The guiding-center dynamics of charged particles moving in a doubly symmetric screw-pinch magnetic field is investigated. In particular, we verify that Kruskal's adiabatic-invariant series expansion of the radial action integral associated with the reduced full-orbit radial motion matches the perturbation expansion of the magnetic-moment gyroaction up to first order in magnetic-field non-uniformity. Because the radial action integral is an exact invariant of the full-orbit dynamics, the magnetic moment is therefore represented as a non-perturbative integral expression, which can be used to test the validity of the guiding-center approximation.
A.J. Brizard (Fri,) studied this question.