Is There an Exact Magnetic-Moment for Charged Particle Motion in a Homogeneous, Time-Dependent Magnetic Field?

The nonperturbative guiding-center model provides an exact alternative to full-orbit simulations of charged particle dynamics in situations where traditional guiding-center theory may fail. We demonstrate that the charged particle motion in a homogeneous, time-varying magnetic field is a solvable example of the nonperturbative guiding-center model. This entails showing that the exact magnetic moment of Qin and Davidson can be constructed to be asymptotic to the adiabatic invariant series of Kruskal. In contrast to the perturbative invariant, the exact invariant contains information about parametric resonances. These resonances destroy the conservation of the usual magnetic moment over very long times. This refutes some previous claims about the all-time invariance of the magnetic moment.