Killing Magnetic Field Research Articles

A physical classification of Killing magnetic fields in Thurston geometries

In recent years, numerous studies have appeared that considered Killing vectors of three‐dimensional Riemannian manifolds as magnetic fields, since these vector fields are divergenceless by definition. The existence of adivergenceless vector field modeled as a magnetic field does not imply that it is physically realizable. In this study, we propose a physical classification scheme based on the divergences of the integral that defines the energy of a Killing magnetic field. We consider all the Killing magnetic fields of Thurston geometries studied in the literature and classify them as either physical or nonphysical.

Fractional approach to evolution of the magnetic field lines near the magnetic null points

In this work, the magnetic reconnection model near null points in 3D space will be investigated using fractional calculations in the 3D magnetohydrodynamic framework. For the initial magnetic configuration (without external currents), we reformulated the numerically solved boundary initial value problem using fractional calculations. We studied the 3D Magnetic reconnection states and the behavior of the magnetic field around the null point and the null line. We also analyzed the fractional significance of charged particle motions in Killing magnetic fields. Moreover, the obtained results were visualized, and a comparison was made between the results obtained from integer and fractional calculations.

Fermi–Walker magnetic curves and Killing trajectories in 3D Riemannian manifolds

This paper deals with obtaining some new characterizations on the dynamics of the moving charged particle acting under particular forces in the three‐dimensional space. In this perspective, we have enlarged the research of magnetic curves and associated Killing magnetic fields by adding the Fermi–Walker parallelism and derivative. By doing this, we investigate the Hall–Killing–Fermi effect on the charged particle, then we define new types of Fermi–Walker magnetic curves corresponding to the Killing magnetic field. We carefully construct the Fermi–Walker character of the magnetic field to prove the parallelism of the magnetic trajectories and uniformly accelerated motion of the charged particle. It leads to defining a very special family of magnetic fields, that is, Fermi–Walker uniform magnetic fields. Moreover, we use the Killing feature of the magnetic field to show that the Frenet–Serret scalars and quasislopes of the magnetic curves must meet particular differential equations that lead to obtaining some geometric interpretations of the magnetic trajectories. Last but not the least, both of these two characters allow us to define Killing and Fermi–Walker electric vector fields influencing the motion of the charged particle experiencing the Lorentz force in the three‐dimensional setting.

Killing Magnetic Curves in $\: \mathbb{H}^{3}$

We consider magnetic curves corresponding to the Killing magnetic fields in hyperbolic 3-space.

On Killing Magnetic Fields of Magnetic Pseudo Null Curves in R_1^3

In present paper, we study the properties of the Killing magnetic field of a pseudo null magnetic curve in R_1^3 . We investigate for the T, N and B-magnetic cases of a pseudo null curve. We find the conditions for the Killing magnetic field to be pseudo null and derive the Frenet formulas for this field. Finally, we obtain the conditions for the transformation of the pseudo null curve to its pseudo null Killing magnetic field with equal torsions.

On characterizations on the surfaces of rotation formed by magnetic curves in Q2 ⊂ E31

In this paper, the di erent magnetic elds on the setting out particle trajectories according to the magnetic ow attached by the Killing magnetic field are given in Q2 ⊂ E31. The surfaces generated by the Killing magnetic field of the y􀀀magnetic curve are expressed according to asymptotic orthonormal frame field {h, b, y} in Q2 ⊂ E31 and, using Clairaut's theorem the conditions being geodesic on these surfaces of rotation are given. Also, the curvatures and the kinetic energy and angular momentum of the surfaces of rotation formed by y-magnetic curve are examined.

Geometrical Aspects of Motion of Charged Particles in Magnetic and Killing Magnetic Fields and Their Corresponding Trajectories in Anti-De Sitter 3-Space

We inquire the motion of charged particles varying under the effect of Lorentz force produced by magnetic and Killing magnetic fields in anti-de Sitter 3-space . Primarily, we characterize magnetic trajectories in in terms of their Frenet apparatus. Thereafter, utilizing a geometrical model of split-quaternions for (where corresponds to a subspace of the Lie group H of split-quaternions) we find 6 independent unit Killing vector fields on which constitutes a basis for the corresponding 6D Lie algebra i(). We conclude with characterizations of Killing magnetic trajectories in with respect to quasi-slope, curvature and torsion or pseudo-torsion (depending on the situation).

On Magnetic Curves According to Killing Vector Fields in Euclidean 3-Space

In the geometric theory of space curves, a magnetic field generates magnetic flow. The trajectories of magnetic flow are called magnetic curves. In the present paper, we obtain magnetic curves corresponding to killing magnetic fields in Euclidean 3-space E3. The magnetic curves of the spherical indicatrices of the tangent, principal normal and binormal for a regular space curve are said to be meant curves. Also, we investigate the magnetic curves of the tangent indicatrix and obtain the trajectories of the magnetic fields called TT-magnetic, NT-magnetic and BT-magnetic curves. Finally, some computational examples in support of our main results are given and plotted.

Magnetic trajectories corresponding to Killing magnetic fields in a three-dimensional warped product

The aim of this paper is to investigate Killing magnetic trajectories of varying electrically charged particles in a three-dimensional warped product [Formula: see text] with positive warping function [Formula: see text], where [Formula: see text] is an open interval in [Formula: see text] equipped with an induced semi-Euclidean metric on [Formula: see text]. First, Killing vector fields on [Formula: see text] are characterized and it is observed that lifts to [Formula: see text] of Killing vector fields tangent to [Formula: see text] are also Killing on [Formula: see text]. Now, any Killing vector field on [Formula: see text] corresponds to a Killing magnetic field on [Formula: see text]. Magnetic trajectories (also known as magnetic curves) of charged particles which move under the influence of Lorentz force generated by Killing magnetic fields on [Formula: see text] are obtained in both Riemannian and Lorentzian cases. Moreover, some examples are exhibited with pictures determining Killing magnetic trajectories in hyperbolic [Formula: see text]-space [Formula: see text] modeled by the Riemannian warped product [Formula: see text]. Furthermore, some examples of spacelike, timelike and lightlike Killing magnetic trajectories are given with their possible graphs in the Lorentzian warped product [Formula: see text].

A Different Interpretation on Magnetic Surfaces Generated by Special Magnetic Curve in Q2 ⊂ E 3 1

By thinking the magnetic flow connected by the Killing magnetic field, the magnetic field on the setting out particle orbit is investigated in 𝐐2 ⊂ 𝐸31 . Clearly, dealing with the Killing magnetic field of 𝛼 -magnetic curve, the rotational surface generated by 𝛼 -magnetic is expressed in 𝐐2 ⊂ 𝐸31, and the variant kinds of axes of rotation in lightlike cone 𝐐2 ⊂ E31 is given. Furthermore, the specific kinetic energy, specific angular momentum and conditions being geodesic on rotational surface generated by α-magnetic curve are expressed with the help of Clairaut's theorem.

Magnetic Trajectories on Lightlike Surfaces

Divergence free vector fields are called magnetic vector fields in three-dimensional semi-Riemannian manifolds. When a charged particle enters the magnetic vector field, it traces a new trajectory called magnetic curve by the influenced of magnetic field. In the present paper, we investigate the magnetic curves on the lightlike surfaces corresponding to the Killing magnetic fields in 3D semi-Riemannian manifolds. Moreover, we give some characterizations of these curves. As an application, we determine all magnetic curves on the lightlike cone. Finally, we give various examples to confirm the main results.

On Killing Magnetic Curves in Sl(2, ℝ) Geometry

Killing magnetic curves are trajectories of charged particles on a Riemannian manifold under action of a Killing magnetic field. In this paper we study Killing magnetic curves in SL(2, ℝ) geometry.

The Equations and Characteristics of the Magnetic Curves in the Sphere Space

We investigate some geometrical properties of magnetic curves in S3 under the action of the Killing magnetic field V=a∂x+ b∂y+ c∂z. The other main result is provided about the classification of the equations of the geodesics in S3. Moreover, some most relevant graphs of the main results were drawn in this paper.

Killing Magnetic Curves in Sol Space

We determine magnetic curves corresponding to the Killing magnetic fields in the 3-dimensional homogenous Riemannian space Sol3.

Some Properties of Special Magnetic Curves

In the theory of curves, a magnetic field generates a magnetic flow whose trajectories are curves called magnetic curves. This paper aims at studying some properties for these curves which corresponding to the Killing magnetic fields in the 3-dimensional Euclidean space. We investigate the trajectories of the magnetic fields called $T$-magnetic and $e$-magnetic curves, also we give some characterizations of these curves. In addition, we determine all magnetic curves for new spherical images of a spherical curve and finally, we defray some examples to confirm our main results.

Notes on magnetic curves in 3D semi-Riemannian manifolds

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 $\gamma $, 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.

A new approach for magnetic curves in 3D Riemannian manifolds

A magnetic field is defined by the property that its divergence is zero in a three-dimensional oriented Riemannian manifold. Each magnetic field generates a magnetic flow whose trajectories are curves called as magnetic curves. In this paper, we give a new variational approach to study the magnetic flow associated with the Killing magnetic field in a three-dimensional oriented Riemann manifold, (M3, g). And then, we investigate the trajectories of the magnetic fields called as N-magnetic and B-magnetic curves.

Magnetic fields in 2D and 3D sphere

In this note we study the Landau–Hall problem in the 2D and 3D unit sphere, that is, the motion of a charged particle in the presence of a static magnetic field. The magnetic flow is completely determined for any Riemannian surface of constant Gauss curvature, in particular, the unit 2D sphere. For the 3D case we consider Killing magnetic fields on the unit sphere, and we show that the magnetic flowlines are helices with the given Killing vector field as its axis.

Killing magnetic curves in a Minkowski 3-space

We classify all magnetic curves in the 3-dimensional Minkowski space corresponding to the Killing magnetic field V=a∂x+b∂y+c∂z, with a,b,c∈R. We find that they are helices in E13 and we draw the most relevant of them in the end.

Magnetic curves corresponding to Killing magnetic fields in ${\mathbb {E}}^3$E3

We explicitly determine all magnetic curves corresponding to the Killing magnetic fields on the 3-dimensional Euclidean space.