Hamiltonian Formalism of Extended Magnetohydrodynamics

TL;DR

This paper develops a Hamiltonian framework for extended MHD systems, including Hall and electron inertia effects, by identifying the underlying Lie algebra structure that ensures the Poisson algebra's consistency.

Contribution

It uncovers the Lie algebra generating the Poisson bracket for extended MHD, providing a unified Hamiltonian formulation for Hall and inertial MHD systems.

Findings

Established the Hamiltonian structure for extended MHD.

Proved Jacobi's identity for the Poisson bracket.

Identified the Lie algebra underlying the Poisson algebra.

Abstract

The extended magnetohydrodynamics (MHD) system, including the Hall effect and the electron inertia effect, has a Hamiltonian structure embodied by a noncanonical Poisson algebra on an infinite-dimensional phase space. A nontrivial part of the formulation is the proof of Jacobi's identity for the Poisson bracket. We unearth a basic Lie algebra that generates the Poisson bracket. A class of similar Poisson algebra may be generated by the same Lie algebra, which encompasses the Hall MHD system and inertial MHD system.