Morse index of approximating periodic solutions for the billiard problem. Application to existence results This paper deals with periodic solutions for the billiard problem in a bounded open set of $\hbox{\Bbbvii R}^N$ which are limits of regular solutions of Lagrangian systems with a potential well. We give a precise link between the Morse index of approximate solutions (regarded as critical points of Lagrangian functionals) and the properties of the bounce trajectory to which they converge. Categories:34C25, 58E50