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

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

## References

1. ^ Woltjer, L. (1958). A theorem on force-free magnetic fields. Proceedings of the National Academy of Sciences, 44(6), 489-491.
2. ^ Chiuderi, C., & Velli, M. (2016). Basics of Plasma Astrophysics. Springer.
3. ^ Moffatt, H. K. (1978). Field generation in electrically conducting fluids. Cambridge University Press, Cambridge, London, New York, Melbourne.
4. ^ Sturrock, P. A. (1994). Plasma Physics: an introduction to the theory of astrophysical, geophysical and laboratory plasmas. Cambridge University Press.
5. ^ Solov'ev, A. A. (1985). Woltjer's theorem and the force-free magnetic field stability problem. Byulletin Solnechnye Dannye Akademie Nauk SSSR, 1985, 55-62.
6. ^ Kholodenko, A. L. (2013). Applications of contact geometry and topology in physics. World Scientific.