Electron–positron planar orbits in a constant magnetic field

The different types of orbits in the classical problem of two particles with equal masses and opposite charges on a plane under the influence of a constant orthogonal magnetic field are classified. The equations of the system are reduced to the problem of a Coulomb center plus a harmonic oscillator. The associated bifurcation diagram is fully explained. Using this information the dynamics of the two particles is described. •The dynamics of a pair electron–positron in a constant magnetic field is described.•The original system is reduced to a Coulomb center plus a harmonic oscillator.•Separability in Euler elliptic coordinates is constructed.•The bifurcation diagram unveils the behavior of the different types of orbits.•A complete classification of the orbits for both problems is performed.