Period function for a class of Hamiltonian systems (Q1592716)

The authors study a planar Hamiltonian system with a Hamiltonian of the form \(H(x,y)=F(x)+G(y)\) with analytic functions \(F(x), G(y)\) at the origin which is supposed to be the minimum for both functions (the origin is a nondegenerate center). There is some neighborhood of the origin filled with periodic orbits, so the period function \(T(h)\) is determined there. The authors solve the inverse problem of characterizing all Hamiltonians of the abovementioned form which have a given periodic function \(T(h)\). When periodic orbits fill all the plane one theorem characterizes the dominant term of the period function at infinity. The problem of isochronicity for the center is also treated.