The Hamiltonian Dynamics of Planar Magnetic Confinement

TL;DR

This paper investigates the behavior of a charged particle in a bounded planar domain with a magnetic field that becomes infinite at the boundary, proving conditions under which the particle cannot reach the boundary and the flow remains complete.

Contribution

It establishes new blow-up conditions on the magnetic field ensuring the particle's confinement and flow completeness in planar magnetic systems.

Findings

Particle never reaches boundary under specified conditions

Magnetic flow is complete with non-integrable boundary magnetic field

Provides criteria for magnetic confinement in planar domains

Abstract

Inspired by a question of Colin de Verdi\`{e}re and Truc we study the dynamics of a classical charged particle moving in a bounded planar domain $\Omega$ under the influence of a magnetic field $\mathbf{B}$ which blows up at the boundary of the domain. We prove that under appropriate blow-up conditions the particle will never reach the boundary. As a corollary we obtain completeness of the magnetic flow. Our blow-up condition is that $\mathbf{B}$ should not be integrable along normal rays to the boundary, while its tangential derivative should be integrable along those same rays.