On the Integrability of Tonelli Hamiltonians

Details On the Integrability of Tonelli Hamiltonians On the Integrability of Tonelli Hamiltonians

2009-03-25

arxiv.org/abs/0903.4300v2

Mathematics

Dynamical Systems

19 pages. Version accepted by Trans. Amer. Math. Soc

Scientific paper

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.

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