The Hamiltonian dynamics of planar magnetic confinement

Inspired by a question of Colin de Verdière and Truc we study the dynamics of a classical charged particle moving in a bounded planar domain Ω under the influence of a magnetic field $B$ which blows up at the boundary of the domain. We prove that under appropriate blow-up conditions the particle will never reach the boundary. As a corollary we obtain completeness of the magnetic flow. Our blow-up condition is that $B$ should not be integrable along normal rays to the boundary, while its tangential derivative should be integrable along those same rays. (paper)