Lie symmetries for two-dimensional charged particle motion

TL;DR

This paper identifies Lie point symmetries in two-dimensional charged particle motion, revealing specific electromagnetic field classes that admit these symmetries, thus enhancing understanding of symmetry properties in classical mechanics.

Contribution

It explicitly solves the PDE system for electromagnetic fields compatible with Lie symmetries, classifying four distinct field types for charged particle motion.

Findings

Four classes of electromagnetic fields compatible with Lie symmetries

Explicit solutions to the PDE system governing symmetries

Identification of symmetry transformations including rotations and dilations

Abstract

We find the Lie point symmetries for non-relativistic two-dimensional charged particle motion. These symmetries comprise a quasi-invariance transformation, a time-dependent rotation, a time-dependent spatial translation and a dilation. The associated electromagnetic fields satisfy a system of first-order linear partial differential equations. This system is solved exactly, yielding four classes of electromagnetic fields compatible with Lie point symmetries.