Lifting the Vlasov-Maxwell bracket by Lie-transform methods

The Vlasov-Maxwell equations possess a Hamiltonian structure expressed in terms of a Hamiltonian functional and a functional bracket [1]. In the present work, the transformation (``lift'') of the Vlasov-Maxwell bracket induced by the dynamical reduction of single-particle particle dynamics is investigated when the reduction is carried out by Lie-transform perturbation methods. A formal proof of the Jacobi identity for the reduced Vlasov-Maxwell bracket is presented. The ultimate goal of this work is to derive explicit Hamiltonian formulations for the guiding-center and gyrokinetic Vlasov-Maxwell equations that have important applications in our understanding of turbulent magnetized plasmas. A comparison with a bracket structure [2] for the gyrokinetic Vlasov-Poisson equations derived by Dirac-constraint method will be presented. \\[4pt] [1] See P.J. Morrison, RMP {\bf 70} 467 (1998) and references therein.\\[0pt] [2] J. Squire, H. Qin, W.M. Tang, and C. Chandre, PoP {\bf 20}, 022501 (2013).