Insolubility of trapped particle motion in a magnetic dipole field

Abstract

Topological and numerical techniques are used to show that the problem of trapped charged particle motion in a magnetic dipole field is insoluble. Similar results hold for motion in the earth's magnetic field and are of interest for radiation belt phenomena. Pedagogical discussion is devoted to the subject of how it can happen that a classical mechanics problem is insoluble and in what sense. It is shown that the complete adiabatic magnetic moment series is divergent and that due to the existence of homoclinic points the solutions to the equations of motion are too complicated to be written in closed form. As a consequence, there is currently no rigorous theoretical explanation for the empirical success of adiabatic orbit theory, and a completely satisfactory mathematical justification will be far from easy.