A multidimensional birkhoff theorem for time-dependent tonelli hamiltonians

Un théorème de Birkhoff multidimensionnel pour les hamiltoniens de Tonelli dépendant du temps

Marie-Claude Arnaud (1) , Andrea Venturelli (1)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon

Let M be a closed and connected manifold, H : T * M × R/Z → R a Tonelli 1-periodic Hamiltonian and L ⊂ T * M a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if L is invariant by the time-one map of H, then L is a graph over M. An interesting consequence in the autonomous case is that in this case, L is invariant by all the time t maps of the Hamiltonian flow of H.

Keywords: generating functions.; Weak KAM Theory; Lagrangian submanifolds; Lagrangian Dynamics

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

Marie-Claude Arnaud, Andrea Venturelli. A multidimensional birkhoff theorem for time-dependent tonelli hamiltonians: Birkhoff theorem. Calc. Var. Partial Differential Equations, 2017, 56 (4), pp.122. ⟨hal-01309652v2⟩