Application of Lagrangian techniques for calculating the on-axis rotational transform

TL;DR

This paper introduces a Lagrangian-based formalism to compute the on-axis rotational transform in magnetic fields, linking Floquet exponents to a coordinate-independent eigenvalue problem suitable for numerical analysis.

Contribution

It combines the near-axis formalism with a Lagrangian approach to derive a discrete eigenvalue method for Floquet exponents and rotational transform calculation.

Findings

Floquet exponents equal the on-axis rotational transform

Eigenvalues are obtained from a 6x6 matrix eigenproblem

Method is suitable for numerical implementation

Abstract

The Floquet exponents of periodic field lines are studied through the variations of the magnetic action on the magnetic axis, which is assumed to be elliptical. The near-axis formalism developed by Mercier, Solov'ev and Shafranov is combined with a Lagrangian approach. The on-axis Floquet exponent is shown to coincide with the on-axis rotational transform, and this is a coordinate-independent result. A discrete solution suitable for numerical implementation is introduced, which gives the Floquet exponents as solutions to an eigenvalue problem. This discrete formalism expresses the exponents as the eigenvalues of a $6\times 6$ matrix.