Notes on magnetic curves in 3D semi-Riemannian manifolds

Abstract

A magnetic field is defined by the property that its divergence is zero in three-dimensional semi-Riemannian manifolds. Each magnetic field generates a magnetic flow whose trajectories are curves γ , called magnetic curves. In this paper, we investigate the effect of magnetic fields on the moving particle trajectories by variational approach to the magnetic flow associated with the Killing magnetic field on three-dimensional semi-Riemannian manifolds. We then investigate the trajectories of these magnetic fields and give some characterizations and examples of these curves.

On Energies of Charged Particles with Magnetic Field

“…Magnetic fields and magnetic curves are studied interdisciplinary, especially in physics and differential geometry. The Lorentz force Equation (5) can be applied in some areas such as in protons, cancer therapy, and velocity selectors [18]. Firstly, we mention about what they mean in physics.…”

Magnetic trajectories in three‐dimensional Lie groups

“…Also, magnetic fields and their corresponding magnetic curves on different spaces are one of the important research topics between physics and differential geometry. So magnetic curves have been studied and developed by a lot of researches under various point of view . Then, Xu and Mould studied magnetic field from a different perspective.…”

“…So magnetic curves have been studied and developed by a lot of researches under various point of view. [1][2][3][4][5] Then, Xu and Mould studied magnetic field from a different perspective. They have used to magnetic curves for create abstract forms and creating description of classes of objects and phenomena that include trees, hair, and conventionally drawn cartoon flames.…”

Magnetic Curves Associated to Killing Vector Fields in a Galilean Space

“…[8,27]), then in 3-dimensional context, on E 3 [16], E 3 1 [17], S 3 [9], S 2 × R [23] etc. For the study of the magnetic curves associated to magnetic fields on arbitrary dimensional Kähler, contact manifolds and Walker 3-manifolds, we also refer the reader to [1,2,7,10,12,13,17,18,21,22,25].…”