Three charges on a plane in a magnetic field: Special trajectories

TL;DR

This paper investigates special, superintegrable trajectories of three Coulomb charges in a plane under a magnetic field, extending previous work and analyzing specific physical systems like electrons and Helium-like atoms.

Contribution

It introduces new special trajectories with conserved distances, identifies their integrals of motion, and extends the analysis to various physical charge configurations and the general n-body case.

Findings

Identified superintegrable trajectories with conserved inter-charge distances.

Analyzed three specific physical systems: electrons, neutral, and Helium-like.

Discussed the general n-body case and its implications.

Abstract

As a generalization and extension of JMP 54 (2013) 022901, the classical dynamics of three non-relativistic Coulomb charges $(e_1, m_1)$, $(e_2, m_2)$ and $(e_3, m_3)$ on the plane placed in a constant magnetic field perpendicular to the plane is considered. Special trajectories for which the distances between the charges remain unchanged are presented and their corresponding integrals of motion are indicated. For these special trajectories the number of integrals of motion is larger than the dimension of the configuration space and hence they can be called \emph{particularly superintegrable}. Three physically relevant cases are analyzed in detail, namely that of three electrons, a neutral system and a Helium-like system. The $n$-body case is discussed as well.