The Kepler Problem with Anisotropic Perturbations

Abstract

We study a 2-body problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree $-β$, $β\ge 2$. For $β>2$, the sets of initial conditions leading to collisions/ejections and the one leading to escapes/captures have positive measure. For $β>2$ and $β\ne 3$, the flow on the zero-energy manifold is chaotic. For $β=2$, a case we prove integrable, the infinity manifold of the zero-energy level is a disconnected set, which has heteroclinic connections with the collision manifold.