STUDY OF A NONLINEAR 2ND-ORDER DIFFERENTIAL-EQUATION OF THE ENERGY OF A SINGLE-PARTICLE IN A HELICOIDAL WIGGLER WITH INVARIANT GUIDING FIELD AND TRAVELING PLASMA-WAVE

In a preceding publication [1] the author generalized the axial invariant of Freund-Drobot and Davidson-Uhm [2] for a single charged particle in a helicoidal wiggler and, further, submitted partly to a constant longitudinal guiding magnetic field B(g), partly to a longitudinally travelling plasma wave field E(z) = E0 cos-PHI = E0 cos (Kz - omega-t + phi-0) owing to a coherent longitudinal plasma wave of assumed constant amplitude and phase (transverse electromagnetic wave being neglected). In the present Note the non-linear second order differential equation, verified by way of the energy, is studied for itself. Results are presented more fully in [3], along with study of the particular case E(z) = const., which cannot be dealt with directly in the case reported in this Note.