Motions of a charged particle in the electromagnetic field induced by a non-stationary current

TL;DR

This paper investigates the non-relativistic motion of a charged particle in a time-dependent electromagnetic field generated by a periodically varying current in a wire, revealing stable, periodic, and quasiperiodic trajectories.

Contribution

It extends the understanding of particle dynamics in non-stationary electromagnetic fields by proving the existence of stable, radially periodic solutions similar to the stationary case.

Findings

Existence of non-resonant radially periodic motions

Lyapunov stability of these motions

Presence of subharmonic and quasiperiodic trajectories

Abstract

In this paper we study the non-relativistic dynamic of a charged particle in the electromagnetic field induced by a periodically time dependent current J along an infinitely long and infinitely thin straight wire. The motions are described by the Lorentz-Newton equation, in which the electromagnetic field is obtained by solving the Maxwell's equations with the current distribution J as data. We prove that many features of the integrable time independent case are preserved. More precisely, introducing cylindrical coordinates, we prove the existence of (non-resonant) radially periodic motions that are also of twist type. In particular, these solutions are Lyapunov stable and accumulated by subharmonic and quasiperiodic motions.