On the integrability of Tonelli Hamiltonians

Alfonso Sorrentino (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DPMMS - Department of Pure Mathematics and Mathematical Statistics

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.

Mots clés: Aubry-Mather theory; Tonelli Hamiltonians; Liouville theorem; Integrals of motion; commuting Hamiltonians; Invariant Lagrangian foliations

Domaines: Systèmes dynamiques [math.DS]

Alfonso Sorrentino. On the integrability of Tonelli Hamiltonians. 2009. ⟨hal-00458379⟩