We show that certain mechanical systems, including a geodesic °ow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy. The assumptions we make in the case of geodesic °ows are: a) The metric and the external perturbation are smooth enough. b) The geodesic °ow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection. c) The frequency of the external perturbation is Diophantine. d) The external potential satisØes a generic condition depending on the periodic orbit considered in b). The assumptions on the metric are C2 open and are known to be dense on many manifolds. The assumptions on the potential fail only in inØnite codimension spaces of potentials. The proof is based on geometric considerations of invariant manifolds and their intersections. The main tools include the scattering map of normally hyperbolic invariant manifolds, as well as standard perturbation theories (averaging, KAM and Melnikov techniques). We do not need to assume that the metric is Riemannian and we obtain results for Finsler or Lorentz metrics. Indeed, there is a formulation for Hamiltonian systems satisfying scaling hypotheses. We do not need to make assumptions on the global topology of the manifold nor on its dimension.