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

TL;DR

This paper investigates how the Hamiltonian structure of the Vlasov-Maxwell equations is preserved under Lie-transform reductions, aiming to derive explicit Hamiltonian formulations for plasma models like guiding-center and gyrokinetics.

Contribution

It demonstrates that the reduced Vlasov-Maxwell equations retain a Hamiltonian structure with a properly transformed bracket and Hamiltonian functional.

Findings

Reduced equations maintain Hamiltonian properties

Explicit form of the reduced bracket derived

Facilitates Hamiltonian formulations of plasma models

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 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.