Periodic bounce solutions of dynamical conservative systems and their minimal period

Abstract

SEVERAL important studies have been made on the problem of finding periodic solutions of a conservative dynamical system. An important part of such a study is concerned with the case in which the minimal period of the trajectory is given (see e.g. [l, 3, 6, 9, 14)). In this paper we consider the problem of existence of periodic trajectories with prescribed minimal period, when the material point moves in a convex billiard in I?” under the action of a conservative field and it bounces in a completely elastic way against the boundary, hitting it orthogonally. If Q is a bounded open convex subset of I?” (not necessarily regular), T > 0 and V is a real Cz-function in a neighbourhood of 0, we consider the nonconstant periodic orbits q : R” -+ Q (which are called “principal bounce trajectories” @.b.t.)) having period 2T such that: