The Hamiltonian Dynamics of Magnetic Confinement in Toroidal Domains

TL;DR

This paper studies magnetic confinement in toroidal domains, showing that particles are confined or escape only through specific boundary conditions related to a closed 1-form, with implications for magnetic confinement stability.

Contribution

It introduces a new class of magnetic fields with boundary behavior controlled by a closed 1-form, analyzing particle dynamics and confinement properties in these settings.

Findings

Particles can only escape through the zero locus of the boundary 1-form.

If the boundary 1-form is nowhere vanishing, particles remain confined indefinitely.

The results connect boundary geometry with particle confinement in magnetic fields.

Abstract

We consider a class of magnetic fields defined over the interior of a manifold $M$ which go to infinity at its boundary and whose direction near the boundary of $M$ is controlled by a closed 1-form $\sigma_\infty \in \Gamma(T^*\partial M)$. We are able to show that charged particles in the interior of $M$ under the influence of such fields can only escape the manifold through the zero locus of $\sigma_\infty$. In particular in the case where the 1-form is nowhere vanishing we conclude that the particles become confined to its interior for all time.