Laser-driven charged particles as a dynamical system.

摘要

Based on Lagrangian formulations of the Lorentz equations of motion, we investigate the relativistic orbit of a classical charged particle in response to a generic electromagnetic field in the four dimensional Minkowski space. Within the context of classical mechanics, the results are relativistically and mathematically exact. With the application to laser-particle interaction in mind, our primary focus is on the particle dynamics in a generic plane wave field.;Taking advantage of the fact that the particle motion in the direction transverse to the wave propagation direction is cyclic, we use the classical Routh's procedure to reduce the number of degrees of freedom of the motion and to manifest the observation that the longitudinal motion of the particle controls every aspect of the particle dynamics under the influence of a generic plane wave field. In fact, we show that the particle longitudinal motion is a generalized natural mechanical system in the sense that it has a Lagrangian consists of the difference of a metric based kinetic energy and a potential function. A corollary of this is the culmination of this work, that is, the geodesic variational principle.;The geodesic variational principle implies that longitudinally, the particle moves along a timelike geodesic in a curved two dimensional Lorentzian spacetime whose metric is determined by the plane wave field. In other words, the effect of the field on the particle dynamics gets replaced by the effect of the geometry and its curvature on the geodesics of a two-dimensional manifold. This gives rise to a geometrization of the laser-particle interaction.;We also use the geodesic variational principle to establish a Lorentzian law of refraction in which the particle, in response to the plane wave field, gets refracted by the field in the same way that light rays get refracted by a medium permeating Euclidean space. The plane wave field acts as a refractive medium with a characteristic Lorentzian refractive index. Introducing the notion of the rapidity of the particle, this gives rise to a Lorentzian Snell's law.;Finally, we apply the law of refraction to study particle scattering process in two counter-propagating trains of square pulses. We discretize the particle dynamics by introducing the notions of the itinerary and the refraction sequence of a particle orbit. As illustrations, we discuss various properties of periodic orbits and construct a class of periodic orbits and a class of unbounded orbits explicitly with the aid of these notions.;Keywords. Dynamical system; Calculus of variations; Lagrangian Mechanics; Geodesic variational principle; Geometrization of electromagnetism; Laser-matter interaction; Law of refraction in spacetime; Snell's law in spacetime; Particle scattering