Abstract
We investigate some geometrical properties of magnetic curves in under the action of the Killing magnetic field . The other main result is provided about the classification of the equations of the geodesics in . Moreover, some most relevant graphs of the main results were drawn in this paper.
1. Introduction
The study of magnetic fields and their corresponding magnetic curves on different manifolds is one of the important research topics between differential geometry and physics. The magnetic curves on the Riemannian manifolds are trajectories of charged particles moving on under the magnetic field. Meanwhile, the different magnetic fields were extended to different ambient spaces [1–12]. Corresponding to parallel Lorentz forces, the magnetic trajectories are obtained on some 2-dimensional space [1, 2]. In [3, 4], the authors had researched the magnetic fields in complex space, which are called Khler form, and in Sasakian 3-manifold. The classification of the magnetic curves in 3-dimensional Minkowski space with Killing magnetic field and in three-dimensional almost paracontact manifolds was given in [5, 6]. The authors obtained the magnetic trajectories as solutions of a variational problem that neither involves any local potential nor constraints the topology given a magnetic field in 3D [7]. And the classification for the Killing magnetic trajectories in two special 3-dimensional manifolds, namely, and , was studied in [8, 11], respectively.
However, if we want to extend this concept to other ambient spaces, it is necessary to distinguish between the manifolds and the tangent vector spaces. In this regard, the sphere spaces play an important role among these manifolds, for their normal vectors direct to the original, and there is a smooth deformation through constant mean curvature surfaces with the same topology, which can be expressed in terms of changing radii. The authors extend the rectifying theory and the relative results in the 3-dimensional sphere [13].
Looking over all these results obtained in classification of magnetic trajectories corresponding to magnetic fields in different ambient spaces, until recently, and to the best of our knowledge, there has been little information available about the magnetic curves in the 3-dimensional sphere. In the present paper, we give some geometrical properties of magnetic curves in , especially, the magnetic curves corresponding to .
The outlines of this work are as follows: we introduce the magnetic curves in in Section 2. In Section 3, for particular geodesics, we adopt the first approach to classify the equation of the geodesics in (Theorem 1). Then, we deal with the magnetic vector field tangent to the factor, which generates the magnetic trajectories described in Theorem 2. In Section 4, as an application, we give some examples and graphs to certify our conclusions. And then, we investigate the trajectories of the magnetic fields called N-magnetic curves. Moreover, we obtain some solutions of the Lorentz force equation and give an example of this curve by drawing their pictures using Mathematica.
2. Preliminaries
Let be a n()-dimensional oriented Riemannian manifold. A represents the trajectory of a charged particle moving in the manifold under the action of a magnetic field. A in is a closed two-form . The corresponding of a magnetic field on is a skew symmetric tensor field defined byfor any
The of are curves on which satisfy the The curve is also known as the of the dynamical system associated with the magnetic field . When the magnetic curve is arc length parametrized , it is called a .
Let be a magnetic curve; if the curve satisfies the equation , we call the curve . Therefore, from the point of view of dynamical systems, a geodesic corresponds to a trajectory of a particle when .
A field vector field on is Killing if and only if it satisfies the Killing equation: for any vector fields on , where is the Levi-Cicita connection on .
Let be a Killing vector field on and the corresponding Killing magnetic field, where the inner product is denoted by . Then, the Lorentz force of the is given by [10]Consequently, the Lorentz force equation may be written asIn this paper, we will introduce some characteristics of magnetic curves in the three-dimensional sphere.
Let denote the three-dimensional unit sphere in centered at the origin and defined by Let and denote the Levi-Civita connections in and , respectively. If and are vector fields tangent to , then and are related by the Gauss formula as follows:where denotes the position vector.
In three-dimensional manifold , the mixed product of the vector fields is defined by In the sphere space , we can define a cross product as follows. Consider a point and take two tangent vectors ; the cross product of the and is the unique tangent vector in such thatwhere denotes the induced metric in and the vectors are considered as column vectors in [13].
Consider a unit speed curve , where is a real open interval and assume that is not a geodesic curve. Let , and then, there is a unique vector field and a positive function so that . Here, denotes the covariant derivative of in , and is called the principal normal vector fields and , the curvature of the given curve. Given a unit speed curve in , the binormal vector field of the curve is defined by , which is a unit vector field orthogonal to both and . Since is collinear with , we can write , and the differentiable function is called the torsion of . There exists a Frenet frame satisfying [13]By using the Gauss formula, we can rewrite the equations in the Euclidean connection as follows:
3. Some Characteristics of the Magnetic Curves in 3D Sphere
By the Gauss formula , where and denote the Levi-Civita connections in and , respectively. Suppose the curve is the geodesics in three-dimensional sphere, and , the tangent vector . We construct a mapping , which is the second fundamental form of in , and the position vector
Let be a smooth curve in . The metric in is given by the restriction of the usual scalar product in and the Levi-Civita connections and in and respectively. We can consider the natural projection The coordinates for the arc length parameter curve in are parameterized by such that and , satisfying the conditions with and
When the Lorentz force vanished, the geodesics may be considered as particular magnetic trajectories. Hence, at first, we consider the classification of the geodesics in the .
Theorem 1. The expression form of the geodesics in the manifold is one of the following three cases:
Case 1. ;
Case 2.
Case 3 where .
Proof. By Gauss formula (7), where and are the second fundamental form of in , and is the projection in , we obtainwhere is the natural projection. Substituting expression (7) in (14), we can getand if the curve is the geodesics in , we know [11] andEquivalently,At the initial conditions, we solve the fourth equation of this system, and we can obtain , where .
Let us consider the following cases:Case 1. When , we can find , and And . Also, there are two equal solutions , which stand for one fixed point in .
Case 2. When , we can findand , and solving the equations above, we obtainand if , we can find the is the great circle in .
If , is included in a plane passing through the origin and parameterized as Case 3. When , the general case, we can find the , and is a circle passing through the original point and parameterized bywhere , and We conclude the proof noticing that indeed the geodesics in are parameterized by (20)-(23).
Motivated by the fact that the equations of the geodesics are particular Lorentz equations, when the Lorentz force vanishes identically, the geodesics may be regarded as particular magnetic trajectories, the magnetic field in is a closed 2-form , and the Lorentz force corresponding to is a (1,1)-type tensor field as (1).
A toy example for a Killing vector field is on the upper half plane equipped matric . The pair is called the hyperbolic plane and has Killing vector field . This should be clear since the covariant derivative transports the metric along an integral curve genenrated by the vector field. In this paper, we mention the basis for vector fields , for any , and the Killing magnetic field determined by is .
Theorem 2. Let be a magnetic trajectory corresponding to the Killing vector field in , and then is where .
Proof. In this proof, we can choose a special parameter , satisfying . Then, we know and are linearly independent, and we can construct a new vector Andand meanwhile,Case 1. When , we know Hence, , which is lineally dependent between and .
Case 2. When , , we know are linearly independent, supposing is the frame in Thus, in the following, we only consider the case and are lineally independent. Let the curve where are functions satisfying the original conditions , .
Then, the Lorentz equation is equivalent to which can be written as follows: And we knowand for the original conditions, we know By solvingwe knowHence, we can get the equation of the curve Remark. When , , the equation of the curve iswhich is a special case of above equation.
4. Some Examples
We will give some projected graphs to appear in the proof of the Theorem 2. For the dimension, we only give the projection of the curve and vectors to three dimensions.
Example 1. The expression of curve is Case 1. When the magnetic fields are and satisfying the figures are Figures 1 and 2, respectively.
Case 2. When the magnetic field is satisfying , the figure is Figure 3.
Example 2. The expression of curve is Case 1. When the magnetic field is satisfying , the figure is Figure 4.
Case 2. When the magnetic field is satisfying , the figure is Figure 5.
Case 3. When the magnetic field is satisfying , the figure is Figure 6.
5. A New Kind of Magnetic Curves in Three-Dimensional Sphere
As we all know, the Lorentz force is always perpendicular to both the velocity of the particle and the magnetic field that created it. When a charged particle moves in a static magnetic field, it traces a helical path in which the helix axis is parallel to the magnetic field. Also, if the charged particle moves parallel to magnetic field, the Lorentz force acts to be zero. Two vectors are perpendicular to the Lorentz force at the largest value. But we know that when a charged particle moves along a curve in a magnetic field V besides the velocity vector, the normal vector is also expressed according to the magnetic field V. Hence, the trajectories of the charged particle are changed. For example, when a charged particle moves in a static magnetic field in and the normal vector is exposed to this field, it traces a slant helical path in which the slant-helix axis is parallel to the magnetic field V[14]. We give an example of the charged particle whose trajectories are N-magnetic curve in a magnetic field V.
Definition 3. Let be a curve in and a magnetic field. We call the curve -magnetic curves if the normal vector field of the curve satisfies the Lorentz force equation; that is,
In three-dimensional Euclidean space [8], a unit speed curve is a magnetic trajectory of a magnetic field if and only if can be written along aswhere the function associated with each magnetic curve will be called its quasislope measured with respect to the magnetic field , and is the tangent vector, and is the binormal vector of the curve [10, 14]. In this section, we give a new kind of magnetic curve called N-magnetic curve in three-dimensional sphere. Moreover, we obtain some characterizations and an example of this kind of curve.
Theorem 4. Let be a unit speed -magnetic curve in with the Lorentz force in the Frenet frame being written asand the magnetic field iswhere
Proof. Since is a unit speed -magnetic curve on three-dimensional sphere and there exist three functions , we have and .
Meanwhile,andHence,
As the same methods, we suppose and can obtain , Hence,
We suppose and can obtain , And we can obtain Let be a unit speed -magnetic curve, and we suppose the magnetic field , by the definition of the -magnetic curve, . We know , the same as . Hence,
Example 3. Let curve be a -magnetic curve as and the Frenet frames as follows: and When , at the original point, we can draw the projection figure of the curve and the vector in Figure 7. When , at the original and points, we can draw the projection figure of the curve and the vector in Figure 8.
Data Availability
The data supporting the conclusions of this manuscript are some open-access articles that have been properly cited, and the readers can easily obtain these articles to verify the conclusions, replicate the analysis, and conduct secondary analysis. Therefore, a publicly available data repository was not created.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The author is very grateful to Professor Lyle Noakes for his effective suggestions on this paper. The first author was supported by NSF of China No. 11601520 and the Fundamental Research Funds for the Central Universities No. 17CX02049.