Magnetic vortex filament flows

Abstract

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.

Magnetic curves in Sasakian manifolds

“…Studying Killing magnetic curves in 3-dimensional space forms, Barros and Romero showed in [10] that these curves are centerlines of Kirchhoff elastic rods. Further on, Barros et al in [9] showed that these curves are solitons of the localized induction equation (LIE). A more global and geometric view on the connection between solitons of LIE and magnetic curves are provided in [29].…”

“…Let us denote the corresponding divergence-free vector field of F by V and the 1-form that is metrically dual to V by ω. If V is a unitary vector field, then (φ ,V, ω) is an almost contact structure on M compatible with the metric g. Hence, an oriented Riemannian 3-manifold (M, g) together with a magnetic field F whose corresponding divergence-free vector field has unit length is regarded as an almost contact metric manifold with closed fundamental 2-form, see [9].…”

Magnetic non-null curves according to parallel transport frame in Minkowski 3-space

“…It is clear that, any Killing vector …eld on (M n ; g) is divergence-free. In particular, if n = 3, then every Killing vector …eld de…nes a magnetic …eld that will be called a Killing magnetic …eld [1].…”

Notes on magnetic curves in 3D semi-Riemannian manifolds

“…If the charged particle moves parallel to magnetic field, the Lorentzian force acting on the particle is zero. When the two vectors (velocity and the magnetic field) are perpendicular to each other, the Lorentz force is maximum (for details, see [4,2,3,6,7,9,8]). …”

“…It is clear that any Killing vector field on (M n , g) is divergencefree. In particular, if n = 3, then every Killing vector field defines a magnetic field that will be called a Killing magnetic field (for details, see [2]). …”