Comment on "Symplectic integration of magnetic systems": a proof that the Boris algorithm is not variational

TL;DR

This paper clarifies that the Boris algorithm is not variational and attributes its long-term numerical fidelity to volume preservation rather than symplecticity, resolving a controversy in the field.

Contribution

It provides a proof that the Boris algorithm cannot be derived from a variational principle, explaining its fidelity through volume preservation instead.

Findings

Boris algorithm is not variational.

Long-term fidelity is due to volume preservation.

Discrepancy about symplecticity is resolved.

Abstract

The Boris algorithm for integrating charged particle trajectories in electric and magnetic fields is popular due to its simple implementation, rapid iteration, and observed long-term numerical fidelity. The underlying cause of this long-term fidelity has become a matter of controversy, with one article claiming the method to be symplectic [S. D. Webb, J. Comput. Phys. 270 (2014) 570], and others claiming the method to be volume preserving but not symplectic [e.g. H. Qin et al., Phys. Plasmas 20 (2013) 084503]. To resolve the discrepancy, this letter leverages a discrete Helmholtz condition to demonstrate that no variational formulation of the Boris algorithm exists, indicating that the long-term fidelity should be attributed to the volume-preserving properties of the algorithm.