Countable periodic solutions of the Lorentz force equation under a time-dependent current

TL;DR

This paper investigates the existence and structure of periodic solutions in a charged particle's motion under a time-dependent electromagnetic field, revealing conditions for harmonic and subharmonic orbits and their phase space implications.

Contribution

It provides a rigorous analysis of countable periodic solutions of the Lorentz force equation with a time-dependent current, using Melnikov method and Hamiltonian dynamics.

Findings

Existence of a unique harmonic periodic solution with the current's period.

Presence of subharmonic solutions with periods multiple of the fundamental period.

Periodic orbits form invariant cylinders that confine the particle's radial motion.

Abstract

The resonant dynamics of a charged particle, governed by the Lorentz force equation in an electromagnetic field generated by a current-carrying wire with a small harmonic modulation, is considered in this study. When regarded as a Hamiltonian system with periodic perturbation, the resonance of periodic orbits in the unperturbed system is analyzed by the Melnikov method. The existence of exactly one harmonic radial periodic solution with period $T_1$ is confirmed, matching the period of the current. Moreover, it is established that any other radial periodic solution must be subharmonic with period $nT_1$ for some integer $n > 1$, with at most one such solution for each $n$. Dynamically, these surviving periodic orbits correspond to invariant cylinders that partition the phase space and globally confine the particle's radial motion.