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

M.-C Arnaud (1, 2)

1 IUF - Institut Universitaire de France
2 LMA - EA2151 Laboratoire de Mathématiques d'Avignon

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}.

Keywords: Lagrangian functions 2010; Hamiltonian dynamics; Poisson bracket; Tonelli Hamiltonians

Domains: Dynamical Systems [math.DS]; Symplectic Geometry [math.SG]

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