A. Delshams, R. de la Llave, T. M. Seara A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of ${\bf T} ^{2}$ (263K, latex) ABSTRACT. We give a proof based in geometric perturbation theory of a result proved by J.N.~Mather using variational methods. Namely, the existence of orbits with unbounded energy in perturbations of a generic geodesic flows in ${\bf T}^2$ by a generic periodic potential.