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Journal Articles Nonlinearity Year : 2015

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

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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Dates and versions

hal-01154913 , version 1 (25-05-2015)

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M.-C Arnaud. Rigidity in topology C^0 of the Poisson bracket for Tonelli Hamiltonians. Nonlinearity, 2015, 28 (8), pp.2731-2742. ⟨hal-01154913⟩
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