Fine tuning of resonances and periodic solutions of Hamiltonian systems near equilibrium (Q1120718)

A Hamiltonian system which has the origin as isolated equilibrium is considered. It is proved that the number of families of periodic solutions near the origin is linked to the number of critical points of a suitable function F defined in a manifold in phase space. A basic role in the analysis described in the paper is held by a lemma on commuting vector fields. The results of the lemma are specialized to the case of Hamiltonian vector fields, using also the theory of normal forms. The families of periodic solutions are connected to the critical points of a suitable function in phase space. The stability of these families of solutions is discussed too.