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For a separable Hamiltonian, there are two fundamental, time-symmetric,second-order velocity-Verlet (VV) and position-Verlet (PV) symplecticintegrators. Similarly, there are two VV and PV version of exact energyconserving algorithms for solving magnetic field trajectories. For a constantmagnetic field, both algorithms can be further modified so that theirtrajectories are exactly on the gyro-circle. The magnetic PV integrator thenbecomes the well known Boris solver, while VV yields a second, previouslyunknown, Boris-type algorithm. Remarkably, the required on-orbit modificationis a reparametrization of the time step, reminiscent of the Ge-Marsden theorem.

A fundamental derivation of two Boris solvers and the Ge-Marsden theorem