The Gyro-Kinetic Approximation. An attempt at explaining the method based on Darboux Algorithm and Lie Transform

TL;DR

This paper introduces a method to derive the gyro-kinetic approximation for particles in strong magnetic fields using Darboux algorithms and Lie transforms, simplifying the particle dynamics by coordinate transformation.

Contribution

It presents a novel approach combining Darboux algorithms and Lie transforms to systematically obtain the gyro-kinetic approximation.

Findings

Successfully derives the gyro-kinetic equations for particle trajectories.

Provides a systematic coordinate transformation method.

Enhances understanding of particle dynamics in strong magnetic fields.

Abstract

This Proceeding presents the method that allows us to get the Gyro-Kinetic Approximation of the Dynamical System satisfied by the trajectory of a particle submitted to a Strong Magnetic Field. The goal of the method is to build a change of coordinates in order to make the dynamic of two components of the trajectory to disappear. This change of coordinates is based on a Darboux mathematical Algorithm and on a Lie Transform