Two body problem on a sphere in the presence of a uniform magnetic field

TL;DR

This paper studies the dynamics of charged particles on a sphere under a uniform magnetic field, identifying equilibrium types and their stability, with implications for understanding magnetic effects in curved geometries.

Contribution

It introduces two families of relative equilibria for two charged particles on a sphere in a magnetic field, analyzing their existence and stability.

Findings

Type I equilibria exist for all magnetic field strengths.

Type II equilibria only exist when the magnetic field is sufficiently strong.

The stability of these equilibria varies with field strength.

Abstract

We investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles, interacting via a cotangent potential, we show there are two families of relative equilibria, called Type I and Type II. The Type I relative equilibria exist for all strengths of the magnetic field, while those of Type II exist only if the field is sufficiently strong. The same is true if the particles are of equal mass but opposite charge. We also determine the stability of the two families of relative equilibria.