Symmetries of charged particle motion under time-independent electromagnetic fields

TL;DR

This paper analyzes the symmetries of charged particle trajectories in stationary electromagnetic fields, identifying classes of solutions, invariants, and relations to magnetic field line symmetries.

Contribution

It provides a comprehensive symmetry classification for charged particle motion in stationary fields, including Lie and Noether symmetries, and links these to magnetic field line symmetries.

Findings

Identified five classes of electromagnetic fields respecting certain symmetries.

Derived invariants and conserved quantities, including a second integral of motion.

Established a connection between particle motion symmetries and magnetic field line symmetries.

Abstract

A symmetry analysis is presented for the three-dimensional nonrelativistic motion of charged particles in arbitrary stationary electromagnetic fields. The general form of the Lie point symmetries is found along with the fields that respect them, considering non-trivial cases of physical interest. The restrictions placed upon the electromagnetic field yield five classes of solutions, expressed in terms of the vector and scalar potential. The Noether type symmetries are also investigated and their corresponding invariants are found. A second integral of motion, besides the Hamiltonian, results in three general cases. Finally, a relation between the symmetries of the charged particle motion and the symmetries of the magnetic field lines is established.