Periodic orbits of a one dimensional non autonomous Hamiltonian system

In this paper we study the properties of the periodic orbits of \"x + V'_x(t, x) = 0 with x \in S1 and V(t, x) a T0 periodic potential. Called {\rho} \in (1/T0)Q the frequency of windings of an orbit in S1 we show that exists an infinite number of periodic solutions with a given {\rho}. We give a lower bound on the number of periodic orbits with a given period and {\rho} by means of the Morse theory.