A Hamiltonian and geometric formulation of general Vlasov-Maxwell-type models

TL;DR

This paper develops three geometric Hamiltonian formulations of Maxwell equations and extends them to Lorentz covariant kinetic theories coupled with nonlinear electrodynamics, emphasizing a metric-free Poisson structure.

Contribution

It introduces three new geometric Hamiltonian frameworks for Maxwell equations and applies them to Lorentz covariant kinetic theories with nonlinear electrodynamics.

Findings

Three geometric Hamiltonian formulations of Maxwell equations.

A metric-free Poisson bracket for kinetic theories.

Lorentz covariant kinetic theory coupled with nonlinear electrodynamics.

Abstract

Three geometric formulations of the Hamiltonian structure of the macroscopic Maxwell equations are given: one in terms of the double de Rham complex, one in terms of L2 duality, and one utilizing an abstract notion of duality. The final of these is used to express the geometric and Hamiltonian structure of kinetic theories in general media. The Poisson bracket so stated is explicitly metric free. Finally, as a special case, the Lorentz covariance of such kinetic theories is investigated. We obtain a Lorentz covariant kinetic theory coupled to nonlinear electrodynamics such as Born-Infeld or Euler-Heisenberg electrodynamics.