On the Integrability of Tonelli Hamiltonians

TL;DR

This paper explores a weaker form of integrability for Tonelli Hamiltonians, showing that involution of integrals can be omitted while still deriving significant dynamical properties, especially on the torus.

Contribution

It demonstrates that for Tonelli Hamiltonians, the involution condition on integrals can be dropped, leading to classical integrability on the torus without this assumption.

Findings

Weaker integrability condition implies classical integrability on the torus.

Existence of independent integrals relates to Mather and Aubry sets.

Non-trivial common invariant sets exist for Hamiltonians Poisson-commuting with a Tonelli Hamiltonian.

Abstract

In this article we discuss a weaker version of Liouville's theorem on the integrability of Hamiltonian systems. We show that in the case of Tonelli Hamiltonians the involution hypothesis on the integrals of motion can be completely dropped and still interesting information on the dynamics of the system can be deduced. Moreover, we prove that on the n-dimensional torus this weaker condition implies classical integrability in the sense of Liouville. The main idea of the proof consists in relating the existence of independent integrals of motion of a Tonelli Hamiltonian to the size of its Mather and Aubry sets. As a byproduct we point out the existence of non-trivial common invariant sets for all Hamiltonians that Poisson-commute with a Tonelli one.