Weak Liouville-Arnold Theorems & Their Implications

TL;DR

This paper extends Liouville-Arnold theorems to certain Hamiltonian systems with symmetries, revealing new insights into invariant Lagrangian graphs and configuration space structures.

Contribution

It introduces weak Liouville-Arnold theorems for Hamiltonians with non-involutive constants of motion, expanding the understanding of integrability in symmetric systems.

Findings

Existence of invariant smooth Lagrangian graphs under new conditions

Structural results on configuration spaces of these Hamiltonian systems

Generalization of classical integrability theorems to broader settings

Abstract

This paper studies the existence of invariant smooth Lagrangian graphs for Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli Hamiltonians with n independent but not necessarily involutive constants of motion and obtain two theorems reminiscent of the Liouville-Arnold theorem. Moreover, we also obtain results on the structure of the configuration spaces of such systems that are reminiscent of results on the configuration space of completely integrable Tonelli Hamiltonians.