Mechanical systems with polynomial potentials of degree 4

We study mechanical systems defined by homogeneous polynomial po-tentials 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 ob-tained 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 of Morales-Ruiz we check the non–integrability of the Hamiltonian systems in terms of the potential parameters. Homogeneous potentials appear also in the modelling of natural phe-nomena or processes. Along this line we may mention Contopoulos [1] and references therein who has considered third or fourth degree homogeneous perturbations of homogeneous quadratic polynomials as models in the dy-namics of galaxies