Collision orbits in the anisotropic Kepler problem

Abstract

The anisotropic Kepler problem is a one-parameter family of classical mechanical systems with two degrees of freedom. When the parameter # = 1, we have the well known Kepler or central force problem. As # increases beyond 1, we introduce more and more anisotropy into the Kepler problem. As we show below, this changes the orbit structure of the system dramatically. When p = 1, the system is completely integrable and the orbit structure is well understood. With the exception of certain collision orbits, all orbits are closed and lie on two dimensional tori in the case of negative total energy. For p > 1, we keep the same potential energy, but make the kinetic energy anisotropic, i.e. the kinetic energy becomes