Periodic trajectories with fixed energy on Riemannian and Lorentzian manifolds with boundary

We state some existence and multiplicity results for periodic solutions with prescribed energy of a Lagrangian system and for closed geodesics on Riemannian manifolds convex close to their boundary. As an application we get the existence and multiplicity of periodic trajectories with fixed energy on a class of physically important Lorentzian manifolds with boundary. Entrata in Redazione il 23 aprile 1998. Part of the Ph.D. thesis of this author is contained in this paper. Partially supported by MEC grant PB97-0784-C03-01.