Timelike spatially closed trajectories under a potential on splitting Lorentzian manifolds

We study the periodic motions of a relativistic particle submitted to the action of an external potential V . On a wide class of Lorentzian manifolds, we find timelike solutions of a differential equation (depending on V ) closed in the spatial component and satisfying a Dirichlet condition in the temporal one. We prove a multiplicity result for the critical points of the (strongly indefinite) functional associated to the problem, using a saddle type theorem based on the notion of relative category. The periodicity of the problem, the non–compactness of the manifold and the lack of some assumptions involving the relative category make necessary to use a suitable penalization scheme and a Galerkin approximation.