Response to "Comment on `Modification of Lie's transform perturbation theory for charged particle motion in a magnetic field'" [Phys. Plasmas 30, 104701 (2023)]

TL;DR

This paper critically addresses misconceptions in a prior comment, defending the author's modification of Lie's transform theory for charged particle motion in magnetic fields, highlighting inconsistencies in the standard approach.

Contribution

The paper clarifies and defends a modified Lie's transform theory, identifying issues in the standard formulation and providing a more consistent and efficient approach.

Findings

Identifies inconsistencies in the standard Lie's transform theory.

Highlights issues with derivative and limit commutation.

Proposes a more straightforward derivation process.

Abstract

Dr. Brizard's comment on my work is based on a conceived procedure that does not come from my work. The defense of his claim that the modification of the so-called standard Lie's transform theory is unnecessary is also unsupported. This response reveals in detail the inconsistency issues in the so-called standard Lie's transform theory by analyzing both its results and root causes. The problem in the so-called standard Lie's transform theory is beyond the issue to take into account the ordering difference between the guiding center motion and gyromotion. The inconsistent commutation of derivative and limit causes another issue. Besides, the so-called standard Lie's transform formulation leads to an unnecessarily lengthy and tedious derivation process for a one or two page task under the singular (or renormalized) formalism described in my paper.