Global dynamics under a weak potential on a sphere

TL;DR

This paper provides a comprehensive analytical study of a point mass's motion on a sphere influenced by a logarithmic potential, including handling singularities and describing the global dynamics.

Contribution

It introduces a novel analytical framework using McGehee-type blow-up to analyze the global dynamics under a logarithmic potential on a sphere.

Findings

Identification of rest-points and their stability

Description of invariant manifolds and flow structure

Complete characterization of the motion dynamics

Abstract

We give a detailed analytical description of the global dynamics of a point mass moving on a sphere under the action of a logarithmic potential. After performing a McGehee-type blow-up in order to cope with the singularity of the potential, we investigate the rest-points of the flow, the invariant (stable and unstable) manifolds and we give a complete dynamical description of the motion.