A slowly moving particle in a two-dimensional magnetic field

We consider the behavior of a slowly moving classical point particle in a magnetic field in two dimensions, and show that, although energy conservation would permit the particle to escape to infinity, it in fact does not escape but is permanently trapped in the field. For any given magnetic field, this is true for particles of slow enough velocity. For such motion the magnetic flux enclosed by the Larmor orbits is an adiabatic invariant. Our results may be described by saying the deviations from conservation of this invariant are not cumulative but remain bounded over arbitrary time intervals, and are small if the velocity is small.