The Hamiltonian Dynamics of Magnetic Confinement in Toroidal Domains

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 ${\sigma }_{\mathrm{\infty }}$. 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.