The Biot-Savart operator for application to knot theory, fluid dynamics, and plasma physics. (English) Zbl 1006.78003
Summary: The writhing number of a curve in 3-space is the standard measure of the extent to wich the curve wraps and coils around itself; it has proved its importance for molecular biologists in the study of knotted DNA and of the enzymes which affect it. The helicity of a vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid dynamics and plasma physics. The Biot-Savart operator associates with each current distribution on a given domain the restriction of its magnetic field to that domain. When the domain is simply connected, the divergence-free fields which are tangent to the boundary and which minimize energy for given helicity provide models for stable force-free magnetic fields in space and laboratory plasmas; these fields appear mathematically as the extreme eigenfields for an appropriate modification of the Biot-Savart operator. Information about these fields can be converted into bounds on the writhing number of a given piece of DNA. The purpose of this paper is to reveal new properties of the Biot-Savart operator which are useful in these applications.
MSC:
| 78A25 | Electromagnetic theory (general) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
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