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We prove a variational principle for geodesies on a Lorentzian manifold M. admitting a timelike Killing vector field. Using this principle and standard techniques of global nonlinear analysis we establish the existence of geodesies that join two fixed points of .M, under a suitable coercivity assumption on M. Whenever M is non contractible, we also get a multiplicity result for geodesies in M. joining two fixed points.

An intrinsic approach to the geodesical connectedness of stationary Lorentzian manifolds