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On the integrability of Tonelli Hamiltonians


Author: Alfonso Sorrentino
Journal: Trans. Amer. Math. Soc. 363 (2011), 5071-5089
MSC (2010): Primary 37J50, 37J35, 37J15; Secondary 53D12, 53D25
Published electronically: May 20, 2011
MathSciNet review: 2813408
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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 Hamiltonian.


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Additional Information

Alfonso Sorrentino
Affiliation: Ceremade, UMR du CNRS 7534, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France
Address at time of publication: Department of Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: A.Sorrentino@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2011-05492-9
Received by editor(s): March 27, 2009
Published electronically: May 20, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.