A General Theory for Gauge-Free Lifting

Given a Hamiltonian set of orbit equations, defined on a phase space of arbitrary dimension, with `forces' that depend explicitly on given electric and magentic fields and possibly all of their derivatives, how does one \underline{lift} to a Hamiltonian kinetic theory coupled to Maxwell's equations? A general theory that answers this question will be presented. The theory produces magnetization and polarization effects in Maxwell's equations via a Poisson bracket that generalizes that for the Vlasov-Maxwell system.\footnote{P.J.~Morrison, Phys.Lett.\ {\bf 80A}, 383 (1980); AIP Conference Proceedings {\bf 88}, 13 (1982).} Several examples will be treated, including the generalized guiding-center kinetic theory of Pfirsch and the author,\footnote{D.~Pfirsch and P.J.~Morrison, Phys.\ Rev.\ \textbf{32A}, 1714 (1985); Phys.\ Fluids \textbf{3B}, 271 (1991).} which relies on the introduction of redundant variables via Dirac constraint theory, and a theory that incorporates spin in the Vlasov context.\footnote{M.~Marklund and P.J.~Morrison, Phys.\ Lett.\ \textbf{375A}, 2362 (2011).}