Commun. Math. Phys. 209, 353 – 392 (2000) Communications in Mathematical Physics © Springer-Verlag 2000 A Geometric Approach to the Existence of Orbits with Unbounded Energy in Generic Periodic Perturbations by a Potential of Generic Geodesic Flows of T 2 Amadeu Delshams 1 , Rafael de la Llave 2 , Tere M. Seara 1 1 Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. E-mail: 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 Received: 22 September 1998 / Accepted: 2 August 1999 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 T 2 by a generic periodic potential. 1. Introduction The goal of this paper is to give a proof, using geometric perturbation methods, of a result proved by J.N. Mather using variational methods [Mat95]. We will prove: Theorem 1.1. Let g be a C r generic metric on T 2 , U : T 2 × T → R a generic C r function, r sufficiently large. Consider the time dependent Lagrangian L(q, ˙ q,t) = 1 2 g q ( ˙ q, ˙ q) − U(q,t), where g q denotes the metric in T q T 2 . Then, the Euler–Lagrange equation of L has a solution q(t) whose energy E(t) = 1 2 g q ( ˙ q(t), ˙ q(t)) + U(q(t),t), tends to infinity as t →∞. Remark 1.2. Note that, in fact, the only unbounded part in E(t) is ˙ q(t), so that the theorem could be expressed as unbounded growth in the velocity. Remark 1.3. As it is usually the case in problems of diffusion, one not only constructs orbits whose energy grows unbounded, but also orbits whose energy makes more or less arbitrary excursions. We formulate this precisely in Theorem 4.26, and deduce Theorem 1.1 from it.