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arXiv · 1505.06866

Rigidity in topology C^0 of the Poisson bracket for Tonelli Hamiltonians

M. -C Arnaud
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Abstract

We prove the following rigidity result for the Tonelli Hamiltonians. Let T * M be the cotangent bundle of a closed manifold M endowed with its usual symplectic form. Let (F\_n) be a sequence of Tonelli Hamiltonians that C^0 converges on the compact subsets to a Tonelli Hamiltonian F. Let (G\_n) be a sequence of Hamiltonians that that C^0 converges on the compact subsets to a Hamiltonian G. We assume that the sequence of the Poisson brackets ({F\_n , G\_n }) C^0-converges on the compact subsets to a C^1 function H. Then H = {F, G}.

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M. -C Arnaud. 2015-05-26. Rigidity in topology C^0 of the Poisson bracket for Tonelli Hamiltonians. https://arxiv.org/abs/1505.06866

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