# Taylor state

In plasma physics, a Taylor state is the minimum energy state of a plasma satisfying the constraint of conserving magnetic helicity.[1]

## Derivation

Consider a closed, simply-connected, flux-conserving, perfectly conducting surface $S$ surrounding a plasma with negligible thermal energy ($\beta \rightarrow 0$).

Since $\vec{B}\cdot\vec{ds}=0$ on $S$. This implies that $\vec{A}_{||}=0$.

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 $\delta \vec{B}\cdot\vec{ds}=0$ and $\delta\vec{A}_{||}=0$ on $S$.

We formulate a variational problem of minimizing the plasma energy $W=\int d^3rB^2/2\mu_\circ$ while conserving magnetic helicity $K=\int d^3r\vec{A}\cdot\vec{B}$.

The variational problem is $\delta W -\lambda \delta K = 0$.

After some algebra this leads to the following constraint for the minimum energy state $\nabla \times \vec{B} = \lambda \vec{B}$.