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 σ∞εΓ(T*δM). 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 σ∞. Inparticular in the case where the 1-form is nowhere vanishing we conclude that the particles become confined to its interior for all time.