On generic properties of Lagrangians on surfaces: the Kupka-Smale theorem

Abstract

We consider generic properties of Lagrangians. Our main result is the Theorem of Kupka-Smale, in the Lagrangian setting, claiming that, for a convex and superlinear Lagrangian defined in a compact surface, for each $k\in \mathbb{R}$, generically, in Ma\~n\'e's sense, the energy level, $k$, is regular and all periodic orbits, in this level, are nondegenerate at all orders, that is, the linearized Poincar\'e map, restricted to this energy level, does not have roots of the unity as eigenvalues. Moreover, all heteroclinic intersections in this level are transversal. All the results that we present here are true in dimension $n \geq 2$, except one, whose proof we are able to obtain just for dimension 2.