The minimal period problem of classical Hamiltonian systems with even potentials

Abstract

In this paper, we study the existence of periodic solutions with prescribed minimal period for even superquadratic autonomous second order Hamiltonian systems defined on Rn with no convexity assumptions. We use a direct variational approach for this problem on a W1, 2 space of functions invariant under the action of a transformation group isomorphic to the Klein Fourgroup V4 = Z2⊕Z2 to find symmetric periodic solutions, and prove a new iteration inequality on the Morse index by iterating such functions properly. Using these tools and the Mountain-pass theorem, we show that for every T > 0 the abobe mentioned system possesses a T-periodic solution x(t) with minimal period T or T/3, and this solution is even about t = 0, T/2 and odd about t = T/4, 3 T/4.