Hamiltonian description for magnetic field lines: a tutorial

TL;DR

This tutorial explains how magnetic field lines in plasmas can be described using Hamiltonian mechanics, aiding analysis of magnetic confinement devices like tokamaks through a variational approach and coordinate applications.

Contribution

It provides a comprehensive review of the Hamiltonian formalism for magnetic field lines, highlighting variational principles and gauge invariance in plasma physics applications.

Findings

Hamiltonian formalism effectively models magnetic field lines.

Application to tokamak geometries demonstrates practical utility.

Variational principles facilitate analysis of magnetic confinement.

Abstract

Under certain circumstances, the equations for the magnetic field lines can be recast in a canonical form, after defining a suitable field line Hamiltonian. This analogy is extremely useful for dealing with a variety of problems involving magnetically confined plasmas, like in tokamaks and other toroidal devices, where there is usually one symmetric coordinate which plays the role of time in the canonical equations. In this tutorial paper we review the basics of the Hamiltonian description for magnetic field lines, emphasizing the role of a variational principle and gauge invariance. We present representative applications of the formalism, using cylindrical and magnetic flux coordinates in tokamak plasmas.