Geometrical Aspects of Integrable Systems

Abstract

We review some basic theorems on integrability of Hamiltonian systems, namely the Liouville-Arnold theorem on complete integrability, the Nekhoroshev theorem on partial integrability and the Mishchenko-Fomenko theorem on noncommutative integrability, and for each of them we give a version suitable for the noncompact case. We give a possible global version of the previous local results, under certain topological hypotheses on the base space. It turns out that locally affine structures arise naturally in this setting.

Lectures on integrable Hamiltonian systems

“…Given the superintegrable system (F i ) (2.1) on (Z, Ω), the well known Mishchenko -Fomenko theorem (Theorem 2.6) states the existence of (semi-local) generalized actionangle coordinates around its connected compact invariant submanifold [9,18,61]. The Mishchenko -Fomenko theorem is extended to superintegrable systems with non-compact invariant submanifolds (Theorem 2.5) [22,24,41,74]. These submanifolds are diffeomorphic to a toroidal cylinder…”

“…Let us consider a non-autonomous mechanical system on a configuration space Q → R in Section 4.2. Its phase space is the vertical cotangent bundle 24).A Hamiltonian of a non-autonomous mechanical system is a section h (4.30) of the one-dimensional fibre bundle (4.20) -(4.23):…”

“…In these Lectures, we consider integrable Hamiltonian systems in a general setting of invariant submanifolds which need not be compact [20,22,23,24,35,41,74,86]. These invariant submanifolds are proved to be diffeomorphic to toroidal cylinders R m−r ×T r (Theorem 1.10).…”