Geometrical aspects of integrable systems

TL;DR

This paper reviews fundamental theorems on the integrability of Hamiltonian systems, extending classical results to noncompact cases and exploring the emergence of affine structures in the global geometric context.

Contribution

It provides adapted versions of key integrability theorems for noncompact systems and introduces a global perspective with topological and geometric insights.

Findings

Extended Liouville-Arnold, Nekhoroshev, Mishchenko-Fomenko theorems to noncompact cases

Identified conditions for global integrability with topological hypotheses

Discovered natural appearance of affine structures in the global setting

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.