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## 2. Coordinates and Fluxes 

We use a general curvilinear coordinate system  ; GLYPH<18>; GLYPH<16> , as shown in Fig. 1, where GLYPH<18> and GLYPH<16> are poloidal and toroidal angles, respectively, and is a label for the coordinate sur- faces that has the dimension of magnetic flux and is such that B = r GLYPH<2>r GLYPH<18> + r GLYPH<16> GLYPH<2>r GLYPH<31> . 1 

The standard linear flux $^{R}$dS n GLYPH<1> B through surface GLYPH<0> : =  $_{GLYPH<0>}$ ( GLYPH<18>; GLYPH<16> ), where dS is an element of surface area and n is the unit normal, is independent of choice of coordi- nates and vanishes identically. Thus it is independent of the choice of GLYPH<0> also, whether it be a magnetic surface (in- variant torus of the field-line flow) or otherwise. 

To measure the amount by which GLYPH<0> departs from being a magnetic surface, we are instead led to define the positive definite quadratic flux [3] 

where GLYPH<2> is an auxiliary poloidal angle, not necessarily the coordinate angle GLYPH<18> . Setting GLYPH<14>’$_{2}$ = 0 for arbitrary variations  $_{GLYPH<0>}$ we find the Euler–Lagrange equation for QFMin pseudo-orbits [3, 4] is B$_{GLYPH<23>}$ GLYPH<1>r GLYPH<23> = 0, where GLYPH<23> GLYPH<17> n GLYPH<1> B = n GLYPH<1>r GLYPH<2> GLYPH<2>r GLYPH<16> and the “pseudo magnetic field” B$_{GLYPH<23>}$ is de- fined by 

Thus GLYPH<23> parametrizes a continuous family of pseudo field lines, which can be thought of as orbits of a dynamical system. We define a p ; q -periodic pseudo-orbit ( p and q being integers) as a path that is a closed loop, with GLYPH<18> in- creasing by 2 GLYPH<25> p when GLYPH<16> increases by 2 GLYPH<25> q , so the average 

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rate of increase of GLYPH<18> along the path is the rational rotational transform GLYPH<19> -$_{p}$$_{;}$$_{q}$ = p = q . 

For a given GLYPH<23> there are generally two distinct p ; q - periodic orbits. In particular, when GLYPH<23> = 0 we find the true periodic orbits associated with the X- and O-points of a p ; q island chain. 

The family of p ; q -periodic QFMin pseudo-orbits parametrized by GLYPH<23> defines an “almost invariant” surface with rotational transform GLYPH<19> -$_{p}$$_{;}$$_{q}$, which passes through the p ; q island chain as depicted in Fig. 2. 

## 3. Action and ghost surfaces 

Using the vector potential representation B = rGLYPH<2> A , the magnetic action of a closed p ; q -periodic path C is 

where ˙ r GLYPH<17> d r = d GLYPH<16> along the path. Hamilton’s Principle, GLYPH<14> S = 0, gives the Euler–Lagrange equation GLYPH<14> S =GLYPH<14> r = ˙ r GLYPH<2> B = 0, which implies ˙ r is parallel to B as required. 

Closed paths making S stationary are isolated, but the Poincar´e–Birkho GLYPH<11> theorem implies there are at least two associated with each island chain, as illustrated in Fig. 2, with the X-point orbits being minima of S , while the O- point orbits are minimax (saddle) points of S . The ghost- curve strategy for “joining the dots” is to generate a family of pseudo-orbits, labelled by a continuous parameter GLYPH<28> , by flowing down the action gradient from minimax orbits to minimizing orbits, 

where P$_{ghost}$ is a symmetric nonnegative dyadic whose choice is discussed below. The union of this set of pseudo- orbits defines a ghost surface (a ghost curve being the Poincar´e section of a ghost surface). 

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