Lifting of the Vlasov–Maxwell bracket by Lie-transform method

Abstract

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 provide an explicit pathway to the Hamiltonian formulations for the guiding-centre and gyrokinetic Vlasov–Maxwell equations, which have found 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.