  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.
