Lifting of the Vlasov-Maxwell Bracket by Lie-transform Method

The Vlasov-Maxwell equations possess a Hamiltonian structure expressed in terms of a Hamiltonian functional and a functional bracket. The transformation (``lift'') of the Vlasov-Maxwell bracket [1,2] induced by the dynamical reduction of single-particle dynamics is investigated when the reduction is carried out by Lie-transform perturbation methods. 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. \newline In particular, we investigate how the Hamiltonian properties of the reduced Vlasov-Maxwell bracket survive (1) the {\it closure} problem: the process of truncation of the guiding-center Vlasov-Maxwell bracket at a finite order in $\epsilon$ (so far expressions have been derived at all orders in $\epsilon$) and (2) the {\it averaging} problem: the process by which which the gyroangle is eliminated from the guiding-center Vlasov-Maxwell bracket (since guiding-center Vlasov-Maxwell equations do not involve the fast gyromotion time scale). \newline [1] P.J.~Morrison, PoP 20 (2013) 012104. \newline [2] A.J.~Brizard, P.J.~Morrison, J.W.~Burby, {\it et al.}, submitted for publication.