Planar undulator motion excited by a fixed traveling wave: Quasiperiodic averaging, normal forms, and the free electron laser pendulum

We present a mathematical analysis of planar motion of energetic electrons moving through a planar dipole undulator, excited by a fixed planar polarized plane wave Maxwell field in the X-Ray Free Electron Laser (FEL) regime. Our starting point is the 6D Lorentz system, which allows planar motions, and we examine this dynamical system as the wavelength of the traveling wave varies. By scalings and transformations the 6D system is reduced, without approximation, to a 2D system in a form for a rigorous asymptotic analysis using the Method of Averaging (MoA), a long time perturbation theory. The two dependent variables are a scaled energy deviation and a generalization of the so-called ponderomotive phase. As the wavelength varies the system passes through resonant and nonresonant (NR) zones and we develop NR and near-to-resonant (NtoR) normal form approximations. For a special initial condition and on resonance, the NtoR normal form reduces to the well-known FEL pendulum system. We then state and prove NR and NtoR first-order averaging theorems which give near optimal error bounds for the MoA approximations. The proofs are novel in that they do not use a near identity transformation and they use a system of differential inequalities. The NR case is an example of quasiperiodic averaging where the small divisor problem enters in the simplest possible way. To our knowledge the planar problem has not been analyzed with the generality we aspire to here nor has the standard FEL pendulum system been derived with associated error bounds as we do here.