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Magnetic vortex filament flows. (English) Zbl 1152.81329
Summary: We exhibit a variational approach to study the magnetic flow associated with a Killing magnetic field in dimension 3. In this context, the solutions of the Lorentz force equation are viewed as Kirchhoff elastic rods and conversely. This provides an amazing connection between two apparently unrelated physical models and, in particular, it ties the classical elastic theory with the Hall effect. Then, these magnetic flows can be regarded as vortex filament flows within the localized induction approximation. The Hasimoto transformation can be used to see the magnetic trajectories as solutions of the cubic nonlinear Schrödinger equation showing the solitonic nature of those.

MSC:
76M30 Variational methods applied to problems in fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
78A25 Electromagnetic theory, general
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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