Multi-Symplectic Magnetohydrodynamics: II, Addendum and Erratum

TL;DR

This paper extends previous work on multi-symplectic magnetohydrodynamics by clarifying conservation laws, correcting earlier formulations, and applying Cartan's geometric theory to represent the system.

Contribution

It relates symplecticity conservation laws to vorticity conservation and corrects prior formulations using Cartan-Poincaré forms.

Findings

Vorticity-symplecticity law is equivalent to curl of momentum conservation.

Corrected form of Poincaré-Cartan differential form for the system.

Established a geometric representation using Cartan's theory.

Abstract

A recent paper arXiv:1312.4890 on multi-symplectic magnetohydrodynamics (MHD) using Clebsch variables in an Eulerian action principle with constraints is further extended. We relate a class of symplecticity conservation laws to a vorticity conservation law, and provide a corrected form of the Poincar\'e-Cartan differential form formulation of the system. We also correct some typographical errors (omissions) in arXiv:1312.4890. We show that the vorticity-symplecticity conservation law, that arises as a compatibility condition on the system, expressed in terms of the Clebsch variables is equivalent to taking the curl of the conservation form of the MHD momentum equation. We use the Cartan-Poincar\'e form to obtain a class of differential forms that represent the system using Cartan's geometric theory of partial differential equations.