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A magnetic arcade is a family of arch-shaped magnetic field lines joining two regions of a boundary. The boundary intersections are its footpoints. In a simple line-tied family, paired footpoints can be labelled by Euler potentials. A decaying linear force-free magnetic arcade is one explicitly solvable planar example; a general arcade need not be linear or force-free.

Choose two independent labels for the magnetic field lines, constant along each line. These are Euler potentials: $\mathbf{B}\cdot \nabla \alpha =\mathbf{B}\cdot \nabla \beta =0$. Where the two gradients are independent, the vector perpendicular to both is $\nabla \alpha \times \nabla \beta$, so initially writeThe solenoidal condition gives $0=\nabla \cdot \mathbf{B}=(\nabla \alpha \times \nabla \beta )\cdot \nabla f$. Thus $f$ is constant along a field line and is locally a function of its labels. Redefining one label, for example $\tilde{\beta }(\alpha ,\beta )={\int }^{\beta }f(\alpha ,b)db$, absorbs this factor. Dropping the tilde givesThis representation automatically has zero divergence, since it is also $\nabla \times (\alpha \nabla \beta )$.

For an magnetic arcade, assign the labels on one polarity of the footpoint surface and transport them along each line to its other footpoint. They specify both line identity and magnetic flux: on a transverse surface, flux coordinates can be chosen so that the directed flux element is $d\alpha d\beta$. In particular,The boundary labels at paired footpoints must agree, so prescribed boundary functions encode a compatible footpoint mapping, not independently arbitrary values at the two ends.

The construction assumes a smooth family of nonintersecting lines that can be labelled from a transverse boundary patch, with suitable patches at nulls or coordinate singularities. A local Euler-potential representation does not establish the existence of two globally single-valued regular labels for every possible solenoidal field; closed, recurrent or more complicated field topology can obstruct such a global choice. The line-tied magnetic arcade geometry supplies the relevant simplifying assumption here.

Use the magnetic energy $E_B=(2{\mu }_0{)}^{-1}{\int }_V|\mathbf{B}{|}^{2}dV$. Vary the Euler potentials while keeping their boundary values fixed. The field variation isLet $\delta \mathbf{A}=\delta \alpha \nabla \beta -\delta \beta \nabla \alpha$. Integration by parts givesbecause $\delta \mathbf{A}=0$ on the boundary. ThereforeThe two interior variations are independent. At a smooth minimum their coefficients must vanish, so the curl is perpendicular to both potential gradients and hence parallel to $\mathbf{B}$ wherever $\mathbf{B}\ne 0$:This is a force-free magnetic field: the Lorentz force density vanishes. Taking the divergence also gives $\mathbf{B}\cdot \nabla \lambda =0$, so the proportionality factor is constant along a field line but need not be the same on different lines.

The fixed-label variation preserves the encoded field-line connection rather than allowing arbitrary reconnection during minimization. The argument proves a necessary condition for a smooth minimum, not that every force-free stationary field is a stable energy minimum. It also does not include gas pressure or gravity in the minimized functional.