Application of Lagrangian integration to locate the magnetic axis

Lagrangian variational methods are applied to the magnetic axis of stellarator vacuum fields. The magnetic fieldline action [1] is given by S[C] = \oint A(x) · dl, where A(x) is a magnetic vector potential and C is a smooth, closed trial curve, which we represent as x(ζ). For a given vector potential, the first variation resulting from a variation, δx, in the trial curve is δS = \oint δx · x′ × B dζ, which shows that stationary curves are magnetic fieldlines,x′ × B=0. The behavior of fieldlines nearby the magnetic axis are governed by the second variation of the action. By exploiting periodicity and using Floquet theory, the on-axis Floquet exponent is shown [2] to coincide with the on-axis rotational transform, and an expression for the rotational transform is derived, which agrees with Mercier’s [3]. Allowing for changes in the vector potential, the second mixed variation of the action determines the sensitivity of the geometry of the magnetic axis to variations in the magnetic field, and singular value decomposition can be used to identify the important “error fields” that drive deformations to the axis. The Lagrangian integration methods can be applied to any closed, periodic fieldline; including for example, the unstable periodic fieldline at the separatrix or at the island island divertor.

[1] Cary & Littlejohn, Ann. Phys. 151:1 (1983)

[2] Guinchard, Sengupta & Hudson, arXiv:2404.17531

[3] Mercier, (1964) Nucl. Fusion 4:213 (1984)