, compact subalgebra. For details, see the works [93, 96, 97]. 4.3. The Equations of Motion of an n-Dimensional Riig~_d_~o_q_~ with a Fixed Point in a Gravitational Field. A completely integrable case of the equations of motion of an n-dimensional rigid body with a fixed point in a gravitational field was first found by Belyaev (see [5]). The case found is a generalization generalization of the Lagrange case to ther, as A. V. Bolsinov has noted, the pletely integrable Hamiltonian systems

of the classical Lagrange case. A somewhat different arbitrary dimension was found by Ratiu [163]. FurLagrange case can be included in a series of comon Lie algebras of the form G| ~)=Q(G).

We associate with an n-dimensional rigid body an orthoframe 0, e~,_ e 2 .... ~en !0 is they fixed point). Then the motion of the body in the fixed coordinate system 0, e I, e2,...,e n can be represented as a path in the group SO(n). The phase space of this system is T'SO(n), while the motion is given by a vector field sgradH, w h e r e H : T * S O ( n ) - + R is the function of total energy which has the f o r m / - ] = T - ~ ( U ) , where T is the left-invariant part of the Hamiltonian, ~:T*~-+@ is the projection of the cotangent bundle, and U : ~ - + R is the potential. The potential U possesses the property that under the action of the group ~ by shifts U generates a finite-dimensional space V. The semidiect sum G I = G + V is defined, where V is considered as an Abelian Lie algebra. As A. Vo Bocharov has noted (see the appendix to [16]), the original system is equivalent to the Hamiltonian system sgradHl, where H I = T I + U l and T I is a function on the component G* equal to the restriction of T to G* and U I is a function on the component V* equal to the value of the functional ~ V ~ on u~Vi Thus, the motion of a rigid body with a fixed point in a gravitational field is described by a Hamiltonian equation on the space (so(n)@Rn) *. As coordinates on (so(n)-t-R~)*~ we take the linear forms ~ij, 7k on (so(n)@Rn) *, which form bases of so(n) and R n, respectively. We choose them so that {vii, v~}=vi~, {v~, 79}=~. The Hamiltonian H in the variables vij, 7k has the form 2 i,]

i

here ~i are diagonal elements of the matrix I of the moments of inertia (all the remaining elements are equal to zero) and r i are the coordinates of the center of gravity in the basis 0,

el,

e z , . . . , e n.

By a generalization of the Lagrange ease Belyaev [5] means the case where

I=

Let

~r_-tl~l[,

g, j 4 n - l ,

we d e f i n e

"'c the

,

r~ . . . . .

functions

r._~----O,

r~O.

rct, W ~~ = ( W ~ : ) ~ ]

2727

THEOREM 4.3.1 (see [5]). For almost all initial conditions (~0ij, V0r) the motion along the trajectories in the Lagrange case occurs along invariant tori of dimension n - i, whereby the tori are given as level sets of the functions ~ij, Vr on orbits of the coadjoint representation. In the work of Ratiu on (so(n)+ R~) * with the real skew-symmetric n • n fixed matrix in so(n), M,

[163, 164], by a Lagrange gyroscope is meant a Hamiltonian system Hamiltonian H = 1/2 <M, ~> + <X, F>, where M = (mij)C SO(n) is a matrix, M = ~J + J~ = C-I~, J = diag(l I ..... In), I~ > 0, X is a F ~SO(n).

By a Lagrange gyroscope Ratiu means the case where:Ii=f2=a, I~=I4 .... = l ~ = b and X~2 = -X2~ = X; all the remaining Xij = 0. The symmetric gyroscope is completely determined as a system for which I I = I~ = ... = In = a and X is an arbitrary element of the algebra so(n). THEOREM 4.3.2 (see [163~ 164]). We suppose that the quantity X~2 ~ 0. equations of motion of an n-dimensional rigid body

1~ ~- [F, ~2l, can be rewritten

Then the Ruler

3~!~ [M, ~1-~ [I', X]

in the form ~t ( F - ~ % M + h 2 C ) = [ F + ~ M ~ - % 2 C ,

~+X%]

( 1) if and only if (i) describes

an n-dimensional L a g r a n g e gyroscope or a completely symmetric gyroscope. The quantities I~=tr(F+iM+%=C) and p f ( F + k M + % ~ C ) = [ d e i ( F + E M + k 2 C ) ] ~ / 2 (for even n) give a complete commutative collection of integrals. Let G be a semisimple Lie algebra, and let e(O)=O| be the tensor extension of the Lie algebra G by means of ~R[x]/(x2)~-~-8R. On ~(O)*~.Q(O) we consider the Hamiltonian 1 F(xi~-ey)--~-~ < ~abD(X), X

> --(b, y).

Under the identification

~(O)*_~-~(O),the Euler equations

2----ad~p(x) (X) take the form

..# = v=

(x), xl-10, (x), vl,

vl,

(2)

where x,g, ~a,b,o(X),b@O. In the case G = so(3) the system (2) takes the form of the classical Lagrange case of the motion of a rigid body in a gravitational field. This remark is due to A. V. Bolsinov. Thus, the Lagrange case turned out to be included in a family of Hamiltonian systems on dual spaces to Lie algebras of type ~(G). THEOREM 4.3.3 (A. V. Bolsinov). The system of equations (2) is completely integrable in the Liouville sense on all orbits of general position of the representation Ad~(D). The first integrals both in Ratiu's Theorem 4.3.2 and in Theorem 4.3.3 of A. V. Bolsinov are constructed by means of the same procedure based on Theorem 4.1.2 of Chap. 2. L-A

Regarding the Kovalevskaya case (see [3, 23]) we note here only that for it a so-called pair has been found (see [54, 30] and also the work [160]).

4.4. Complete Collections of Functions on Some Semidirect Sums. Let p be a representation of a Lie algebra; we denote by Akp and Skp the k-th exterior and k-th symmetric powers of the representation p. THEOREM 4.4.1 (T. A. Pevtsova). Suppose G is a semidirect sum of a simple Lie algebra H and an Abelian algebra with respect to a representation ~. If a) the Lie algebra is H = gl(2n) and the representation ~ = A2p, b) the Lie algebra H = sl(2n) and the representation =smp, c) the Lie algebra is H = sp(2n) and the representation is ~ = p + ~, where p is a minimal representation and T is the one-dimensional trivial representation, then the procedure of chains of subalgebras makes it possible to explicitly construct a complete, involutive family of rational functions on the space G*. A. V. Brailov constructed a complete, involutive family of functions on the space (n)| n , where ~ is a minimal representation. For the case su(~)GC n, where ~ is a minimal representation, two distinct complete involutive collections of functions have been constructed by A. V. Brailov and A. V. Bolsinov, respectively. Each irreducible representation k of the Lie algebra el(2,C) is given by its numerical indices o on the scheme of simple roots;

2728

we denote it by ~k" E. G. Shuvalov showed that shifts . . . of. invariants give a complete, v• family of functions for the Lie algebra si(2, C ) | k+1 for k = i, 2, 3, 4.

in-

~k

THEOREM 4.4.2 (see [9]).

Let L be the space dual to one of the Lie algebras L12, L 9, Then on L there exists a complete, involutive family of functions which are first integrals of the Euler equations described in parts 3.4-3.9 of Chap. i, i.e., these equations are completely integrable in the Liouville sense.

so(3)OE3, Ah, m, LmGA0,n (corresponding notation was introduced in Sec. 3, Chap. i).

CHAPTER 4 QUESTIONS OF NONINTEGRABILITY IN HAMILTONIAN MECHANICS I.

Poincarg Method of Proving Nonintegrability

I.i. Perturbation Theory and the Investigation of Systems Close to Integrable Systems. In the space of all Hamiltonians open regions are distinguished which sometimes fill out almost the entire space and consist of Hamiltonians f of "general position" for which the corresponding Hamiltonian systems v = sgrad f are not Liouville integrable (or not integrabie in some other more general sense). The picture we have described is not a rigorously proved theorem, since in the formulation presented above too many objects are in need of a correct specification which is not always possible. Nevertheless, results presently known of "negative character," i.e., results asserting the nonintegrability of many concrete types of systems, make it possible to view the principle formulated above as some experimental observation which may serve as a guide in the study of concrete systems. Thus, integrable cases fill a set of "measure zero" in the space of all systems. It is already clear from this that the search for integrable Hamiltonian systems is a very difficult problem, since it is necessary to somehow "guess" or algorithmically discover in the immense set of all possible Hamiltonians those rare cases when some additional symmetries cause the appearance of a sufficient number of integrals. In Chap. 2 regular methods were indicated for constructing functions on homogeneous symplectic manifolds which in practice, as a rule, give integrable Hamiltonian systems which are interesting from the viewpoint of mechanics (see Chap. 3). We shall now demonstrate to the reader that a Hamiltonian "taken at random" most often generates a nonintegrable system. Let (M, m) be a symplectic manifold, let v 0 = s g r a d H 0 be a completely integrable Hamiltonian system, and let T n be one of the compact, connected level surfaces of a collection of first integrals fl, f2,..-,fn which are functionally independent and pairwise in involution in a neighborhood of T n. As we know, T n is diffeomorphic to an n-dimensional torus. In an open neighborhood U of the torus T n we consider curvilinear "action-angle" coordinates sl,..., an, 91 ..... ~n, where ~ are angular coordinates on the torus and s i are coordinates normal to the torus. For brevity we introduce the vector notation s=(s1,.:., an), 9=(~i,-i., ~n). We represent the neighborhood U as a direct product U = D n • T n, where D n = D is an open domain in Rn(s), for example, homeomorphic to a sufficiently small disk. Thus, we have separated the regular coordinates in the neighborhood U into two groups: U(s, 9) =Dn(S) xTn(~) 9 The integral trajectories of the system v 0 = s g r a d H 0 are distributed on the torus T n (and on tori s • T n, where s6Dn), near to it), forming a rectilinear winding (see Fig. i0). As we know (see the Liouville theorem), in the coordinates (s, ~) the Hamiltonian H ~ depends only on the variables s, i.e., H = H(s) in the neighborhood U. We now consider a perturbation of the original Hamiltonian system by means of a perturbation of its Hamiltonian H 0. We consider a family of Hamiltonian systems v~=sgradH(s, ~, s), where H(s, 9, s) is a real analytic function defined on the direct product ! U X (--e0, so) and such that for E = 0 we obtain the original Hamiltonian, i.e., H(s, % 0)=H0(s). The Hamiltonian equations

OH~

~=~-~(s),

v0-----sgrad/40 in the coordinates

where ~(s)=(~1(s) .... ,~n(s))i

(s,~} can be written thus:

As we see, these equations are ex-

plicitly integrable. If the point s = s o is fixed, it determines a torus in the neighborhood U on which the vector ~(s ~ is constant (does n o t depend on ~), and hence the equations of motion can be integrated as follows: s(t)~s0, q(t)=~0-b~(s0)t. The equations of the perturbed Hamiltonian system v = s g r a d H can be written in the following form: s ~

O0~' H ~ _---~, 0H . . . . ~.) + .... where f-/(s,~)=/-/0(s)~-sff1(s,

Assuming that the parameter

2729