Woltjer's theorem

Remarks

The proof uses a Lagrange multiplier. Write $B = \nabla \times A$ with a vector potential $A$ and vary $A$ by a smooth field $a$ with compact support in $V$. Integration by parts gives $\delta E = 2 \int_V a \cdot (\nabla \times B) \, dV$ and $\delta \mathcal{H} = 2 \int_V a \cdot B \, dV$. At a constrained minimum $\delta E = \alpha \, \delta \mathcal{H}$ for some constant $\alpha$ and all $a$, so $\nabla \times B = \alpha B$. The constant is nonzero, since $\alpha = 0$ would make $B$ curl-free, and a curl-free, divergence-free field tangent to the boundary of a simply connected region is zero, which has helicity $0 \neq h$.

Woltjer also showed that the magnetic helicity of a perfectly conducting fluid in a region bounded by a magnetic surface is constant in time. Together the two results suggest that a turbulent plasma loses energy while keeping its helicity, and settles into a linear force-free state. This idea was developed into Taylor's theory of plasma relaxation.

Going deeper

For a student

The theorem is a variational characterization of linear force-free fields, also called Beltrami fields, as eigenfields of the curl operator. In the language of operators, minimizing $\int \abs{B}^2$ with $\int A \cdot B$ fixed is a Rayleigh quotient problem for the inverse of the curl on divergence-free fields tangent to the boundary. Helicity conservation in ideal magnetohydrodynamics makes this the natural end state of relaxation in solar flares, spheromaks and reversed-field pinches.