Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities

TL;DR

This paper extends the classical Arnold-Liouville theorem to nondegenerate singular level sets in integrable Hamiltonian systems, providing a topological and local geometric description crucial for understanding their global structure.

Contribution

It introduces a detailed topological and geometric framework for neighborhoods of singular level sets, including action-angle functions and decomposition into simple singularities.

Findings

Neighborhoods admit a finite cover with action-angle coordinates

Singular level sets decompose into products of simple singularities

Results are foundational for global topological analysis of integrable systems

Abstract

The classical Arnold-Liouville theorem describes the geometry of an integrable Hamiltonian system near a regular level set of the moment map. Our results describe it near a nondegenerate singular level set: a tubular neighborhood of a connected singular nondegenerate level set, after a normal finite covering, admits a non-complete system of action-angle functions (the number of action functions is equal to the rank of the moment map), and it can be decomposed topologically, together with the associated singular Lagrangian foliation, to a direct product of simplest (codimension 1 and codimension 2) singularities. These results are essential for the global topological study of integrable Hamiltonian systems.