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.
Categories: physics.plasm-ph