Chaos in the Planar Two-Body Coulomb Problem with a Uniform Magnetic Field

The dynamics of the classical two-body Coulomb problem in a uniform magnetic field are explored numerically in order to determine when chaos can occur. The analysis is restricted to the configuration of planar particles with an orthogonal magnetic field, for which there is a four-dimensional phase space. Parameters of mass and charge are chosen to represent physically motivated systems. To check for chaos, the largest Lyapunov exponent and Poincaré section are determined for each case. We find chaotic solutions when particles have equal signs of charge. We find cases with opposite signs of charge to be numerically unstable, but a Poincaré section shows that chaos occurs in at least one case.