Smoothed dynamics in the central field problem

Abstract

Consider the motion of a material point of unit mass in a central field determined by a homogeneous potential of the form (- 1 / rα), α > 0, where r being the distance to the centre of the field. Due to the singularity at r = 0, in computer-based simulations, usually, the potential is replaced by a similar potential that is smooth, or at least continuous. In this paper, we compare the global flows given by the smoothed and non-smoothed potentials. It is shown that the two flows are topologically equivalent for α < 2, while for α ≥ 2, smoothing introduces fake orbits. Further, we argue that for α ≥ 2, smoothing should be applied to the amended potential c / (2 r2) - 1 / rα, where c denotes the angular momentum constant. © 2008 Elsevier Ltd. All rights reserved.