Smoothed dynamics in the central field problem

TL;DR

This paper compares the global flows of a central potential system with and without smoothing, revealing topological equivalence for certain parameters and the emergence of fake orbits when smoothing is applied to the original potential.

Contribution

It demonstrates the conditions under which smoothing preserves the flow's topology and proposes a modified smoothing approach for specific potential parameters.

Findings

Flows are topologically equivalent for α<2

Smoothing introduces fake orbits for α≥2

Modified smoothing is recommended for α≥2

Abstract

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.