Eight Lectures on Integrable Systems

Abstract

These lectures provide an introduction to the theory of integrable systems from the point of view of Poisson manifolds. In classical mechanics, an integrable system is a dynamical system on a symplectic manifold which admits a complete set of constants of motion which are in involution. These constants are usually constructed by means of a symmetry group G acting symplectically on the phase-space. The point of view adopted in these lectures is to replace the group G by a “Poisson action” of the algebra of observables on de.ned by a second Poisson bracket on . The development of this idea leads to the concept of a bihamiltonian manifold, which is a manifold equipped with a pair of “compatible” Poisson brackets. The geometry of these manifolds builds upon two main themes: the Marsden-Ratiu (henceforth, MR) reduction process and the concept of generalized Casimir functions. The MR reduction clari.es the complicated geometry of a bihamiltonian manifold and de.nes suitable reduced phase-spaces, while the generalized Casimir functions constitute the Hamiltonians of the integrable systems. These three concepts (bihamiltonian manifolds, MR reduction, and generalized Casimir functions) are introduced, in this order, in the .rst three lectures. A simple example (the KdV theory) is used to show how they may be applied in practice.