  The Vlasov-Maxwell equations possess a Hamiltonian structure expressed in
terms of a Hamiltonian functional and a functional bracket. In the present
paper, the transformation ("lift") of the Vlasov-Maxwell bracket 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. Here, it is
shown that the general form of the reduced Vlasov-Maxwell equations possesses a
Hamiltonian structure defined in terms of a reduced Hamiltonian functional and
a reduced bracket that automatically satisfies the standard bracket properties.
