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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https://hal-univ-avignon.archives-ouvertes.fr/hal-01154913
Contributor : Marie-Claude Arnaud <>
Submitted on : Monday, May 25, 2015 - 4:47:20 PM
Last modification on : Tuesday, January 14, 2020 - 10:38:15 AM
Long-term archiving on: Tuesday, September 15, 2015 - 6:49:36 AM

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  • HAL Id : hal-01154913, version 1
  • ARXIV : 1505.06866

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

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