Equatorial charged particle motion around a homogeneously magnetized Newtonian center
We study the motion of a charged massive particle in the presence of a constant magnetic field and a Newtonian center. We start with the Lagrangian, its equations of motion are generally non-integrable unless we restrict the motion to be equatorial. Upon nondimensionalizing its Lagrangian, we discover that, as long as the magnetic field is non-zero, the combined effect of gravitational and magnetic fields is mainly to adjust the characteristic scales of the system, leading to a kind of universality in its dynamics that is much easier to fully analyze. Upon converting to the Hamiltonian, we characterize the possible equatorial orbits based on its effective potential and numerically calculate its precession as a function of its conserved quantities.