Periodic orbits and non-integrability of Hénon-Heiles systems

We apply the averaging theory of second order to study the periodic orbits for a generalized Hénon-Heiles system with two parameters, which contains the classical Hénon-Heiles system. Two main results are shown. The first result provides sufficient conditions on the two parameters of these generalized systems, which guarantee that at any positive energy level, the Hamiltonian system has periodic orbits. These periodic orbits form in the whole phase space a continuous family of periodic orbits parameterized by the energy. The second result shows that for the non-integrable Hénon-Heiles systems in the sense of Liouville-Arnol'd, which have the periodic orbits analytically found with averaging theory, cannot exist any second first integral of class C1. In particular, for any second first integral of class C 1, we prove that the classical Hénon-Heiles system and many generalizations of it are not integrable in the sense of Liouville-Arnol'd. Moreover, the tools we use for studying the periodic orbits and the non-Liouville-Arnol'd integrability can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom. © 2011 IOP Publishing Ltd.