Woltjer's theorem

{{Short description|Theorem in plasma physics}}

In plasma physics, Woltjer's theorem states that force-free magnetic fields in a closed system with constant force-free parameter $\backslash alpha$ represent the state with lowest magnetic energy in the system and that the magnetic helicity is invariant under this condition. It is named after Lodewijk Woltjer who derived it in 1958.{{cite journal |last1=Woltjer |first1=L. |title=A Theorem on Force-Free Magnetic Fields |journal=Proceedings of the National Academy of Sciences |date=June 1958 |volume=44 |issue=6 |pages=489–491 |doi=10.1073/pnas.44.6.489 |doi-access=free |pmid=16590226 |bibcode=1958PNAS...44..489W|pmc=528606 }}{{cite book |last1=Chiuderi |first1=C. |last2=Velli |first2=Marco |last3=Chiuderi |first3=Claudio |title=Basics of Plasma Astrophysics |date=2015 |publisher=Springer |location=Milan |isbn=978-88-470-5280-2}}{{cite book |last1=Moffat |first1=H. K. |title=Magnetic Field Generation in Electrically Conducting Fluids |date=1978 |publisher=Cambridge University Press |location=Cambridge |url=https://www.researchgate.net/publication/229086771}}{{cite book |last1=Sturrock |first1=P. A. |title=Plasma Physics: An Introduction to the Theory of Astrophysical, Geophysical and Laboratory Plasmas |date=1994 |publisher=Cambridge University Press |location=Cambridge |isbn=9780521448109 |url=https://archive.org/details/plasmaphysicsint0000stur}}{{cite journal |last1=Solov'ev |first1=A. A. |title=Woltjer's Theorem and the Force-Free Magnetic Field Stability Problem |journal=Byulletin Solnechnye Dannye Akademie Nauk USSR |date=1985 |volume=1985 |pages=55–62 |bibcode=1985BSolD1985...55S}}{{cite book |last1=Kholodenko |first1=Arkady L. |title=Applications of Contact Geometry and Topology in Physics |date=2013 |publisher=World Scientific |location=New Jersey |doi=10.1142/8514 |bibcode=2013acgt.book.....K |isbn=9789814412087 |url=https://www.worldscientific.com/worldscibooks/10.1142/8514}} A force-free magnetic field with flux density $\backslash mathbf\{B\}$ satisfies

:$\backslash nabla\; \backslash times\; \backslash mathbf\{B\}\; =\; \backslash alpha\; \backslash mathbf\{B\}$

where $\backslash alpha$ is a scalar function that is constant along field lines. The helicity $\backslash mathcal\{H\}$ invariant is given by

:$\backslash frac\{d\backslash mathcal\{H\}\}\{d\; t\}\; =\; 0$

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

:$\backslash mathcal\{H\}\; =\; \backslash int\_V\; \backslash mathbf\{A\}\backslash cdot\backslash mathbf\{B\}\backslash \; dV\; =\; \backslash int\_V\; \backslash mathbf\{A\}\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash mathbf\{A\})\; \backslash \; dV.$