Hamiltonian group actions and dynamical systems of calogero type

Abstract

In recent years there has been a renewed interest in completely integrable Hamiltonian systems, particularly in conjunction with the study of certain non-linear partial differential equations such as the Korteweg-de Vries equation and their "soliton" solutions. For example, see the paper by Moser[12] where several of these completely integrable systems are studied by the "Lax method" of relating these problems to isospectral families, that is, to curves of matrices with the same eigenvalues. In this paper we wish to establish the complete integrability of certain of these systems by a method which is more directly related to group theory and to a familiar procedure in classical mechanics. The procedure that we have in mind is the one of "quotienting out" variables associated to known integrals so as to reduce the order of the system. For example, starting out with a system of several particles in R3 for which the total linear momentum is conserved, one obtains a system with three fewer degrees of freedom by looking at the submanifold of phase space consisting of all states with a definite value of the total linear momentum and then ignoring the position of the center of mass of the system. In physical language, one usually denotes this process as introducing "co-ordinates relative to the center of mass". In more mathematical terms, we can express the procedure as follows: let "I' denote the submanifold in question. Then the restriction of the symplectic form of phase space to "I' is singular and has a three-demensional null space at each point of "I' This defines a three-dimensional foliation of "I'; the quotient space of this foliation is a symplectic manifold and the original Hamiltonian defines a Hamiltonian on this quotient symplectic manifold. Now, in the usual applications, applying this method of reduction simplifies the equations of motion. However, it is conceivable that quite the reverse might be the case-that the original equations of motion are quite transparent, but the equations of motion of the quotient system appear more complicated. Indeed, we shall show that the Calogero system (cf.[3]) of n particles on the line moving under the inverse square potential, and the corresponding Sutherland system (cf.[15]> of n particles on the circle moving under the sin-* potential are examples of mechanical systems which arise as quotients of much simpler looking mechanical systems.

New integrable systems from unitary matrix models

“…Instead, we shall work with the canonical structure of the original matrix problem and make use of the fact that the projection from the full matrix phase space of U to the phase space of its eigenvalues is a hamiltonian reduction [6,7]. To see this, define from (1) a canonical momentum P U P U = δL δU = −U −1U U −1 (28)…”

On collective complete integrability according to the method of Thimm

“…[8], and is an immediate consequence of (1.1) that $~1(<5 > ) is a co-isotropic submanifold, and that the null foliation through p of <&~l (0) This is a generalization of a theorem of Duflo and Vergne [2] which asserts that if € is a maximal dimensional orbit in g* and a e.6 then g a is abelian. The result (1.12) was obtained independently by Mishchenko in [12].…”

“…Proof of (1.12). From the definition of g a it follows that if £eg a then [1][2][3][4][5][6][7][8][9][10][11][12][13][14] [ga,g«]< By (1.2) this implies that (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) [ga,ga] = gp, which gives (1.12).…”

Dualities in Integrable Systems: Geometrical Aspects

“…The first one involves the Hamiltonian reduction procedure for the group like symplectic manifolds. The group approach to the integrable many body system is known for a while [5] and the idea to use the Hamiltonian reduction procedure for the finite dimensional groups to derive the Calogero type systems was invented in [6]. The generalization along this line involves the affine groups and algebras [7,8] which amounts to the more general Calogero and Ruijsenaars systems while the double affine structures amount to the elliptic Calogero and Ruijsenaars models [9,10].…”