Lie transforms and motion of a charged particle in periodic magnetic field (Q1067106)

The authors use the Lie series averaging method to find a complete second order solution for motion of a charged particle in a constant magnetic field on which a weak spatially periodic magnetic field (of order \(\epsilon\), a small parameter) is superimposed. The first order solution (O(\(\epsilon)\)) coincides with that of \textit{T. P. Coffey} [J. Math. Phys. 10, 1362-1368 (1969)], while the second order solution (of order \(O(\epsilon^ 2))\) is not negligible and confirms the periodic behavior of the secular motions.