Smoothed dynamics in the central field problem

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 nonsmoothed 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/(2r2) − 1/rα, where c denotes the angular momentum constant.