Charged Particle Motion in Spherically Symmetric Distributions of Magnetic Monopoles

TL;DR

This paper analyzes the motion of charged particles in spherically symmetric magnetic monopole distributions, revealing integrability and solutions in special cases, with implications for nonassociative algebraic structures and plasma physics.

Contribution

It transforms the classical equations into linear systems, demonstrating integrability and providing explicit solutions for specific monopole distributions.

Findings

Equations of motion can be linearized, showing integrability.

Solutions for single monopole and uniform distributions are derived.

Relevance to nonassociative star products and plasma models.

Abstract

The classical equations of motion of a charged particle in a spherically symmetric distribution of magnetic monopoles can be transformed into a system of linear equations, thereby providing a type of integrability. In the case of a single monopole, the solution was given long ago by Poincar\'e. In the case of a uniform distribution of monopoles the solution can be expressed in terms of parabolic cylinder functions (essentially the eigenfunctions of an inverted harmonic oscillator). This solution is relevant to recent studies of nonassociative star products, symplectic lifts of twisted Poisson structures and fluids and plasmas of electric and magnetic charges.