Lectures on integrable Hamiltonian systems

G.Sardanashvily, Lectures on integrable Hamiltonian systems, arXiv: 1303.5363 

Abstract. We consider integrable Hamiltonian systems in a general setting of invariant submanifolds which need not be compact. For instance, this is the case a global Kepler system, non-autonomous integrable Hamiltonian systems and integrable systems with

time-dependent parameters.

Introduction

The Liouville -- Arnold theorem for completely integrable systems, the Poincar\'e -- Lyapounov -- Nekhoroshev theorem for partially integrable systems and the Mishchenko -- Fomenko theorem for the superintegrable ones state the existence of action-angle coordinates around a compact invariant submanifold of a Hamiltonian integrable system which is a torus. However, it is well known that global extension of these action-angle coordinates meets a certain topological obstruction.

Note that superintegrable systems sometimes are called non-commutative (or non-Abelian) completely integrable systems.

In these Lectures, we consider integrable Hamiltonian systems in a general setting of invariant submanifolds which need not be compact. These invariant submanifolds are proved to be diffeomorphic to toroidal cylinders. A key point is that a fibred manifold whose fibres are diffeomorphic either to a compact manifold or an Euclidean space is a fibre bundle, but this is not the case of toroidal cylinders.

In particular, this is the case of non-autonomous integrable Hamiltonian systems and Hamiltonian mechanics with time-dependent parameters.

It may happen that a Hamiltonian system on a phase space Z falls into different integrable Hamiltonian systems on different open subsets of Z. For instance, this is the case of the Kepler system. It contains two different globally superintegrable systems on different open subsets of a phase space Z. Their integrals of motion form the Lie algebras so(3) and so(2,1) with compact and non-compact invariant submanifolds, respectively.

Geometric quantization of completely integrable and superintegrable Hamiltonian systems with respect to action-angle variables is considered. The reason is that, since a

Hamiltonian of an integrable system depends only on action variables, it seems natural to provide the Schrodinger representation of action variables by first order differential

operators on functions of angle coordinates.

Throughout the Lectures, all functions and maps are smooth, and manifolds are real smooth and paracompact. We are not concerned with the real-analytic case because a paracompact real-analytic manifold admits the partition of unity by smooth functions. As a consequence, sheaves of modules over real-analytic functions need not be acyclic that is essential for our consideration.