Hamiltonian formulation of reduced Vlasov-Maxwell equations

We present a Hamiltonian formulation of the reduced Vlasov- Maxwell equations which is expressed in terms of the macroscopic fields ${\bf D}$ and ${\bf H}$. These macroscopic fields are themselves expressed in terms of the Lie-transform operator $\exp \pounds_{{\cal S}}$ generated by the functional ${\cal S}$, where $\pounds_{{\cal S}}{\cal F} \equiv [{\cal S},\;{\cal F}]$ is expressed in terms of the Poisson bracket $[\;,\;]$ for the exact Vlasov-Maxwell equations. Hence, the polarization vector ${\bf P} \equiv ({\bf D} - {\bf E})/4\pi$ and the magnetization vector ${\bf M} \equiv ({\bf B} - {\bf H})/ 4\pi$ are defined in terms of the expressions $4\pi\,{\bf P} \equiv [{\cal S},\;{\bf E}] + \cdots$ and $4\pi\,{\bf M} \equiv -\; [{\cal S},\;{\bf B}] + \cdots$, where lowest-order terms yield dipole contributions.