Collision trajectories and regularisation of two-body problem on $S^2$

TL;DR

This paper analyzes collision trajectories of two bodies on a sphere under a specific potential, describing their long-term behavior, regularizing the system, and exploring dynamics near collisions using advanced geometric methods.

Contribution

It provides a geometric and dynamical systems analysis of two-body collisions on a sphere, including regularization techniques and the study of degenerate equilibria.

Findings

Characterization of collision orbits and their omega-limit sets.

Geometric description of the dynamics on the sphere.

Regularization of the system near collisions using high-dimensional blow-ups.

Abstract

In this paper, we investigate collision orbits of two identical bodies placed on the surface of a two-dimensional sphere and interacting via an attracting potential of the form $V(q)=-\cot(q)$, where $q$ is the angle formed by the position vectors of the two bodies. We describe the $\omega$-limit set of the variables in the symplectically reduced system corresponding to initial data that lead to collisions. Furthermore we provide a geometric description of the dynamics. Lastly, we regularise the system and investigate its behaviour on near collision orbits. This involves the study of completely degenerate equilibria and the use of high-dimensional non-homogenous blow-ups.