Adiabatic invariants and trapping of point charge in a strong non-uniform magnetic field

We consider here the classical problem of charge trapping by strong nonuniform magnetic fields (Van Allen belts, magnetic bottles), and study it by means of the averaging methods of classical perturbation theory. At variance with the usual cases, this problem has three (in place of two) different time scales, which are associated to the Larmor rotation around field lines (fast motion), to the motion along field lines (intermediate scale motion), and to the drift across field lines (slow motion). Such time scales get well separated for strong magnetic fields, so that Nekhoroshev-Neishtadt perturbation methods can be applied. As a result, one finds that the system admits two adiabatic invariants (namely, the magnetic moment and a second quantity, related to the energy of the motion along the field lines), which turn out to be preserved for time scales growing exponentially with the field intensity. In fact, the system can be given an integrable form, up to an exponentially small remainder.