Two charges on plane in a magnetic field: special trajectories

TL;DR

This paper analyzes the classical dynamics of two Coulomb charges in a plane under a perpendicular magnetic field, identifying special superintegrable trajectories with constant inter-charge distance and classifying charge pairs for which these occur.

Contribution

It introduces a classification of charge pairs and explicitly describes special trajectories, expanding understanding of superintegrability in charged particle systems under magnetic fields.

Findings

Identification of superintegrable circular and linear trajectories.

Larger number of constants of motion for special trajectories.

Explicit description of trajectories for specific charge pairs.

Abstract

A classical mechanics of two Coulomb charges on a plane $(e_1, m_1)$ and $(e_2, m_2)$ subject to a constant magnetic field perpendicular to a plane is considered. Special "superintegrable" trajectories (circular and linear) for which the distance between charges remains unchanged are indicated as well as their respectful constants of motion. The number of the independent constants of motion for special trajectories is larger for generic ones. A classification of pairs of charges for which special trajectories occur is given. The special trajectories for three particular cases of two electrons, (electron - positron), (electron - $\alpha$-particle) are described explicitly.