TITLE: A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of ${\bf T}^{2}$ AUTHORS: Amadeu Delshams(1), Rafael de la Llave(2), Tere M. Seara(1) (1) Departament de Matem\`atica Aplicada I, Universitat Polit\`ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain E-mails: amadeu@ma1.upc.es, tere@ma1.upc.es (2) Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA E-mail: llave@math.utexas.edu 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 flow in ${\bf T}^{2}$ by a generic periodic potential. KEYWORDS: geodesic flow, a priori chaotic systems, Melnikov method, normal hyperbolicity. 1991 MSC numbers: 58F17, 34C37