Action-Angle variables defined on island chains

Straight-field-line coordinates are a particular case of action-angle variables, which, in standard Hamiltonian mechanics, are defined only for integrable systems. In order to describe 3-D magnetic field systems, a generalization of this concept was proposed in [1] that unified the concepts of ghost surfaces (almost-invariant tori defined by an action-gradient flow between O and X points of an island chain) and quadratic-flux-minimizing surfaces (QFMin tori, which minimize a weighted mean of the square of the normal component of \textbf{B}). This was based on a simple canonical transformation, generated by a change of variable $\theta = \theta(\Theta)$, where $\theta$ is the old poloidal angle and $\Theta$ a new one giving straight pseudo-orbits (approximate field lines [2]). This was illustrated using a perturbative construction of the transformation. Investigations of this idea using the Standard Map [3], with the analog of the same constraint as used implicitly in [1] to make $\Theta$ unique, show this constraint is not optimal in that $\theta(\Theta)$ ceases to be monotone beyond a certain nonlinearity.\\ \noindent[1] R.L. Dewar, S.R. Hudson and A.M. Gibson JPFR (2010) http://arxiv.org/abs/1001.0483; [2] R.L. Dewar, S.R. Hudson and A.M. Gibson CNSNS in press (2011) DOI:10.1016/j.cnsns.2011.04.022; [3] R.L. Dewar and A.B. Khorev, Physica D \textbf{85}, 66 (1995)