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

The motion of a charged particle in straight magnetic field B=B(y)z? with constant perpendicular gradient is solved exactly terms elliptic functions and integrals. can be decomposed periodic along the y-axis drift x-axis. described as trapped symmetric quartic potential y. canonical transformation from coordinates (y,Py) to action–angle (?,J) explicitly generating function S(?,J) that expressed Jacobi functions. presence weak electric E=E0y? introduces an asymmetric component potential, associated perturbatively up second order.