A fundamental derivation of two Boris solvers

For a separable Hamiltonian, there are two fundamental, time-symmetric, second-order velocity-Verlet (VV) and position-Verlet (PV) symplectic integrators. Similarly, there are VV and PV version of exact energy conserving algorithms for solving magnetic field trajectories. For a constant magnetic field, both algorithms can be further refined so that their trajectories are exactly on the gyro-circle. The magnetic PV integrator then becomes the well known Boris solver, while VV yields a second, previously unknown, Boris-type algorithm. The original Boris solver is unique in that its trajectory is always on the gyro-circle for $all$ $\Delta t$. For the second Boris solver, its time step is restricted to $|\Delta t|\le|2/\omega|$, same as that of a harmonic oscillator. Higher order schemes can then be constructed from these basic algorithms.