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

Authors: Jianguo Sun
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

We investigate some geometrical properties of magnetic curves inS3 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.


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## Body

## 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 onM 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 Ka¨hler 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, E3 and S2×R, 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 inS3, especially, the magnetic curves corresponding to V=a∂x+b∂y+c∂z.The outlines of this work are as follows: we introduce the magnetic curves inS3 in Section 2. In Section 3, for particular geodesics, we adopt the first approach to classify the equation of the geodesics in S3 (Theorem 1). Then, we deal with the magnetic vector field V=a∂x+b∂y+c∂z tangent to the R 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

LetM be a n(n≥2)-dimensional oriented Riemannian manifold. A magneticcurve represents the trajectory of a charged particle moving in the manifold under the action of a magnetic field. A magneticfield in (M,g) is a closed two-form F. The corresponding Lorentzforce of a magnetic field F on (M,g) is a skew symmetric (1,1)-tensor field ϕ defined by(1)gϕX,Y=FX,Y,for any X,Y∈χ(M).Themagnetictrajectories of F are curves γ on M which satisfy the Lorentzequation(2)∇γ′γ′=ϕγ′.The curve γ(s) is also known as the flowline of the dynamical system associated with the magnetic field F. When the magnetic curve γ(s) is arc length parametrized v0=1, it is called a normalmagneticcurve.Letγ(s) be a magnetic curve; if the curve satisfies the equation ∇γ′γ′=0, we call the curve geodesic. Therefore, from the point of view of dynamical systems, a geodesic corresponds to a trajectory of a particle when F=0.A field vector fieldV on M is Killing if and only if it satisfies the Killing equation:(3)g∇YV,Z+g∇ZV,Y=0 for any vector fields Y,Z on M, where ∇ is the Levi-Cicita connection on M.LetV be a Killing vector field on M and FV=ιVdvg the corresponding Killing magnetic field, where the inner product is denoted by ι. Then, the Lorentz force of the FV is given by [10](4)ϕX=V×X.Consequently, the Lorentz force equation may be written as(5)∇γ′γ′=V×γ′.In this paper, we will introduce some characteristics of magnetic curves in the three-dimensional sphere.LetS3 denote the three-dimensional unit sphere in R4 centered at the origin and defined by(6)S3=x=x1,x2,x3,x4∣x,x=x12+x22+x32+x42=1. Let ∇¯ and ∇0 denote the Levi-Civita connections in S3 and R4, respectively. If X and Y are vector fields tangent to S3, then ∇¯ and ∇0 are related by the Gauss formula as follows:(7)∇X0Y=∇¯XY-hX,YG,where G:S3→R4 denotes the position vector.In three-dimensional manifold(M3,g), the mixed product of the vector fields X,Y,Z∈χ(M3) is defined by g(X∧Y,Z)=Ω3(X,Y,Z). In the sphere space S3, we can define a cross product as follows. Consider a point q∈S3 and take two tangent vectors v1,v2∈Tq(S3); the cross product of the v1 and v2 is the unique tangent vector v1×v2 in Tq(S3) such that(8)v1×v2,ω=det⁡v1,v2,ω,q,∀ω∈TqS3,where 〈,〉 denotes the induced metric in S3 and the vectors are considered as column vectors in R4 [13].Consider a unit speed curveγ:I→S3, where I is a real open interval and assume that γ is not a geodesic curve. Let T(s)=γ′(s), and then, there is a unique vector field N(s) and a positive function κ(s) so that ∇¯T(s)T(s)=κ(s)N(s). Here, ∇¯T(s)T(s) denotes the covariant derivative of γ(s) in S3, and N(s) is called the principal normal vector fields and κ(s), the curvature of the given curve. Given a unit speed curve in S3, the binormal vector field of the curve γ(s) is defined by B(s)=T(s)×N(s), which is a unit vector field orthogonal to both T(s) and N(s). Since ∇¯T(s)B(s) is collinear with N(s), we can write ∇¯T(s)B(s)=-τ(s)N(s), and the differentiable function τ(s) is called the torsion of γ(s). There exists aFrenet frame{T(s),N(s),B(s)} satisfying [13](9)∇¯TsTs=κsNs∇¯TsNs=-κsTs+τsBs∇¯TsBs=-τsNs.By using the Gauss formula, we can rewrite the equations in the Euclidean connection ∇0 as follows:(10)∇Ts0Ts=κsNs-γs∇Ts0Ns=-κsTs+τsBs∇Ts0Bs=-τsNs.


## 3. Some Characteristics of the Magnetic Curves in 3D Sphere

By the Gauss formula∇X0Y=∇¯XY-h(X,Y)G, where ∇¯ and ∇0 denote the Levi-Civita connections in S3 and R4, respectively. Suppose the curve γ(s) is the geodesics in three-dimensional sphere, and γ(s)=(x(s),y(s),z(s),t(s)), the tangent vector T(s)=γ′(s)={x˙(s),y˙(s),z˙(s),t˙(s)}. We construct a mapping h:S3→R4, which is the second fundamental form of S3 in R4, and the position vector G={x,y,z,0}.Letγ:I⊂R→S3⊂R4 be a smooth curve in S3. The metric in S3 is given by the restriction of the usual scalar product 〈,〉 in R4 and the Levi-Civita connections ∇0 and ∇¯ in S3 and R4 respectively. We can consider the natural projection (11)π:R3×R→R3⊂R4:πv1,v2,v3,v4=v1,v2,v3,0.The coordinates for the arc length parameter curve γ(s) in S3 are parameterized by γ(s)=(x(s),y(s),z(s),t(s)) such that x2(s)+y2(s)+z2(s)+t2(s)=1 and x˙2(s)+y˙2(s)+z˙2(s)+t˙2(s)=1, satisfying the conditions (12)x0=x0,y0=y0,z0=z0,t0=t0,x˙0=u0,y˙0=v0,z˙0=ω0,t˙0=ξ0, with x02+y02+z02+t02=u02+v02+ω02+ξ02=1 and x0u0+y0v0+z0ω0+t0ξ0=0.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 theS3.Theorem 1.
The expression form of the geodesics in the manifoldS3 is one of the following three cases:
Case 1.γ(s)=γ(s0)=(x0,y0,z0,t0);
Case 2.γ(s)=(x0cos⁡s+u02-x02sin⁡s,y0cos⁡s+v02-y02sin⁡s,z0cos⁡s+ω02-z02sin⁡s,t0).
Case 3(13)γs=ma2-x02sin⁡1-ξ02s+x0cos⁡1-ξ02s,ma2-y02sin⁡1-ξ02s+y0cos⁡1-ξ02s,ma2-z02sin⁡1-ξ02s+z0cos⁡1-ξ02s,ξ0s+t0;where m=(1-ξ02)y02+v02/1-ξ02.Proof.
By Gauss formula (7), where X,Y∈χ(S3) and h:S3→R4 are the second fundamental form of S3 in R4, and π({x,y,z,t})={x,y,z,0} is the projection in S3, we obtain(14)γ¨=∇¯γ˙γ˙-πγ˙,πγ˙πγ,where π is the natural projection. Substituting expression (7) in (14), we can get(15)γ¨=∇¯γ˙γ˙-1-t˙2x,y,z,0,and if the curve γ(s) is the geodesics in S3, we know ∇¯γ˙γ˙=0 [11] and(16)γ¨=-1-t˙2x,y,z,0.Equivalently,(17)x¨=-1-t˙2x,y¨=-1-t˙2y,z¨=-1-t˙2z,t¨=0.At the initial conditions, we solve the fourth equation of this system, and we can obtain t(s)=ξ0s+t0, where ξ0∈[-1,1].
Let us consider the following cases:(18)x¨=-1-ξ02x,y¨=-1-ξ02y,z¨=-1-ξ02z,t¨=0.
Case
1. When ξ0=±1, we can find x¨=y¨=z¨=0, and (19)γs=x,y,z,t=u0s+x0,v0s+y0,ω0s+z0,ξ0s+t0.And (u0s+x0)2+(v0s+y0)2+(ω0s+z0)2+(ξ0s+t0)2=1. Also, there are two equal solutions s1=s2=0, which stand for one fixed point γ0 in S3.
Case
2. When ξ0=0, we can find(20)x¨=-x,y¨=-y,z¨=-z,and t=t0, and solving the equations above, we obtain(21)x=u0sin⁡±s+arcsin⁡x0u0,y=v0sin⁡±s+arcsin⁡y0v0,z=ω0sin⁡±s+arcsin⁡z0ω0,and if x0/u0=y0/v0=z0/ω0, we can find the γ(s) is the great circle in S3.
Ifx0/u0≠y0/v0≠z0/ω0, γ(s) is included in a plane passing through the origin and parameterized as (22)γs=x0cos⁡s+u02-x02sin⁡s,y0cos⁡s+v02-y02sin⁡s,z0cos⁡s+ω02-z02sin⁡s,t0.
Case
3. When ξ0∈(-1,0)∪(0,1), the general case, we can find the t(s)=ξ0s+t0, and γ(s) is a circle passing through the original point and parameterized by(23)x=ma2-x02sin⁡1-ξ02s+x0cos⁡1-ξ02s,y=ma2-y02sin⁡1-ξ02s+y0cos⁡1-ξ02s,z=ma2-z02sin⁡1-ξ02s+z0cos⁡1-ξ02s,t=ξ0s+t0,where m=(1-ξ02)y02+v02/1-ξ02, and (24)γs=ma2-x02sin⁡1-ξ02s+x0cos⁡1-ξ02s,ma2-y02sin⁡1-ξ02s+y0cos⁡1-ξ02s,ma2-z02sin⁡1-ξ02s+z0cos⁡1-ξ02s,ξ0s+t0.We conclude the proof noticing that indeed the geodesics in S3 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 inS3 is a closed 2-form F, and the Lorentz force corresponding to F is a (1,1)-type tensor field ϕ as (1).A toy example for a Killing vector field is on the upper half planeR2,y≥0 equipped matric g=y-2(dx2+dy2). The pair (M,g) is called the hyperbolic plane and has Killing vector field ∂x. This should be clear since the covariant derivative ∇∂zg transports the metric along an integral curve genenrated by the vector field. In this paper, we mention the basis for vector fields V=a∂x+b∂y+c∂z, for any a,b,c∈R, and the Killing magnetic field determined by V is FV=adx∧dy+bdy∧dz+cdz∧dx.Theorem 2.
Letγ(s) be a magnetic trajectory corresponding to the Killing vector field V=a∂x+b∂y+c∂z in S3, and then γ(s) is (25)γs=γ0+γ′0,VsV+1+cos⁡sV×γ′0+sin⁡sW,where W=γ′0-γ′0,VV.Proof.
In this proof, we can choose a special parameters, satisfying 〈V,V〉=1. Then, we know V and γ′0 are linearly independent, and we can construct a new vector W=γ′0-γ′0,VV. And(26)V×γ′0,V×γ′0=1,V,V×γ′0=0,and meanwhile,(27)V,W=V,γ′0-γ′0,VV=0,V×γ′0,W=V×γ′0,γ′0-γ′0,VV=0,W,W=γ′0,γ′0-2V,γ′02-V,γ′02+V,γ′02γ′0,γ′0.
Case
1. When γ′0,γ′0=〈V,γ′(0)〉2, we know (28)W,W=V×γ′0,V×γ′0=0.Hence, W=V×γ′(0)=0, which is lineally dependent between V and γ′(0).
Case
2. When γ′0,γ′0≠〈V,γ′(0)〉2, 〈V,γ′(0)〉2≠0, we know V,V×γ′(0),W are linearly independent, supposing V,V×γ′(0),W} is the frame in S3.Thus, in the following, we only consider the case V and γ′(0) are lineally independent. Let the curve (29)γs=γ0+AsV+BsV×γ′0+CsW,where A(s),B(s),C(s) are functions satisfying the original conditions A(0)=B(0)=C(0)=0, A′(0)=〈γ′(0),V〉,B′(0)=0,C′(0)=1.
Then, the Lorentz equation is equivalent toγ′′s=V×γ′s which can be written as follows: (30)γ′′s=A′′sV+B′′sV×γ′0+C′′sW=V×A′sV+B′sV×γ′0+C′sW=B′sV×V×γ′0+C′sW=C′sV×γ′0+B′s-γ′0+V,γ′0V=C′sV×γ′0-B′sW.And we know(31)A′′s=0,B′′s=C′s,C′′s=-B′s,and for the original conditions, we know A(s)=〈V,γ′(0)〉s. By solving(32)B′′s=C′s,C′′s=-B′s,we know(33)Bs=cos⁡s+1,Cs=sin⁡s.Hence, we can get the equation of the curve (34)γs=γ0+γ′0,VsV+1+cos⁡sV×γ′0+sin⁡sW.
Remark. When 〈γ′(0),γ′(0)〉≠〈V,γ′(0)〉2, 〈V,γ′(0)〉2=0, the equation of the curve is(35)γs=γ0+1+cos⁡sV×γ′0+sin⁡sγ′0,which is a special case of above equation.


## 4. Some Examples

We will give some projected graphs to appear in the proof of the Theorem2. For the dimension, we only give the projection of the curve and vectors to three dimensions.Example 1.
The expression of curve is(36)γs=33cos⁡s+sin⁡s,33cos⁡s-sin⁡s,33cos⁡2s,33sin⁡2s.
Case
1. When the magnetic fields are V={0,1,0,0} and V=3/3{1,0,1,1} satisfying (37)γ′0,γ′0≠V,γ′02≠0,the figures are Figures 1 and 2, respectively.
Case
2. When the magnetic field is V=1/2{1,1,1,1} satisfying 〈γ′(0),γ′(0)〉=〈V,γ′(0)〉2, the figure is Figure 3.Figure 1
Red vector isγ′(0)={1,-1,0,-2}, and green vectors are magnetic fields V={0,1,0,0}.Figure 2
Red vector isγ′(0)={1,-1,0,-2}, and green vectors are magnetic fields V=3/3{1,0,1,1}.Figure 3
Red vector isγ′(0)={1,-1,0,-2}, and green vectors are magnetic fields V=1/2{1,1,1,1}.Example 2.
The expression of curve is(38)γs=22sin⁡2s,22cos⁡2s,22sin⁡s,22cos⁡s.
Case
1. When the magnetic field is V=1/2{1,1,1,1} satisfying 〈γ′(0),γ′(0)〉≠〈V,γ′(0)〉2≠0, the figure is Figure 4.
Case
2. When the magnetic field is V=2/2{0,1,0,1} satisfying 〈γ′(0),γ′(0)〉≠〈V,γ′(0)〉2=0, the figure is Figure 5.
Case
3. When the magnetic field is V=42/212/2,1,22,1 satisfying 〈γ′(0),γ′(0)〉=〈V,γ′(0)〉2, the figure is Figure 6.Figure 4
Red vector isγ′(0)={2,0,1,0}, and green vectors are magnetic fields V=1/2{1,1,1,1}.Figure 5
Red vector isγ′(0)={2,0,1,0}, and green vectors are magnetic fields V=2/2{0,1,0,1}.Figure 6
Red vector isγ′(0)={2,0,1,0}, and green vectors are magnetic fields V=42/21{2/2,1,22,1}.


## 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 inR3 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.
Letγ:I→S3 be a curve in S3 and V a magnetic field. We call the curve γ(s)N-magnetic curves if the normal vector field of the curve satisfies the Lorentz force equation; that is,(39)∇¯TN=ϕN=V×N.In three-dimensional Euclidean space(R3,g) [8], a unit speed curve γ is a magnetic trajectory of a magnetic field V if and only if V can be written along γ as(40)V=ωsTs+κsBs,where the function ω(s) associated with each magnetic curve will be called its quasislope measured with respect to the magnetic field V, and T(s) is the tangent vector, and B(s) 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.
Letγ(s) be a unit speed N-magnetic curve in S3 with the Lorentz force in the Frenet frame being written as(41)ϕT=κsNsϕN=-κsTs+τsBsϕB=ω1sTs-ω2sNsand the magnetic field is(42)V=τsT+ω1sN-κsB,where ω1(s)=〈V×B,T〉,ω2(s)=〈V×B,N〉.Proof.
Sinceγ(s) is a unit speed N-magnetic curve on three-dimensional sphere and there exist three functions λ1(s),η1(s),ζ1(s), we have ϕ(T)=λ1(s)T+η1(s)N+ζ1(s)B and λ1=〈ϕ(T),T〉=0.
Meanwhile,(43)η1s=ϕT,N=--κsTs,Ts=κs,and(44)ζ1s=ϕT,B=V×T,B=0.Hence, ϕ(T)=κ(s)N(s).
As the same methods, we supposeϕ(N)=λ2(s)T+η2(s)N+ζ2(s)B and can obtain η2=0,(45)λ2s=ϕN,T=V×N,T=-κs,ζ2s=τs. Hence, ϕ(N)=-κ(s)T(s)+τ(s)B(s).
We supposeϕ(B)=λ3(s)T+η3(s)N+ζ3(s)B and can obtain ζ3=0,(46)λ3s=ϕB,T=V×B,T=ω1s,ζ3s=ϕB,N=V×B,N=ω2s. And we can obtain (47)ϕB=ω1Ts+ω2Ns.Let γ(s) be a unit speed N-magnetic curve, and we suppose the magnetic field V=λT+ηN+ζB, by the definition of the N-magnetic curve, ϕ(T)=κ(s)N(s)=V×T. We know ζ=-κ(s), the same as λ=τ(s),η=ω1. Hence, (48)V=τsT+ω1sN-κsB.Example 3.
Let curveγ(s) be a N-magnetic curve as (49)γs=cos⁡scos⁡2s,cos⁡ssin⁡2s,sin⁡scos⁡2s,sin⁡ssin⁡3s,and the Frenet frames {T(s),B(s),N(s),κ(s),τ(s)} as follows: (50)Ts=-cos⁡2ssin⁡s-2cos⁡ssin⁡2s,-sin⁡ssin⁡2s+2cos⁡scos⁡2s,-cos⁡scos⁡3s-3sin⁡scos⁡3s,cos⁡ssin⁡3s+3cos⁡3ssin⁡s,Ns=5cos⁡ssin⁡2s+4cos⁡2ssin⁡s,-5cos⁡scos⁡2s,-10sin⁡scos⁡3s,6cos⁡scos⁡3s-10sin⁡ssin⁡3s,Bs=17cos⁡scos⁡3s+cos⁡2ssin⁡s,8cos⁡3ssin⁡s,-15sin⁡2scos⁡s+3cos⁡ssin⁡2s,15cos⁡2scos⁡s,and κ(s)=sin⁡s+cos⁡s,τ(s)=cos⁡s(sin⁡2scos⁡s-cos⁡3s). When ω1(s)=1, at the original point, we can draw the projection figure of the curve γ(s) and the vector V in Figure 7. When ω1(s)=0, at the original and s=π/3 points, we can draw the projection figure of the curve γ(s) and the vector V in Figure 8.Figure 7
Red vector isT(0), blue vector is N(0), green vector is B(0), and black vector is V.Figure 8
Red vector isT, blue vector is N, green vector is B, and black vector is V.



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