A gyro-gauge independent minimal guiding-center reduction by Lie-transforming the velocity vector field

TL;DR

This paper presents a gauge-independent, systematic guiding-center reduction method using Lie-transforming the velocity vector field, simplifying particle dynamics by removing fast time-scales with explicit second-order derivation.

Contribution

It introduces a novel, geometric Lie-transform approach to guiding-center reduction that is gauge-independent and systematically derived up to second order.

Findings

Reduction procedure is algorithmic and explicit up to second order.

The method simplifies guiding-center calculations compared to phase-space Lagrangian approaches.

The approach clarifies the structure of guiding-center expansions.

Abstract

We introduce a gyro-gauge independent formulation of a simplified guiding-center reduction, which removes the fast time-scale from particle dynamics by Lie-transforming the velocity vector field. This is close to Krylov-Bogoliubov method of averaging the equations of motion, although more geometric. At leading order, the Lie-transform consists in the generator of Larmor gyration, which can be explicitly inverted, while working with gauge-independent coordinates and operators, by using the physical gyro-angle as a (constrained) coordinate. This brings both the change of coordinates and the reduced dynamics of the minimal guiding-center reduction order by order in a Larmor radius expansion. The procedure is algorithmic and the reduction is systematically derived up to full second order, in a more straightforward way than when Lie-transforming the phase-space Lagrangian or averaging the equations of motion. The results write up some structures in the guiding-center expansion. Extensions and limitations of the method are considered.