Periodic orbits for anisotropic perturbations of the Kepler problem

We study the Kepler problem perturbed by an anisotropic term, that is a potential conformed by a Newtonian term, 1/r1/r, plus an anisotropic term, b/(r2[1+ϵcos2θ])β/2b/(r2[1+ϵcos2θ])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=2β=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=2β=2, and the cases β=1β=1 and β=3β=3.