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 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.

39 pages

References in corpus (10)

• A general theory for gauge-free lifting
• Hamiltonian formulation of the gyrokinetic Vlasov-Maxwell equations
• Beyond linear gyrocenter polarization in gyrokinetic theory
• Lagrangian and Hamiltonian constraints for guiding-center Hamiltonian theories
• On the Hamiltonian formulation of incompressible ideal fluids and magnetohydrodynamics via Dirac's theory of constraints
• The Hamiltonian description of incompressible fluid ellipsoids
• The Hamiltonian structure and Euler-Poincaré formulation of the Vlasov-Maxwell and gyrokinetic systems
• Variational principles for the guiding-center Vlasov-Maxwell equations
• Energetically-consistent collisional gyrokinetics
• Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas

Cited by in corpus (5)

• Structure and structure-preserving algorithms for plasma physics
• Explicit high-order noncanonical symplectic algorithms for ideal two-fluid systems
• Hamiltonian structure of the gauge-free gyrokinetic Vlasov-Maxwell equations
• Hamiltonian formulations for perturbed dissipationless plasma equations
• Hamiltonian structure of the guiding-center Vlasov-Maxwell equations