Two charges on plane in a magnetic field: special trajectories
Escobar-Ruiz, M. A.
Turbiner, A. V.
Physics - Classical Physics | Mathematical Physics | Condensed Matter - Strongly Correlated Electrons | Nonlinear Sciences - Exactly Solvable and Integrable Systems
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.