A Tutorial on Variational Integrators

TL;DR

This paper provides a tutorial on variational integrators, specifically focusing on multisymplectic particle-in-cell algorithms derived from discretized Lagrangian methods, emphasizing their benefits for electromagnetic simulations.

Contribution

It offers a detailed derivation and explanation of a 1D electrostatic particle-in-cell algorithm using variational integrators, making the approach more accessible.

Findings

Demonstrates how variational integrators can be applied to particle-in-cell algorithms.

Highlights the advantages of symplectic integration in electromagnetic simulations.

Provides a step-by-step derivation for a standard 1D electrostatic PIC algorithm.

Abstract

We present a brief tutorial on the nuts and bolts computation of a multisymplectic particle-in-cell algorithm using the discretized Lagrangian approach. This approach, originated by Marsden, Shadwick, and others, brings the benefits of symplectic integration of Hamiltonian systems to full electromagnetic particle-in-cell algorithms. To make the work more approachable, we present a basic discussion of the philosophy, combined with a detailed derivation of a standard 1-dimensional electrostatic particle-in-cell algorithm.