The motion of a charged particle interacting with a plane electromagnetic wave propagating at an arbitrary angle to a uniform magnetic field

The relativistic equations of motion of a charged particle moving in the field of a plane electromagnetic wave which is propagating at an arbitrary angle to a uniform magnetic induction are considered. It is shown that a system of coordinates exists with respect to which the behavior of the perpendicular component of the particle velocity, as a function of the component parallel to the uniform field, can be characterised by a family of ellipsoidal surfaces in the velocity space and that this family is invariant with respect to coordinate transformations which preserve the axial symmetry of the uniform field. From the known behavior when the angle between the direction of propagation of the wave and the uniform field is zero, the mode of variation of the particle pitch angle when the angle is non-zero is deduced. For the case where motion takes place in a vacuum the predicted behavior is seen to be verified by the numerical solution of the equations of motion.