Periodic perturbations of Hamiltonian systems

We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincar\ue9\u2013Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems