On the dynamics of mechanical systems with homogeneous polynomial potentials of degree 4

Abstract

In this work we study mechanical systems defined by homogeneous polynomial potentials of degree 4 on the plane, when the potential has a definite or semi-definite sign and the energy is non-negative. We get a global description of the flow for the non-negative potential case. Some partial results are obtained for the more complicated case of non-positive potentials. In contrast with the non-negative case, we prove that the flow is complete and we find special periodic solutions, whose stability is analyzed. By using results from Ziglin theory following Morales-Ruiz and Ramis we check the non-integrability of the Hamiltonian systems in terms of the potential parameters.