Action-angle coordinates for motion in a straight magnetic field with constant gradient
arXiv:2203.05343 · doi:10.1016/j.cnsns.2022.106652
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 -axis and a drift motion along the -axis. The periodic motion can be described as a particle trapped in a symmetric quartic potential in . The canonical transformation from the canonical coordinates to the action-angle coordinates is solved explicitly in terms of a generating function 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.
12 pages, 7 figures