A multi-parameter Lie transform method applied to the transform of the Lagrangian differential one-form

TL;DR

This paper generalizes the Lie transform perturbation theory to multiple parameters to better handle multi-scale perturbations and applies it to derive an improved gyrokinetic model that cancels certain terms without gauge functions.

Contribution

It introduces a multi-parameter Lie transform method for transforming Lagrangian forms and applies it to derive a gyrokinetic model with simplified finite Larmor radius terms.

Findings

Finite Larmor Radius terms are canceled in the new model.

The method reduces fast variables order by order.

No gauge function is needed for the cancellation.

Abstract

In this paper, to fit the multi-scale perturbations, the single-parameter Lie transform perturbation theory given in [Ann Phys, 151 1 (1983)] is generalized to a multi-parameter case, which provides a formal solution of the new Lagrangian differential 1-form transformed from the old one. In the new method, the generators and their orders are appropriately chosen under the purpose of reducing the fast variables from the new Lagrangian differential 1-form order by order. As for the application, this multi-parameter Lie transform method is applied to deriving the gyrokinetic model with the presentation of low-frequency electrostatic perturbations. Compared with the conventional model, the Finite Larmor Radius terms in the Lagrangian 1-form of the new model are perfectly cancelled, without using any gauge function.