Woltjer's Theorem
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Woltjer's Theorem

In plasma physics, Woltjer's theorem states that force-free magnetic fields, for a closed system with constant ${\displaystyle \alpha }$, have the minimum magnetic energy and the magnetic helicity is invariant under this condition, named after Lodewijk Woltjer who derived in 1958[1][2][3][4][5][6]. The force-free field strength ${\displaystyle \mathbf {B} }$ equation is

${\displaystyle \nabla \times \mathbf {B} =\alpha \mathbf {B} .}$

The helicity ${\displaystyle {\mathcal {H}}}$ invariant is given by

${\displaystyle {\frac {d{\mathcal {H}}}{dt}}=0.}$

where ${\displaystyle {\mathcal {H}}}$ is related to ${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$ through the vector potential ${\displaystyle \mathbf {A} }$ as below

${\displaystyle {\mathcal {H}}=\int _{V}\mathbf {A} \cdot \mathbf {B} \ dV=\int _{V}\mathbf {A} \cdot (\nabla \times \mathbf {A} )\ dV.}$