Magnetic moment type of lifting from particle dynamics to Vlasov-Maxwell dynamics

TL;DR

This paper develops coordinate transformation techniques for the Maxwell-Vlasov Hamiltonian framework, enabling the use of magnetic moment coordinates and simplifying gyrokinetic models by reducing electric field energy.

Contribution

It introduces a method to lift particle coordinate changes dependent on magnetic fields to the field Hamiltonian structure, facilitating advanced plasma modeling.

Findings

Successfully transforms the noncanonical Poisson bracket for magnetic coordinate changes.

Derives a simplified gyrokinetic model using magnetic moment as a coordinate.

Provides examples including magnetic and spherical velocity coordinates.

Abstract

Techniques for coordinate changes that depend on both dependent and independent variables are developed and applied to the Maxwell-Vlasov Hamiltonian theory. Particle coordinate changes with a new velocity variable dependent on the magnetic field, with spatial coordinates unchanged, are lifted to transform the noncanonical Poisson bracket and, thus, the field Hamiltonian structure of the Vlasov-Maxwell equation. Several examples are given including magnetic coordinates, where the velocity is decomposed into components parallel and perpendicular to the local magnetic field, and the case of spherical velocity coordinates. An example of the lifting procedure is performed to obtain a simplified version of gyrokinetics, where the magnetic moment is used as a coordinate and the dynamics is reduced by elimination of the electric field energy in the Hamiltonian.