  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}.
