Stability analysis of a charged particle subject to two non-stationary currents

TL;DR

This paper investigates the non-relativistic dynamics of a charged particle influenced by two non-stationary currents in parallel wires, establishing conditions for stable vertical motion within the plane of the wires.

Contribution

It provides a rigorous proof of a stable motion region for a charged particle under time-dependent currents, extending stability analysis beyond classical Hamiltonian energy considerations.

Findings

Existence of a vertical stable motion strip for the particle.

Stability holds under specific conditions on the currents.

Stability is stronger than traditional isoenergetic stability.

Abstract

We study the non relativistic motions of a charged particle in the electromagnetic field generated by two parallel electrically neutral vertical wires carrying time depends currents. Under quantitative conditions on the currents we prove the existence of a vertical strip of stable motions of the particle. The stable strip is contained in the plane of the two wires and the stability is understood in a stronger sense than the isoenergetic stability of Hamiltonian systems. Actually, also variations of the integral given by the linear momentum will be allowed.