Period Function for a Class of Hamiltonian Systems

This paper studies the period function of the class of Hamiltonian systems x=−Hy, y=Hx where H(x, y) has the special form H(x, y)=F(x)+G(y) and the origin is a non-degenerate center. More concretely, if T(h) denotes the period of the periodic orbit contained in H(x, y)=h we solve the inverse problem of characterizing all systems with a given function T(h). We also characterize the limiting behaviour of T at infinity when the origin is a global center and apply this result to prove, among other results, that there are no nonlinear polynomial isochronous centers in this family.