Guiding-center dynamics in a screw-pinch magnetic field

TL;DR

This work demonstrates that in a doubly-symmetric screw-pinch magnetic field, Kruskal's adiabatic-invariant expansion of the reduced radial action matches the perturbative expansion of the magnetic moment up to third order in the nonuniformity parameter $\epsilon$, establishing a non-perturbative definition of the magnetic moment via a radial-action invariant. By formulating full-orbit Newtonian and Lagrangian dynamics in screw-pinch geometry and performing a Lie-transform-based guiding-center reduction, the authors construct the reduced radial action $J_r(E,P_\theta,P_z)$ and prove $J_r = \mu_{\rm gc}$ to $O(\epsilon^3)$. The approach provides a stringent test of guiding-center theory in a realistic confinement geometry and offers a pathway to higher-order (e.g., fourth-order) extensions, including potential applications to nonstationary perturbations and bounce-center dynamics. Overall, the results reinforce the non-perturbative consistency between reduced radial invariants and magnetic moment in highly symmetric magnetic geometries, with implications for magnetic confinement and adiabatic-invariant theory.

Abstract

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 non-perturbative integral expression, which can be used to test the validity of the guiding-center approximation.