#### Vol. 304, No. 2, 2020

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The Hamiltonian dynamics of magnetic confinement in toroidal domains

### Gabriel Martins

Vol. 304 (2020), No. 2, 613–628
##### 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
##### Authors
 Gabriel Martins Department of Mathematics and Statistics Sacramento State University Sacramento, CA United States