The Hamiltonian Dynamics of Magnetic Confinement in Toroidal Domains

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

16 pages

References in corpus (2)

• Symplectic and Poisson geometry on b-manifolds
• Examples and counter-examples of log-symplectic manifolds