Fundamental derivation of two Boris solvers and the Ge-Marsden theorem

It has been known for some time that when one uses the Lorentz force law, rather than Hamilton's equation, one can derive two basic algorithms for solving trajectories in a magnetic field formally similar to the velocity-Verlet (VV) and position-Verlet (PV) symplectic integrators independent of any finite-difference approximation. Because the Lorentz force law uses the mechanical rather than the canonical momentum, the resulting magnetic field algorithms are exact energy conserving, rather than symplectic. In general, both types of algorithms can only yield the exact trajectory in the limit of vanishing small time steps. This work shows that, for a constant magnetic field, both magnetic algorithms can be further modified so that their trajectories are exactly on the gyrocircle at finite time steps. The magnetic form of the PV integrator then becomes the well-known Boris solver, while the VV form yields a second, previously unknown Boris-type algorithm, unrelated to any finite-difference scheme. Remarkably, the modification needed for the trajectory to be exact is a reparametrization of the time step, reminiscent of the Ge-Marsden theorem.