Lorentz Covariant Canonical Symplectic Algorithms for Dynamics of Charged Particles

TL;DR

This paper introduces Lorentz covariant canonical symplectic algorithms (LCCSA) that preserve Lorentz invariance and symplectic structure for simulating relativistic charged particles, ensuring accuracy and stability in complex electromagnetic fields.

Contribution

The paper develops a general procedure for constructing Lorentz covariant symplectic algorithms and provides an explicit LCCSA for relativistic charged particle dynamics, including adaptive time-stepping.

Findings

LCCSA maintains Lorentz invariance and symplectic structure.

LCCSA achieves long-term numerical accuracy and stability.

LCCSA effectively handles time-dependent electromagnetic fields.

Abstract

In this paper, the Lorentz covariance of algorithms is introduced. Under Lorentz transformation, both the form and performance of a Lorentz covariant algorithm are invariant. To acquire the advantages of symplectic algorithms and Lorentz covariance, a general procedure for constructing Lorentz covariant canonical symplectic algorithms (LCCSA) is provided, based on which an explicit LCCSA for dynamics of relativistic charged particles is built. LCCSA possesses Lorentz invariance as well as long-term numerical accuracy and stability, due to the preservation of discrete symplectic structure and Lorentz symmetry of the system. For situations with time-dependent electromagnetic fields, which is difficult to handle in traditional construction procedures of symplectic algorithms, LCCSA provides a perfect explicit canonical symplectic solution by implementing the discretization in 4-spacetime. We also show that LCCSA has built-in energy-based adaptive time steps, which can optimize the computation performance when the Lorentz factor varies.