A Hamiltonian formulation for the perturbed Vlasov-Maxwell equations

The Hamiltonian formulation for the perturbed Vlasov-Maxwell equations is expressed in terms of the perturbation derivative $\partial{\cal F}/\partial\epsilon \equiv [{\cal F}, {\cal S}]$ of an arbitrary functional ${\cal F}[f,{\bf E},{\bf B}]$ of the Vlasov-Maxwell fields $(f,{\bf E},{\bf B})$, which are assumed to depend continuously on the (dimensionless) perturbation parameter $\epsilon$. Here, $[\;,\;]$ denotes the standard Vlasov-Maxwell functional bracket, and the perturbation {\it action} functional ${\cal S}$ is said to generate perturbations of the Vlasov-Maxwell fields. The new Hamiltonian perturbation formulation highlights the crucial roles played by polarization and magnetization in Vlasov-Maxwell perturbation theory.