Non-planar elasticae as optimal curves for the magnetic axis of stellarators

TL;DR

This paper derives and analyzes non-planar elasticae as optimal magnetic axis curves for stellarators, providing new solutions with specific geometric properties and implications for magnetic field-line behavior.

Contribution

It introduces a novel class of non-planar elasticae solutions for stellarator magnetic axes, including closed curves and their geometric and topological properties.

Findings

Derived Euler-Lagrange equations for optimal curves

Identified infinite families of closed non-planar curves

Analyzed torsion, linking, and writhe in relation to stellarator fields

Abstract

The problem of finding an optimal curve for the target magnetic axis of a stellarator is addressed. Euler-Lagrange equations are derived for finite length three-dimensional curves that extremise their bending energy while yielding fixed integrated torsion. The obvious translational and rotational symmetry is exploited to express solutions in a preferred cylindrical coordinate system in terms of elliptic Jacobi functions. These solution curves, which, up to similarity transformations, depend on three dimensionless parameters, do not necessarily close. Two closure conditions are obtained for the vertical and toroidal displacement (the radial coordinate being trivially periodic) to yield a countably infinite set of one-parameter families of closed non-planar curves. The behaviour of the integrated torsion (Twist of the Frenet frame), the Linking of the Frenet frame and the Writhe of the solution curves is studied in light of the \Calugareanu theorem. A refreshed interpretation of Mercier's formula for the on-axis rotational transform of stellarator magnetic field-lines is proposed.