On the Geometrical Gyro-Kinetic Theory

TL;DR

This paper develops a sophisticated coordinate transformation for Hamiltonian systems modeling charged particle motion in fusion devices, utilizing advanced mathematical tools like PDEs, differential geometry, and introducing Partial Lie Sums.

Contribution

It introduces a novel coordinate change method employing Partial Lie Sums, enhancing the reduction of Hamiltonian systems in plasma physics.

Findings

Successfully constructs a coordinate transformation for charged particle dynamics.

Integrates PDEs, differential geometry, and Lie transforms in the methodology.

Provides a new mathematical tool: Partial Lie Sums, for system reduction.

Abstract

Considering a Hamiltonian Dynamical System describing the motion of charged particle in a Tokamak or a Stellarator, we build a change of coordinates to reduce its dimension. This change of coordinates is in fact an intricate succession of mappings that are built using Hyperbolic Partial Differential Equations, Differential Geometry, Hamiltonian Dynamical System Theory and Symplectic Geometry, Lie Transforms and a new tool which is here introduced : Partial Lie Sums.