Action-angle coordinates for motion in a straight magnetic field with constant gradient

TL;DR

This paper derives explicit action-angle coordinates for a charged particle in a straight magnetic field with a constant gradient, using elliptic functions, and explores the effects of a weak electric field perturbation.

Contribution

It provides an exact canonical transformation to action-angle variables for this system, including perturbative solutions when a weak electric field is present.

Findings

Exact solution for particle motion using elliptic functions.

Explicit transformation to action-angle coordinates.

Perturbative analysis of electric field effects.

Abstract

The motion of a charged particle in a straight magnetic field ${\bf B} = B(y)\,\wh{\sf z}$ with a constant perpendicular gradient is solved exactly in terms of elliptic functions and integrals. The motion can be decomposed in terms of a periodic motion along the $y$-axis and a drift motion along the $x$-axis. The periodic motion can be described as a particle trapped in a symmetric quartic potential in $y$. The canonical transformation from the canonical coordinates $(y,P_{y})$ to the action-angle coordinates $(J,\theta)$ is solved explicitly in terms of a generating function $S(\theta,J)$ that is expressed in terms of Jacobi elliptic functions. The presence of a weak constant electric field ${\bf E} = E_{0}\,\wh{\sf y}$ introduces an asymmetric component to the quartic potential, and the associated periodic motion is solved perturbatively up to second order.