Travelling waves and a fruitful `time' reparametrization in relativistic electrodynamics

TL;DR

This paper introduces a novel reparametrization technique in relativistic electrodynamics that simplifies equations of motion for charged particles and plasmas, enabling new solutions and insights into wave-particle interactions.

Contribution

The authors develop a light-like coordinate reparametrization that simplifies equations and derives new solutions in relativistic electrodynamics and plasma physics.

Findings

Rigorous formulation of a no-final-acceleration theorem

Derivation of cyclotron autoresonance

Reduction of PDE systems to ODEs in specific cases

Abstract

We simplify the nonlinear equations of motion of charged particles in an external electromagnetic field that is the sum of a plane travelling wave F_t(ct-z) and a static part F_s(x,y,z): by adopting the light-like coordinate ct-z instead of time t as an independent variable in the Action, Lagrangian and Hamiltonian, and deriving the new Euler-Lagrange and Hamilton equations accordingly, we make the unknown z(t) disappear from the argument of F_t. We study and solve first the single particle equations in few significant cases of extreme accelerations. In particular we obtain a rigorous formulation of a Lawson-Woodward-type (no-final-acceleration) theorem and a compact derivation of cyclotron autoresonance, beside new solutions in the presence of uniform F_s. We then extend our method to plasmas in hydrodynamic conditions and apply it to plane problems: the system of partial differential equations may be partially solved and sometimes even completely reduced to a family of decoupled systems of ordinary ones; this occurs e.g. with the impact of the travelling wave on a vacuum-plasma interface (what may produce the slingshot effect). Since Fourier analysis plays no role in our general framework, the method can be applied to all kind of travelling waves, ranging from almost monochromatic to socalled "impulses", which contain few, one or even no complete cycle.