Travelling waves and a fruitful ‘time’ reparametrization in relativistic electrodynamics

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 µν t (ct−z) and a static part F µν s (x, y, z): by adopting the light-like coordinate ξ = 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 µν 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 µν s . We then extend our method to plasmas in hydrodynamic conditions and apply it to plane problems: the system of (Lorentz-Maxwell+continuity) partial differential equations may be partially solved or 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).Our method can be seen as an application of the light-front approach. 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.

Travelling waves and light-front approach in relativistic electrodynamics

“…As fas as we now, such general solutions have not appeared in the literature before Ref. [1]. We next analyze a few special cases (the first two have already appeared in the literature).…”

“…x(t) = cp(t) m 2 c 2 +p 2 (t) (1) in its general form is non-autonomous and highly nonlinear in the unknowns x(t), p(t).…”

“…Here we summarize an approach [1] that systematically applies the light-front formalism [7]; it is especially fruitful if in the spacetime region of interest (where we wish to follow the particles' worldlines) E, B are the sum of static parts and plane transverse travelling waves propagating in the z direction:…”

“…withû z ,γ expressed as in (8). These equations amount [1] to the Euler-Lagrange equations d dξ ∂L ∂x = ∂L ∂x , that are obtained applying Hamilton's principle to the action functional S(λ) with λ parametrized by ξ (instead of t), as well as to the Hamilton equationsx = ∂Ĥ ∂Π ,Π = − ∂Ĥ ∂x , where the Hamiltonian…”

On cold diluted plasmas hit by short laser pulses

“…In section II we discuss the motion of electrons when R = ∞ (χ R ≡ 1) using an improved plane hydrodynamical model [11,12] (for shorter presentations see [13]) that allows to reduce the system of Lorentz-Maxwell and continuity partial differential equations (PDEs) into ordinary differential equations (ODEs), more precisely into a family of decoupled systems of arXiv:1803.02915v2 [physics.plasm-ph] 25 Mar 2018 non-autonomous Hamilton Equations in dim 1 in rational form. In the model we alternatively adopt the light-like coordinate ξ = ct−z or time t to parametrize the electron motion, the transverse and the light-like components p ⊥ e , p 0 e −cp z e ≡ mc 2 s e (instead of the longitudinal one p z e ) of the electron 4-momentum as unknowns, neglect pump depletion, control how long this is valid, how long the hydrodynamical picture holds, when and where it fails (by wave-breaking [4]).…”