Study of periodic solutions of some Hamiltonian systems: systems with non-trivial prime integrals. Billiard problem.

Summary This work consists of two parts. The first part deals with the existence of periodic Hamiltonian trajectories on certain r#2#n subvarieties, intersections of hyper level surfaces of several Hamiltonian functions in involution. We define the notion of periodic trajectory for such subvarieties and we define a contact condition which ensures the existence of at least one periodic trajectory. We also prove that a subvariety satisfying the contact condition has a strictly positive symplectic capacity. The second part studies the periodic solutions for the billiard problem in a boundary open of r#n limits of regular solutions of Lagrangian systems with potential wells. The main result establishes a precise link between the Morse index of approximate solutions (considered as critical points of Lagrangian functionals) and some properties of the periodic trajectory of boundary billiards.