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

Ka Xie, Pengcheng Xu, Zuohuan Zheng

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.

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

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 is confirmed, matching the period of the current. Moreover, it is established that any other radial periodic solution must be subharmonic with period for some integer , with at most one such solution for each . Dynamically, these surviving periodic orbits correspond to invariant cylinders that partition the phase space and globally confine the particle's radial motion.
Paper Structure (6 sections, 5 theorems, 95 equations)

This paper contains 6 sections, 5 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. The Lorentz force equation with an infinite wire
  3. The dynamics of the unperturbed system
  4. The dynamics of the perturbed system
  5. Conclusions
  6. Proof of Lemma 1

Key Result

Lemma 1

Function increases monotonically for $r \in (0,\infty)$.

Theorems & Definitions (12)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 2 more