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 ∈ S1 and V ′x(t, x) a T0 periodic potential. Called ρ ∈ 1T0Q the frequency of windings of an orbit in S1 we show that exists an infinite number of periodic solutions with a given ρ. We give a lower bound on the number of periodic orbits with a given period and ρ by means of the Morse theory. Key Words: Morse theory, periodic orbits, twisting number