Generalized action-angle coordinates defined on island chains

Straight-field-line coordinates are very useful for representing magnetic fields in toroidally confined plasmas, but fundamental problems arise regarding their definition in 3D geometries because of the formation of islands and chaotic field regions, i.e. non-integrability. In Hamiltonian dynamical systems terminology these coordinates are similar to action-angle variables, but these are normally defined only for integrable systems. In order to describe 3D magnetic field systems, a generalization of this concept was proposed recently by the present authors that unified the concepts of ghost surfaces and quadratic-flux-minimizing (QFMin) surfaces. This was based on a simple canonical transformation generated by a change of variable θ = θ(Θ, ζ), where θ and ζ are poloidal and toroidal angles, respectively, with Θ a new poloidal angle chosen to give pseudo-orbits that are (a) straight when plotted in the ζ, Θ plane and (b) QFMin pseudo-orbits in the transformed coordinate. These two requirements ensure that the pseudo-orbits are also (c) ghost pseudo-orbits. In this paper, it is demonstrated that these requirements do not uniquely specify the transformation owing to a relabelling symmetry. A variational method of solution that removes this lack of uniqueness is proposed.