Noether symmetries for two-dimensional charged particle motion

TL;DR

This paper identifies specific symmetries in the motion of charged particles in two dimensions, linking them to electromagnetic fields and deriving invariants, thus enhancing understanding of particle dynamics under these symmetries.

Contribution

The paper explicitly solves for electromagnetic fields compatible with Noether point symmetries in two-dimensional charged particle motion, providing exact solutions and invariants.

Findings

Three classes of electromagnetic fields compatible with symmetries identified.

Exact solutions to the symmetry equations derived.

Noether invariants for each class are obtained and interpreted.

Abstract

We find the Noether point symmetries for non-relativistic two-dimensional charged particle motion. These symmetries are composed of a quasi-invariance transformation, a time-dependent rotation and a time-dependent spatial translation. The associated electromagnetic field satisfy a system of first-order linear partial differential equations. This system is solved exactly, yielding three classes of electromagnetic fields compatible with Noether point symmetries. The corresponding Noether invariants are derived and interpreted.