The Flow on Negative Energy of Mechanical Systems with Homogeneous Polynomial Potentials

In this paper we study the global flow of classical mechanical systems with homogeneous polynomial potentials with two degrees of freedom by using McGehee coordinates. Then the flow is extended to a 3-dimensional manifold whose boundary is known as the infinity manifold. In the negative energy case, this manifold is compact, it can be imbedded in \({{{\mathbb R}^3}}\) , and its shape is generically described for any degree of homogeneity. We study the global flow for the non-trivial case corresponding to a “peapod” shaped manifold which appears for homogeneity degree greater than four.