Guiding center motion for particles in a ponderomotive magnetostatic end plug

TL;DR

This paper develops a perturbative Hamiltonian framework to analyze particle motion in a rotating plasma with magnetic multipoles, revealing conditions for axial confinement and effects of ponderomotive potentials.

Contribution

It introduces a second-order Lie transformation method to derive effective Hamiltonians and pseudopotentials, including mass modifications and confinement criteria in a ponderomotive magnetostatic system.

Findings

Particles can be axially confined by ponderomotive potentials.

Radial excursions depend on the confining potential.

A minimal rotation speed condition for confinement is derived.

Abstract

The Hamiltonian dynamics of a single particle in a rotating plasma column, interacting with an magnetic multipole is perturbatively solved for up to second order, using the method of Lie transformations. First, the exact Hamiltonian is expressed in terms of canonical action-angle variables, and then an approximate integrable Hamiltonian is introduced, using another set of actions and angles. The perturbation introduces an effective ponderomotive potential, which to leading order is positive. At the second order, the pseudopotential consists of a sum of terms of the Miller form, and can have either sign. Additionally at second order, the ponderomotive interaction introduces a modification to the particle effective mass, when considering the motion along the column axis. It is found that particles can be axially confined by the ponderomotive potentials, but acquire radial excursions which scale as the confining potential. The radial excursions of the particle along its trajectory are investigated, and a condition for the minimal rotation speed is derived, in order for particles to remain radially confined. Last, we comment on the changes to the aforementioned solution to the pseudopotintials and particle trajectory in the case of resonant motion, that is, a motion which has the same periodicity as the perturbation.