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 E. 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.


Introduction
The magnetic curves on three dimensional Riemannian manifold (M 3 , g) are trajectories of charged particles moving on M 3 under the action of a magnetic field F . Each trajectory γ may be found by solving the Lorentz equation ∇ γ γ = φ(γ ), where φ is the Lorentz force corresponding to F and ∇ is the Levi Civita connection of g. In particular, the trajectories of (charged) particles moving without the action of a magnetic field are geodesics, which satisfy ∇ γ γ = 0 (see for more details [1,2]). In a three-dimensional space, when a charged particle moves along a regular curve, the tangent, normal and binormal vectors describe the kinematic and geometric properties of this particle. These vectors and the time dimension affect the trajectory of the charged particle during the motion in a magnetic field [3,4]. Moreover, the study of magnetic curves was extended to other ambient spaces, such as complex space forms [5,6], Sasakian 3-manifold [7,8]. Recently, results of classification for the Killing magnetic trajectories on two special 3-dimensional manifolds, namely E 3 and S 2 × R, were obtained in [9] and [10], respectively. Barros and Romero proved that if (M 3 , g) has constant curvature, then the magnetic curves corresponding to a Killing magnetic field are center lines of Kirchhoff elastic rods [11]. The curves and their frames play an important role in differential geometry and in many branches of science such as mechanics and physics. So, we are interested here in studying some of these curves called magnetic curves, which have many applications in modern physics. In this work, we investigate the trajectories of the magnetic fields called as T T -magnetic, N T -magnetic and B Tmagnetic curves and obtain some solutions of the Lorentz force equation. We are looking forward to see that our results will be helpful to researchers who are specialized in mathematical modeling, mechanics and modern physics.

Basic concepts
In this section, we list some notions, formulae and conclusions for curves in three-dimensional Euclidean space which can be found in the text books on differential geometry (see for instance [1,12,13]). Let E 3 denotes the real vector space with its usual vector structure. We denote by (x 1 , x 2 , x 3 ) the coordinates of a vector with respect to the canonical basis of E 3 . For any two vectors x = (x 1 , x 2 , x 3 ) and y = (y 1 , y 2 , y 3 ), the metric g on E 3 is defined by The norm of x is given by and the vector product is denoted by The sphere of radius r > 0 with center at the origin is given by Let γ = γ(s) : I ⊂ R → E 3 be an arbitrary curve in E 3 , s be the arclength parameter of γ. It is well known that each unit speed curve with at least four continuous derivatives, one can associate three mutually orthogonal unit vector fields T , N and B called the tangent, the principal normal and the binormal vector fields, respectively [14]. The Frenet equations for γ are given by [15] where κ(s) and τ (s) are called the curvatures of γ.
For spherical images of a regular curve in Euclidean 3-space, we present the following definition: taking into consideration that , is the geodesic curvature of the principal image of the principal normal indicatrix of the curve γ, s T is a natural representation parameter of the tangent indicatrix of γ and also it is the total curvature of the curve γ and κ T , τ T are the curvature and torsion of γ T . Therefore, we can see that τ T κ T = σ. Let us introduce the following notions [6,11,18].

Definition 2.2.
A magnetic field on a three-dimensional oriented Riemannian manifold (M 3 , g) is defined as a closed 2-form F on M 3 , related to a skew-symmetric (1, 1)−tensor field φ called the Lorentz force of F , and we have The magnetic trajectories of F are curves γ on M 3 which satisfy the Lorentz equation: Let V be a Killing vector field on M 3 , then the Lorentz force can be written as in this case, the Lorentz force equation is given by Note that, for a trivial magnetic field; F = 0, the Lorentz equation becomes ∇ γ γ = 0 and then the solutions are geodesics.
Manifold (M 3 , g) and V be a vector field along the curve γ. Then, one can take a variation of γ in the direction of V , say, a map Π : In this setting, we have the following functions: 1. the speed function v (s, t) = ∂Π ∂s (s, t) ; t is the time dimension, 2. the curvature κ(s, t) and the torsion τ (s, t) are functions of γ(s). The variations of these functions at t = 0 are given as follows: where R is the curvature tensor of M 3 .

Magnetic curves of the tangent indicatrix
Definition 3.1. [11,18] Let γ T : I → S 2 ⊂ E 3 be a tangent indicatrix of a regular curve γ in three-dimensional Euclidean space E 3 , and F be a magnetic field on M 3 , then the curve γ T is In the light of this definition, we can investigate the following.
Then, we have the Frenet formula: and the Lorentz force in the Frenet frame can be written as where Ψ 1 is a certain function defined by Proof. From Definition 3.1, one can write Use the following equalities: to get Hence, Similarly, we can easily obtain

5)
Proof. Let γ T be a T T -magnetic trajectory of a magnetic field F . Then, by using Proposition 3.1 and Eq. (2.3), we can easily have Conversely, we assume that Eq. (3.5) holds, then we get V × T T = φ(T T ) and so the curve γ T is a T T -magnetic curve.
Theorem 3.1. Let γ T be a T T −magnetic curve and V be a Killing vector field on a space form (M 3 (K), g). If γ T is one of the T T −magnetic trajectories of (M 3 (K), g, V ), then its curvatures satisfying the following relations: , where K is the curvature of Riemannian space M 3 and A 1 , A 2 are constants.
Then, the Lorentz force in the Frenet frame can be written as
Proof. From Definition 3.1, one can write Using the following equalities: we get and therefore, Similarly, we can easily obtain that Hence, from Eqs. (3.11), (3.12) and (3.13), the proof is completed.
Corollary 3.1. Let γ T be a curve in E 3 . Then, the curve γ T is a N T -magnetic trajectory of a magnetic field F if and only if the vector field V along γ is written as (3.14) Proof. The proof is similar to that we have considered in Proposition 3.2.
Theorem 3.2. Let γ T be a N T −magnetic curve and V be a Killing vector field on a space form (M 3 (K), g). If the curve γ T is one of the N T −magnetic trajectories of (M 3 (K), g, V ), then its curvatures satisfying the following relations: where A 3 is a constant.
Proof. Differentiating Eq. (3.14) with respect to s, we get Since V is a Killing vector, then from Proposition 3.2 (V (v ) = 0), we have Also, differentiation of Eq. (3.15) together with Eq. (2.2), gives If M 3 has a constant curvature K, then and therefore Using the condition V (σ √ 1 + f 2 ) = 0 in Eqs. (3.15) and (3.16), we obtain Integrating Eq. (3.18) leads to Thus, this completes the proof.
Corollary 3.2. Let γ T be a N T −magnetic curve in Euclidean 3-space with Ψ 2 is zero, then γ T is a circular helix. Moreover, the axis of the circular helix is the vector field.
Proof. It is clear from Theorem 3.2.
Then, the Lorentz force in the Frenet frame can be written as (3.20) where Ψ 3 is given by Ψ 3 = g(φ(T T ), N T ).
Proof. As we mentioned the above, we can write Using the following conditions: we can obtain From this, we get Therefore, the proof is completed. σ 1 + f 2 = const., Proof. Since V is a vector field, differentiating Eq. (3.24) with respect to s, we get Since V is a Killing vector, then we have Again, differentiating Eq. (3.25) and using Eq. (2.2), we get which leads to If K = const., then we have and therefore

Applications
In what follows, we give two computational examples to illustrate our main results. The Frenet vectors of γ T are given as follows B T (s T ) = (0, 0, 1) .
Moreover, the natural representation and the curvature of γ T are respectively, In addition, the torsion and the certain function of γ T are respectively, τ T = 0 and Ψ 2 = 0, it means that γ T is N T -magnetic as well as B T -magnetic curve.
(a) (b) Figure 7. The circular helix γ and its spherical image γ T .

Conclusion
The value of this paper is due to the important and prominent role of the theory of curves in differential geometry as well as magnetic fields that generate magnetic flow whose trajectories give so-called magnetic curves. In this sense, the idea of this work is devoted to examine some conditions to construct special magnetic curves of spherical images for a regular curve γ in Euclidean 3-space.
Some characterizations of magnetic curves of the tangent indicatrix of γ are obtained. An application to confirm our main results is given and plotted.