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

TL;DR

This paper derives two fundamental Boris-type algorithms for magnetic field trajectory integration, revealing a previously unknown second Boris solver and connecting modifications to the Ge-Marsden theorem.

Contribution

It introduces a new Boris-type integrator for magnetic trajectories and links modifications to the Ge-Marsden theorem, expanding the understanding of symplectic integrators.

Findings

Derived two Boris-type algorithms for magnetic trajectories.

Identified a second, previously unknown Boris solver.

Connected modifications to the Ge-Marsden theorem.

Abstract

For a separable Hamiltonian, there are two fundamental, time-symmetric, second-order velocity-Verlet (VV) and position-Verlet (PV) symplectic integrators. Similarly, there are two VV and PV version of exact energy conserving algorithms for solving magnetic field trajectories. For a constant magnetic field, both algorithms can be further modified 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. Remarkably, the required on-orbit modification is a reparametrization of the time step, reminiscent of the Ge-Marsden theorem.