Isochronicity for Several Classes of Hamiltonian Systems

In this paper we study isochronous centers of analytic Hamiltonian systems giving special attention to the polynomial case. We first revisit the potential systems and we show the connection between isochronicity and involutions. We then study a more general system, namely the ones associated to Hamiltonians of the form H(x, y)=A(x)+B(x) y+C(x) y2. As an application we classify the cubic polynomial Hamiltonian isochronous centers and we give examples of nontrivial and nonglobal polynomial Hamiltonian isochronous centers