The Hamiltonian dynamics of magnetic confinement in toroidal domains

Gabriel Martins

Vol. 304 (2020), No. 2, 613–628

DOI: 10.2140/pjm.2020.304.613

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 ${\sigma }_{\infty }\in \Gamma \left(T^{\ast }\partial M\right)$. 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.

Keywords

magnetic confinement, toroidal domains, completeness, magnetic flow, manifolds with boundary

Mathematical Subject Classification 2010

Primary: 53D25, 70H05, 78A35

Milestones

Received: 28 August 2018

Revised: 7 January 2019

Accepted: 1 May 2019

Published: 12 February 2020