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)(−1/rα), α>0α>0, where rr being the distance to the centre of the field. Due to the singularity at r=0r=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<2α<2, while for α≥2α≥2, smoothing introduces fake orbits. Further, we argue that for α≥2α≥2, smoothing should be applied to the amended potential c/(2r2)−1/rαc/(2r2)−1/rα, where cc denotes the angular momentum constant.