On the period map of planar Hamiltonian systems with separated variables

We consider planar Hamiltonian systems with Hamiltonian function of the type $$H(x,y)=F(y)+G(x)$$ having the origin as a global center, and generalize to this setting some known results on the period map. We are thus led to a characterization of isochronism for scalar second order equations involving the p -Laplacian operator.