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 $(-1/r^{\alpha})$, $\alpha>0,$ 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 $\alpha < 2,$ while for $\alpha \geq 2,$ smoothing introduces fake orbits. Further, we argue that for $\alpha\geq 2,$ smoothing should be applied to the amended potential $c/(2r^2)-1/r^{\alpha},$ where $c$ denotes the angular momentum constant.