Algebraization of Hamiltonian systems on orbits of Lie groups

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6~ is the number of nonisomorphic algebras of functions of class i Fig. 9 Let M 2n be a symplectic manifold on which a Hamiltonian Lie group ~ acts. We say that this action is multiplicity-fre~ if the ring of ~-invariant functions on M 2n is commutative relative to the Poisson bracket. I Let ~ :A42~-+G * be the moment mapping corresponding to the given action of the Lie group | on the symplectic manifold M2n:. Following the terminology of Guillemin-Sternberg (see [138])~ we call functions of the form f'#, where f: G*-+R, collective. A completely integrable system whose Hamiltonian is collective and all first integrals are likewise we call a completely integrable system of collective type; see [164, 138, 155]. A necessary condition that the manifold M 2n admit a completely integrable system of collective type is the condition of multiplicity-free action of the corresponding Lie group G 9 For the groups U(n) and O(n) this conditions is also sufficient (see [50]). The following basic theorem holds and establishes a connection between multiplicity-free representations and the existence of completely integrable systems of collective type. THEOREM 1.5.1 (see [137]). A representation of a Lie group ~ in the space L2(M) is multiplicity-free if and only if the action of the Lie group ~ on the sympiectic manifold M is multiplicity-free [here Tgf(m) = f(g-lm)]. 2.

Al~ebraization of Hamiltonian Systems on Orbits of Lie Groups

2.1. General Definition. One of the methods of integrating a given Hamiitonian system is the identification of it with some Euler equation on a suitable Lie algebra. Definition 2.1.1. A dynamical system v on a manifold M n is said to admit imbedding in a Lie algebra if M n can be identified with a submanifold N c G* in the dual space G* to some Lie algebra G of the Lie group ~ in such a way that a) N = U ~ ( f ) is invariant relative to the coadjoint representation of the Lie group ~ , i.e., the submanifold N is the union of orbits G(f) in the coadjoint representation; b) the vector field v is tangent (after this identification) to the orbits ~(f) of the coadjoint representation of the Lie group ~ On G*; c) the vector field v on N is Hamiltonian on the orbits d(f) relative to the canonical Kirillov form and has the form v = sgrad f, where ~6C~(~(~)). The class of such systems Contains important mechanical systems. Below, as an example, we treat the motion of a point on the sphere under the action of a linear force. Other examples (the inertial motion of a rigid body with a fixed point, the Toda lattice) are considered, for example, in [103]. 2711

2.2. Algebraization of the Neumann System. We consider a material point on the sphere S n = {xl[x I = i} in the field of a quadratic potential U(x) = 2-i(x, AE), where A is a real symmetric matrix which may be assumed diagonal with no loss of generality. Such a system is completely integrable which for n = 2 was shown in 1859 by K. Neumann using the method of separation of variables in the Hamilton-Jacobi equation. The equations of motion of this system have the form xj=--ajxT~-vxj, where ~=((x, Ax)--[xI2) is the Lagrange multiplier. They can be rewritten in the form

i~ = ~ ,

bj-- - a j z i +(Ax|

We introduce the matrices X = x |

P=X|174 following matrix system of equations

)r

Pl,

#--[A, X l,

~ H) xj.

Then these equations are equivalent to the

]xl=l,

In these equations it is possible to make the change then be assumed that #~, P, AC sl(n, R).

(x, g)=O.

X-+X--n,II, A-+A--(n:I)(trA)I;

it may

To realize the Neumann system on an orbit of the coadjoint representation we require the following construction. Let G be an arbitrary Lie algebra. We set by definition fi(G) = O @ O with commutator [(~, N~), (~, N~)]=([$~, $~], adglN2"adg=D0. To it there corresponds the Lie group ~ ( ~ ) = ~ X G with multiplication (g~, ~0"(g~, ~):(g~g~, ~q-Adg~(~)). In ~(sl(~, R)) we consider the subalgebra 7 V = s o ( ~ ) X S , where S is the vector space of all symmetric matrices, and the subspace K = S Xs0(~). The Neumann system indicated above can be realized on a special orbit of the coadjoint representation of the Lie group ~, corresponding to the Lie algebra N. The space N* can be identified with K by using the direct sum of the Cartan-Killing forms

< A, B > =--2-~tr(AB): THEOREM 2 . 2 . 1 . We c o n s i d e r an o r b i t of t h e group ~ = S O ( n ) X S , . p a s s i n g t h r o u g h the p o i n t (z| in t h e space N* = S @so (n), z = ( 1 . . . . . 1)/t~n -, I t c o n s i s t s of p a i r s (X, P), X = x | n-~l, P = x | 1 7 4 [x[=l, (x, y)--0. This 2(n - 1 ) - d i m e n s i o n a l o r b i t w i t h t h e s y m p l e c t i c K i r i l l o v s t r u c t u r e i s d i f f e o m o r p h i c to TSn-~ under t h e mapping (X, P) + (x, y) w i t h t h e symp l e c t i c s t r u c t u r e induced on TSn-~ by t h e s t r u c t u r e ~ d x i A d y i in t h e space R=~(x~, yi). The H a m i l t o n i a n H ( x , p)=--2-~ =~ < A, X> d e f i n e s on t h i s o r b i t t h e e q u a t i o n s of motion f o r t h e Neumann system ) ~ = [ P , X], P = [ X , A]~ ] x [ = l , (x, y ) = 0 . For t h e p r o o f , see [162]. l i s h e d in [117]. 3.

The H a m i l t o n i a n p r o p e r t y of t h e s e e q u a t i o n s was a l s o e s t a b -

Extension of a Geodesic Flow

3.1. A Geodesic Flow on the Sphere. We consider the sphere S n imbedded in standard fashion in R~+I: S~={x1!xl~=l}, x=(xo, xl ..... Xn)CR ~+I. A geodesic flow on it is described by the equation R = -vx, where v(x) = (~)2. The equation of the geodesics has the Hamiltonian form x

= ~OH ' Y=

0H Ox

with Hamiltonian H=2-11x[~[y[ ~ on restriction to the tangent bundle

{(x, y) I[xl=1, (x,y)=0}. We thus have a Hamiltonian system in R 2~+2, which after restriction to the tangent bundle becomes a geodesic flow on the sphere S n. It turns out that this model example can be generalized to the case of arbitrary surfaces i n R n 3.2. A Geodesic Flow on Surfaces in Rn§ Suppose an arbitrary n-dimensional surface is isometrically imbedded in R n+~ with the standard metric ds 2. Suppose, further, that f(x) is a smooth, convex function on R n+i such that f(x) ~ ~ as IxI + ~. We define F(x, y)=min t

f(x + ty) for x, yER n§ and consider the Hamiltonian system ~ sesses the integrals I l = lyl 2 and 12 = F(x, y). e q u a t i o n s ~ = 0 F-0y,

~=

0F ~_~,~=

oOF Ox

This system pos-

We can therefore consider solutions of the

- -OF ~ , satisfying the additional conditions [yI=l, F(x, y)=0.

We consider a point (x, y) of phase space as a line l=l(x,y)={x~ty},passing through the point x~6Rn§ in the direction y. Thus, (x, y) is a directed line l=l(x,y) with a distinguished point x on it. We additionally suppose that grad f(x) r 0 on the surface f = 0, i.e., the equation f = 0 defines a smooth submanifold.

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