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

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.