The Motion of a Charged Particle on a Riemannian Surface Under a Non-Zero Magnetic Field

TL;DR

This paper investigates the dynamics of a charged particle on a Riemannian surface under a magnetic field, establishing conditions for perpetual trapping near certain magnetic field level sets using advanced mathematical theorems.

Contribution

It introduces new sufficient conditions for trapping charged particles on Riemannian surfaces influenced by magnetic fields, utilizing Moser's Twist Theorem and symplectic reduction techniques.

Findings

Conditions for trapping are local and near non-degenerate critical points of B.

Results apply to S^1-invariant magnetic fields on R^3.

Provides a mathematical framework for understanding particle confinement.

Abstract

In this paper we study the motion of a charged particle on a Riemannian surface under the influence of a positive magnetic field B. Using Moser's Twist Theorem and ideas from classical pertubation theory we find sufficient conditions to perpetually trap the motion of a particle with a sufficiently large charge in a neighborhood of a level set of the magnetic field. The conditions on the level set of the magnetic field that guarantee the trapping are local and hold near all non-degenerate critical local minima or maxima of B. Using symplectic reduction we apply the results of our work to certain S^1 - invariant magnetic fields on R^3.