Periodic orbits for anisotropic perturbations of the Kepler problem

Abstract

We study the Kepler problem perturbed by an anisotropic term, that is a potential conformed by a Newtonian term, 1 / r , plus an anisotropic term, b / ( r 2 [ 1 + ϵ cos 2 θ ] ) β / 2 . Because of the anisotropic term, although the system is conservative the angular momentum is not a constant of motion. In this work we present an analytic and numerical analysis for the periodic orbits of a particle moving under the influence of the above potential. This is a reversible system with two degrees of freedom; thus the technique of symmetry lines can be used in the search for periodic orbits. For the particular case of β = 2 , there is a second constant of motion, so we can define a special kind of Kepler’s third law. We present comparative results for the integrable case β = 2 , and the cases β = 1 and β = 3 .