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 varies the system passes through resonant and nonresonant (NonR) intervals and we develop NonR and near-to-resonant (NearR) MoA normal form approximations to the exact equations. The NearR normal forms contain a parameter which measures the distance from a resonance. For the planar motion, with the special initial condition that matches into the undulator design trajectory, and on resonance, the NearR normal form reduces to the well-known FEL pendulum system. We then state and prove NonR and NearR first-order averaging theorems which give explicit error bounds for the normal form approximations. We prove the theorems in great detail, giving the interested reader a tutorial on mathematically rigorous perturbation theory in a context where the proofs are easily understood. The proofs are novel in that they do not use a near-identity transformation and they use a system of differential inequalities. The NonR 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. We briefly discuss the low gain theory in light of our NearR normal form. Our mathematical treatment of the noncollective FEL beam dynamics problem in the framework of dynamical systems theory sets the stage for our mathematical investigation of the collective high gain regime.