Note on planar undulator wiggler single electron motion

Single particle motion in a planar undulator wiggler free electron laser magnetic field configuration is set up and examined for the construction of invariants, preparatory to specifying beam equilibria for input to a Maxwell-Vlasov analysis. Two invariants are immediately available, the energy gamma and one component of transverse canonical momentum P sub y. In an effort to find a third invariant, or a basis of determination for an advantageous approximate third invariant, which can be associated with the axial periodicity of the field, a variable change having some interesting features is reported here. The axial motion between Z = + or infinity is converted to the oscillations of a particle in an infinitely long, axially reflecting box. The transit of a half-cycle of wiggler oscillation, absolute value of Delta Z = Pi/k sub 0, corresponds to a single bounce between Z = + or - infinity, where Z is the new coordinate. Transverse coordinates and velocities are continuous at each axial reflection from infinity.