Taylor state

In plasma physics, a Taylor state is the minimum energy state of a plasma satisfying the constraint of conserving magnetic helicity.
Derivation.
Consider a closed, simply-connected, flux-conserving, perfectly conducting surface formula_1 surrounding a plasma with negligible thermal energy (formula_2).
Since formula_3 on formula_1. This implies that formula_5.
As discussed above, the plasma would relax towards a minimum energy state while conserving its magnetic helicity. Since the boundary is perfectly conducting, there cannot be any change in the associated flux. This implies formula_6 and formula_7 on formula_1.
We formulate a variational problem of minimizing the plasma energy formula_9 while conserving magnetic helicity formula_10.
The variational problem is
formula_11.
After some algebra this leads to the following constraint for the minimum energy state
formula_12.