A topological study of the motion of particles trapped in a magnetic field

Abstract

The drift motion of charged particles trapped in a magnetic field takes place along curves of J = constant, where J is the adiabatic integral invariant. These are closed curves in the cases usually thought of in connection with physical systems such as the earth's radiation belts. The present paper reports an exploration of the possibility that there exist magnetic fields such that the curves J = constant are a family of spirals. It is shown that a necessary condition for this to occur is that there exists a J = constant spiral surface on which B = 0, or each line of force lying in it has the property that a particle spiralling about it does not, on the average, drift off it, or J suffers a discontinuity as one crosses it. Whether any of these conditions in fact leads to the curves being a family of spirals is not discussed. It seems to the author to be unlikely that any reasonable model of the earth's field possesses these properties.