Int. J. Anal. Appl. (2022), 20:18 On Magnetic Curves According to Killing Vector Fields in Euclidean 3-Space M. Khalifa Saad1,∗, H. S. Abdel-Aziz2 and Haytham A. Ali2 1Department of Mathematics, Faculty of Science, Islamic University of Madinah, 170 Al-Madinah, KSA 2Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, Egypt ∗Corresponding author: mohammed.khalifa@iu.edu.sa Abstract. 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. 1. Introduction The magnetic curves on three dimensional Riemannian manifold (M3,g) are trajectories of charged particles moving on M3 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 Received: Jan. 22, 2022. 2010 Mathematics Subject Classification. 53A04, 53A17, 53B20. Key words and phrases. magnetic curves; killing vector field; Lorentz force; spherical indicatrices. https://doi.org/10.28924/2291-8639-20-2022-18 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-18 2 Int. J. Anal. Appl. (2022), 20:18 magnetic trajectories on two special 3-dimensional manifolds, namely E3 and S2 ×R, were obtained in [9] and [10], respectively. Barros and Romero proved that if (M3,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 TT-magnetic, NT-magnetic and BT- magnetic 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. 2. 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 E3 denotes the real vector space with its usual vector structure. We denote by (x1,x2,x3) the coordinates of a vector with respect to the canonical basis of E3. For any two vectors x = (x1,x2,x3) and y = (y1,y2,y3), the metric g on E3 is defined by g(x,y) = x1y1 + x2y2 + x3y3. The norm of x is given by ‖x‖ = √ g(x,x), and the vector product is denoted by x×y = ((x2y3 −x3y2), (x3y1−x1y3), (x1y2 −x2y1)). The sphere of radius r > 0 with center at the origin is given by S2 = {x = (x1,x2,x3) ∈E3 : g(x,x) = r2}. Let γ = γ(s) : I ⊂ R →E3 be an arbitrary curve in E3, 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]. Let {T (s),N(s),B(s)} be the moving frame along γ, where these vectors are mutually orthogonal vectors satisfying 〈T (s),T (s)〉 = 〈N(s),N(s)〉 = 〈B(s),B(s)〉 = 1. Int. J. Anal. Appl. (2022), 20:18 3 The Frenet equations for γ are given by [15]  T ′(s) N′(s) B′(s)   =   0 κ(s) 0 −κ(s) 0 τ(s) 0 −τ(s) 0     T (s) N(s) B(s)   , (2.1) 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: Definition 2.1. [16,17] Let γ be a curve in Euclidean 3-space with Frenet vectors T , N and B. The unit tangent vectors along the curve γ(s) generate a curve γT = T on the sphere of radius 1 about the origin. The curve γT is called the spherical indicatrix of T or more commonly, γT is called tangent indicatrix of the curve γ. If γ = γ(s) is a natural representations of the curve γ, then γT = T (s) will be a representation of γT . Similarly, one can consider the principal normal indicatrix γN = N(s) and binormal indicatrix γB = B(s). Let γ be a curve in E3 and consider γT = T (s) as the tangent indicatrix of γ with {TT ,NT ,BT} as its Frenet vectors. Then we have the Frenet formula as follows:  T ′T (sT ) N′T (sT ) B′T (sT )   =   0 κT 0 −κT 0 τT 0 −τT 0     TT (sT ) NT (sT ) BT (sT )   , where   TT = N, NT = −1√ 1+f 2 T + f√ 1+f 2 B, BT = f√ 1+f 2 T + 1√ 1+f 2 B, and sT = ∫ κ(s)ds, κT = √ 1 + f 2, τT = σ √ 1 + f 2, f = τ(s) κ(s) , (2.2) taking into consideration that σ = f ′(s) κ(s) (1 + f 2(s)) 3/2 , is the geodesic curvature of the principal image of the principal normal indicatrix of the curve γ, sT is a natural representation parameter of the tangent indicatrix of γ and also it is the total curvature of the curve γ and κT , τT are the curvature and torsion of γT . Therefore, we can see that τT κT = σ. Let us introduce the following notions [6,11,18]. Definition 2.2. A magnetic field on a three-dimensional oriented Riemannian manifold (M3,g) is defined as a closed 2-form F on M3, related to a skew-symmetric (1, 1)−tensor field φ called the Lorentz force of F, and we have g(φ(X),Y ) = F (X,Y ), ∀ X,Y ∈ χ(M). 4 Int. J. Anal. Appl. (2022), 20:18 The magnetic trajectories of F are curves γ on M3 which satisfy the Lorentz equation: ∇γ′γ′ = φ(γ′). Let V be a Killing vector field on M3, then the Lorentz force can be written as φ(X) = V ×X, (2.3) in this case, the Lorentz force equation is given by ∇γ′γ′ = V ×γ′. Note that, for a trivial magnetic field; F = 0, the Lorentz equation becomes ∇γ′γ′ = 0 and then the solutions are geodesics. Proposition 2.1. Let γ : I ⊂ R → M3 be a curve in the three-dimensional oriented Riemannian Manifold (M3,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 : I × (−�,�) → M3 which satisfies Π(s, 0) = γ(s), ( ∂Π ∂s (s,t) ) = V (s). 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: V (v) = ( ∂v ∂t (s,t) )∣∣∣∣ t=0 = g(∇TV,T ), V (κ) = ( ∂κ ∂t (s,t) )∣∣∣∣ t=0 = g(∇2TV,N) − 2κ g(∇TV,T ) + g(R(V,T )T,N), V (τ) = ( ∂τ ∂t (s,t) )∣∣∣∣ t=0 = [ 1 κ g(∇2TV + R(V,T )T,B) ]′ +g(R(V,T )N,B)+τg(∇TV,T )+2κ g(∇TV,B), where R is the curvature tensor of M3. Corollary 2.1. Let V (s) be a restriction to γ(s) of a Killing vector field V of M3, then V (v) = V (κ) = V (τ) = 0. 3. Magnetic curves of the tangent indicatrix Definition 3.1. [11, 18] Let γT : I → S2 ⊂ E3 be a tangent indicatrix of a regular curve γ in three-dimensional Euclidean space E3, and F be a magnetic field on M3, then the curve γT is (i) TT−magnetic curve if TT satisfies the Lorentz force equation, ∇TT TT = φ(TT ) = V ×TT , (ii) NT−magnetic curve if NT satisfies the Lorentz force equation, ∇TT NT = φ(NT ) = V ×NT , (iii) BT−magnetic curve if BT satisfies the Lorentz force equation, ∇TT BT = φ(BT ) = V ×BT . In the light of this definition, we can investigate the following. Int. J. Anal. Appl. (2022), 20:18 5 3.1. TT-magnetic curve. Proposition 3.1. Let γT be a TT−magnetic curve in E3, with the Frenet apparatus {TT ,NT ,BT ,κT ,τT}. Then, we have the Frenet formula:  T ′T (sT ) N′T (sT ) B′T (sT )   =   0 √ 1 + f 2 0 − √ 1 + f 2 0 σ √ 1 + f 2 0 −σ √ 1 + f 2 0     TT (sT ) NT (sT ) BT (sT )   , and the Lorentz force in the Frenet frame can be written as  φ(TT ) φ(NT ) φ(BT )   =   0 √ 1 + f 2 0 − √ 1 + f 2 0 Ψ1 0 −Ψ1 0     TT NT BT   . (3.1) where Ψ1 is a certain function defined by Ψ1 = g(φ(NT ),BT ). Proof. From Definition 3.1, one can write φ(TT ) = √ 1 + f 2 NT . (3.2) Since φ(NT ) ∈ span{TT ,NT ,BT}, we have φ(NT ) = λ1TT + λ2NT + λ3BT . Use the following equalities: g(φ(NT ),TT ) = −g(φ(TT ),NT ) = − √ 1 + f 2, g(φ(NT ),NT ) = 0, g(φ(NT ),BT ) = Ψ1, to get λ1 = − √ 1 + f 2, λ2 = 0, λ3 = Ψ1. Hence, φ(NT ) = − √ 1 + f 2TT + Ψ1BT . (3.3) Similarly, we can easily obtain φ(BT ) = −Ψ1NT . (3.4) From Eqs. (3.2), (3.3) and (3.4), we get the required result. � Proposition 3.2. The curve γT is a TT-magnetic trajectory of a magnetic field F if and only if the vector field V is given by V = Ψ1TT + √ 1 + f 2 BT . (3.5) 6 Int. J. Anal. Appl. (2022), 20:18 Proof. Let γT be a TT-magnetic trajectory of a magnetic field F. Then, by using Proposition 3.1 and Eq. (2.3), we can easily have V = Ψ1TT + √ 1 + f 2 BT . Conversely, we assume that Eq. (3.5) holds, then we get V ×TT = φ(TT ) and so the curve γT is a TT-magnetic curve. � Theorem 3.1. Let γT be a TT−magnetic curve and V be a Killing vector field on a space form (M3(K),g). If γT is one of the TT−magnetic trajectories of (M3(K),g,V ), then its curvatures satisfying the following relations: Ψ1 = const., (1 + f 2) ( Ψ1 2 −σ √ 1 + f 2 ) = A1, (√ 1 + f 2 )′′ + σ(1 + f 2)Ψ1 −σ2(1 + f 2)3/2 + K √ 1 + f 2 + 1 2 (1 + f 2)3/2 = A2 √ 1 + f 2, where K is the curvature of Riemannian space M3 and A1, A2 are constants. Proof. Let V be a vector field in Riemannian manifold M3, then V satisfies Eq. (3.5). So, differenti- ating Eq. (3.5) with respect to s, we get ∇TV = Ψ′1TT + √ 1 + f 2(Ψ1 −σ √ 1 + f 2)NT + (√ 1 + f 2 )′ BT . (3.6) Since V is a Killing vector then from Corollary 2.1, V (v) = 0 and ∇TV has no tangential component, i.e., Ψ1 = const. Also, the differentiation of Eq. (3.6) and using Eq. (2.3) lead to ∇2TV = (1 + f 2)(σ √ 1 + f 2 − Ψ1)TT + ((√ 1 + f 2 )′′ + σ(1 + f 2)Ψ1 −σ2(1 + f 2)3/2 ) BT + ((√ 1 + f 2 )′( Ψ1 − 2σ √ 1 + f 2 ) − √ 1 + f 2 ( σ √ 1 + f 2 )′) NT . (3.7) Thus, from Eqs. (3.6) and (3.7) and Corollary 2.1, we have (V ( √ 1 + f 2) = 0). So, we get (1 + f 2) ( Ψ1 2 −σ √ 1 + f 2 ) + A1 = 0. (3.8) Similarly, according to Proposition 2.2, when Eqs. (3.6) and (3.7) are considered with the condition V (σ √ 1 + f 2) = 0, we can easily obtain[ 1 √ 1 + f 2 ((√ 1 + f 2 )′′ + σ(1 + f 2)Ψ1 −σ2(1 + f 2)3/2 + g(R(V,TT )TT ,BT ) )]′ + √ 1 + f 2 (√ 1 + f 2 )′ = 0. If M3 has constant curvature K, then g(R(V,TT )TT ,BT ) = Kg(V,BT ) = K √ 1 + f 2, Int. J. Anal. Appl. (2022), 20:18 7 therefore,(√ 1 + f 2 )′′ + σ(1 + f 2)Ψ1 −σ2(1 + f 2)3/2 + K √ 1 + f 2 + 1 2 (1 + f 2)3/2 = A2 √ 1 + f 2. (3.9) Hence, the proof is completed. � 3.2. NT-magnetic curve. Proposition 3.3. Let γT be a NT-magnetic curve in E3 with Frenet apparatus {TT ,NT ,BT ,κT ,τT}. Then, the Lorentz force in the Frenet frame can be written as  φ(TT ) φ(NT ) φ(BT )   =   0 √ 1 + f 2 Ψ2 − √ 1 + f 2 0 σ √ 1 + f 2 −Ψ2 −σ √ 1 + f 2 0     TT NT BT   , (3.10) where Ψ2 is a function defined by Ψ2 = g(φ(TT ),BT ). Proof. From Definition 3.1, one can write φ(NT ) = − √ 1 + f 2TT + σ √ 1 + f 2BT . (3.11) Since φ(TT ) ∈ span{TT ,NT ,BT}, then we have φ(TT ) = µ1TT + µ2NT + µ3BT . Using the following equalities: g(φ(TT ),TT ) = 0, g(φ(TT ),BT ) = Ψ2, g(φ(TT ),NT ) = −g(φ(NT ),TT ) = √ 1 + f 2, we get µ1 = 0, µ2 = √ 1 + f 2, µ3 = Ψ2, and therefore, φ(TT ) = √ 1 + f 2NT + Ψ2BT . (3.12) Similarly, we can easily obtain that φ(BT ) = −Ψ2TT −σ √ 1 + f 2NT . (3.13) Hence, from Eqs. (3.11), (3.12) and (3.13), the proof is completed. � Corollary 3.1. Let γT be a curve in E3. Then, the curve γT is a NT-magnetic trajectory of a magnetic field F if and only if the vector field V along γ is written as V = σ √ 1 + f 2TT − Ψ2NT + √ 1 + f 2BT . (3.14) Proof. The proof is similar to that we have considered in Proposition 3.2. � 8 Int. J. Anal. Appl. (2022), 20:18 Theorem 3.2. Let γT be a NT−magnetic curve and V be a Killing vector field on a space form (M3(K),g). If the curve γT is one of the NT−magnetic trajectories of (M3(K),g,V ), then its curvatures satisfying the following relations: Ψ2 = ( σ √ 1 + f 2 )′ √ 1 + f 2 , Ψ2σ 2(1 + f 2) −σ √ 1 + f 2 (√ 1 + f 2 )′ − Ψ′′2 = KΨ2,(√ 1 + f 2 )′′ − 2Ψ′2σ √ 1 + f 2 − Ψ2 ( σ √ 1 + f 2 )′ + K √ 1 + f 2 + (1 + f 2)3/2(1 + σ) 2 = A3 √ 1 + f 2, where A3 is a constant. Proof. Differentiating Eq. (3.14) with respect to s, we get ∇TV = ( Ψ2 √ 1 + f 2 + ( σ √ 1 + f 2 )′) TT − Ψ′2NT + ((√ 1 + f 2 )′ − Ψ2σ √ 1 + f 2 ) BT . (3.15) Since V is a Killing vector, then from Proposition 3.2 (V (v) = 0), we have Ψ2 = ( σ √ 1 + f 2 )′ √ 1 + f 2 . Also, differentiation of Eq. (3.15) together with Eq. (2.2), gives ∇2TV = Ψ ′ 2 √ 1 + f 2TT + ( Ψ2σ 2(1 + f 2) −σ √ 1 + f 2 (√ 1 + f 2 )′ − Ψ′′2 ) NT + ((√ 1 + f 2 )′′ − 2Ψ′2σ √ 1 + f 2 − Ψ2 ( σ √ 1 + f 2 )′) BT . (3.16) Thus, from Eqs. (3.15) and (3.16) together with Proposition 2.2 (V ( √ 1 + f 2) = 0), we get Ψ2σ 2(1 + f 2) −σ √ 1 + f 2 (√ 1 + f 2 )′ − Ψ′′2 + g(R(V,TT )TT ,NT ) = 0. If M3 has a constant curvature K, then g(R(V,TT )TT ,NT ) = Kg(V,NT ) = −KΨ2, and therefore Ψ2σ 2(1 + f 2) −σ √ 1 + f 2 (√ 1 + f 2 )′ − Ψ′′2 −KΨ2 = 0. (3.17) Using the condition V (σ √ 1 + f 2) = 0 in Eqs. (3.15) and (3.16), we obtain[ 1 √ 1 + f 2 ((√ 1 + f 2 )′′ − 2Ψ′2σ √ 1 + f 2 − Ψ2 ( σ √ 1 + f 2 )′ + K √ 1 + f 2 )]′ + √ 1 + f 2 (√ 1 + f 2 )′ + σ √ 1 + f 2 ( σ √ 1 + f 2 )′ = 0. (3.18) Integrating Eq. (3.18) leads to(√ 1 + f 2 )′′ − 2Ψ′2σ √ 1 + f 2 − Ψ2 ( σ √ 1 + f 2 )′ + K √ 1 + f 2 + (1 + f 2)3/2(1 + σ) 2 = A3 √ 1 + f 2. (3.19) Int. J. Anal. Appl. (2022), 20:18 9 Thus, this completes the proof. � Corollary 3.2. Let γT be a NT−magnetic curve in Euclidean 3-space with Ψ2 is zero, then γT is a circular helix. Moreover, the axis of the circular helix is the vector field. Proof. It is clear from Theorem 3.2. � 3.3. BT-magnetic curve. Proposition 3.4. Let γT be a BT-magnetic curve in E3 with Frenet apparatus {TT ,NT ,BT ,κT ,τT}. Then, the Lorentz force in the Frenet frame can be written as  φ(TT ) φ(NT ) φ(BT )   =   0 Ψ3 0 −Ψ3 0 σ √ 1 + f 2 0 −σ √ 1 + f 2 0     TT NT BT   . (3.20) where Ψ3 is given by Ψ3 = g(φ(TT ),NT ). Proof. As we mentioned the above, we can write φ(BT ) = −σ √ 1 + f 2NT , (3.21) φ(TT ) = υ1TT + υ2NT + υ3BT . Using the following conditions: g(φ(TT ),TT ) = 0, g(φ(TT ),NT ) = Ψ3, g(φ(TT ),BT ) = −g(φ(BT ),TT ) = 0, we can obtain µ1 = 0, µ2 = Ψ3, µ3 = 0. From this, we get φ(TT ) = Ψ3NT . (3.22) Also, we obtain φ(NT ) = −Ψ3TT + σ √ 1 + f 2BT . (3.23) Therefore, the proof is completed. � Corollary 3.3. Let γT be a curve in E3. The curve γT is a BT-magnetic trajectory of a magnetic field F if and only if the vector field V along γ is written as V = σ √ 1 + f 2TT + Ψ3BT . (3.24) 10 Int. J. Anal. Appl. (2022), 20:18 Theorem 3.3. Let γT be a BT−magnetic curve and V be a Killing vector field on a space form (M3(K),g). If the curve γT is one of the BT−magnetic trajectories of (M3(K),g,V ), then its curvatures satisfying the following relations: σ √ 1 + f 2 = const., Ψ′3 = 1 2 (√ 1 + f 2 )′ , Ψ′′3 + σ 2(1 + f 2) (√ 1 + f 2 − Ψ3 ) + KΨ3 + (1 + f 2)3/2 4 = A4 √ 1 + f 2 ; A4 is constant. Proof. Since V is a vector field, differentiating Eq. (3.24) with respect to s, we get ∇TV = ( σ √ 1 + f 2 )′ TT + σ √ 1 + f 2 (√ 1 + f 2 − Ψ3 ) NT + Ψ ′ 3BT . (3.25) Since V is a Killing vector, then we have σ √ 1 + f 2 = const. (3.26) Again, differentiating Eq. (3.25) and using Eq. (2.2), we get ∇2TV = −σ(1 + f 2) (√ 1 + f 2 − Ψ3 ) TT + σ √ 1 + f 2 ((√ 1 + f 2 )′ − 2Ψ′3 ) NT + ( Ψ′′3 + σ 2(1 + f 2) (√ 1 + f 2 − Ψ3 )) BT , (3.27) which leads to Ψ′3 = 1 2 (√ 1 + f 2 )′ . (3.28) Similarly, using the condition V (σ √ 1 + f 2) = 0 in Eqs. (3.25) and (3.27), we obtain[ 1 √ 1 + f 2 ( Ψ′′3 + σ 2(1 + f 2) (√ 1 + f 2 − Ψ3 ) + g(R(V,TT )TT ,BT ) )]′ +Ψ′3 √ 1 + f 2 = 0. (3.29) If K = const., then we have g(R(V,TT )TT ,BT ) = Kg(V,BT ) = KΨ ′ 3, and therefore Ψ′′3 + σ 2(1 + f 2) (√ 1 + f 2 − Ψ3 ) + KΨ′3 + (1 + f 2)3/2 4 = A4 √ 1 + f 2, (3.30) thus, this completes the proof. � Corollary 3.4. Let γT be a BT−magnetic curve in Euclidean 3-space with Ψ3 constant, then γT is a circular helix. Moreover, the axis of the circular helix is the vector field. Proof. It is obvious from Eq. (3.26) and Eq. (3.28). � Int. J. Anal. Appl. (2022), 20:18 11 Using Eq. (3.30), we obtain the following second-order nonlinear ordinary differential equation u′′(s)+σ2(1+f 2)u(s)+Ku′(s)+2u3(s)−2A4u(s) = 0, u(s) = 1 2 √ 1 + f 2; K and σ √ 1 + f 2 = const. Now, we can consider the above differential equation in Euclidean 3- space E3, in 3- sphere S3 and in hyperbolic 3- space H3, respectively. Case 3.1. Euclidean 3- space E3 (K = 0, σ √ 1 + f 2 = 3) : u′′(s) + 2u3(s) + 7u(s) = 0, A sample of individual solutions for this equation is given in the following figures: Figure 1 Sample solution family: Figure 2. Trajectories of the curvature κT of B-magnetic curve in Euclidean 3-space. 12 Int. J. Anal. Appl. (2022), 20:18 Case 3.2. 3-sphere S3 (K = 1, σ √ 1 + f 2 = 3): u′′(s) + u′(s) + 2u3(s) + 7u(s) = 0, A sample of individual solutions for this equation is given in the following figures: Figure 3. Sample solution family: Figure 4. Trajectories of the curvature κT of B-magnetic curve in 3-sphere. Case 3.3. 3- hyperbolic space H3(K = −1, σ √ 1 + f 2 = 3): u′′(s) −u′(s) + 2u3(s) + 7u(s) = 0, K = −1, σ √ 1 + f 2 = 3, A sample of individual solutions for this equation is given in the following figures: Int. J. Anal. Appl. (2022), 20:18 13 Figure 5. Sample solution family: Figure 6. Trajectories of the curvature κT of B-magnetic curve in Hyperbolic 3-space. Remark 3.1. According to the study that we have considered in the case of magnetic curves of the tangent indicatrix of γ, we can do similar study for the other spherical indicatrices, the principal normal indicatrix and the binormal indicatrix. 4. Applications In what follows, we give two computational examples to illustrate our main results. Example 4.1. Let α : I →E3 be a regular curve in the three-dimensional Euclidean space E3, can be written as α = ( s 2 cos[ln[ s 2 ]], s 2 sin[ln[ s 2 ]], s √ 2 ) , 14 Int. J. Anal. Appl. (2022), 20:18 taking the first derivative of the curve α we get T (s) = ( 1 2 ( cos[ln[ s 2 ]] − sin[ln[ s 2 ]] ) , 1 2 ( cos[ln[ s 2 ]] + sin[ln[ s 2 ]] ) , 1 √ 2 ) . Also, we can get the principal normal and binormal vectors of α respectively, N(s) = ( − cos[ln[ s 2 ]] + sin[ln[ s 2 ]] √ 2 , cos[ln[ s 2 ]] − sin[ln[ s 2 ]] √ 2 , 0 ) , B(s) = ( 1 2 ( sin[ln[ s 2 ]] − cos[ln[ s 2 ]] ) , 1 2 ( −sin[ln[ s 2 ]] − cos[ln[ s 2 ]] ) , 1 √ 2 ) , the curvatures of α are κ(s) = τ(s) = 1 √ 2s . It is clear that α is a general helix. The tangent indicatrix of α is obtained as follows αT = ( 1 2 ( cos[ln[ s 2 ]] − sin[ln[ s 2 ]] ) , 1 2 ( cos[ln[ s 2 ]] + sin[ln[ s 2 ]] ) , 1 √ 2 ) , From direct calculations, we can get the Frenet vectors of αT TT (sT ) = ( − cos[ln[ s 2 ]] + sin[ln[ s 2 ]] √ 2 , cos[ln[ s 2 ]] − sin[ln[ s 2 ]] √ 2 , 0 ) , NT (sT ) = ( 1 √ 2 ( sin[ln[ s 2 ]] − cos[ln[ s 2 ]] ) , 1 √ 2 ( −sin[ln[ s 2 ]] − cos[ln[ s 2 ]] ) , 0 ) , BT (sT ) = (0, 0, 1) . The natural representation and the curvatures of αT are respectively, sT = 1 √ 2 ln[s], f = 1, σ = 0, κT = √ 2, τT = 0, In addition, the certain function of αT is Ψ1 = const., it means that αT is a TT-magnetic curve. Example 4.2. We consider the circular helix γ in Euclidean 3− space defined by γ(s) = ( cos [ s √ 2 ] , sin [ s √ 2 ] , s √ 2 ) . Differentiating this equation, we get the tangent vector T as follows: T (s) = ( −1 √ 2 sin [ s √ 2 ] , 1 √ 2 cos [ s √ 2 ] , 1 √ 2 ) . It follows that, the principal normal and binormal vectors of γ respectively, are given by N(s) = ( −cos [ s √ 2 ] ,−sin [ s √ 2 ] , 0 ) , B(s) = ( 1 √ 2 sin [ s √ 2 ] , −1 √ 2 cos [ s √ 2 ] , 1 √ 2 ) , and so, the curvatures of γ are obtained κ(s) = τ(s) = 1 2 . Int. J. Anal. Appl. (2022), 20:18 15 From the above calculations, the tangent indicatrix of γ is given as follows γT (sT ) = ( −1 √ 2 sin [ s √ 2 ] , 1 √ 2 cos [ s √ 2 ] , 1 √ 2 ) . The Frenet vectors of γT are given as follows TT (sT ) = ( −cos [ s √ 2 ] ,−sin [ s √ 2 ] , 0 ) , NT (sT ) = ( sin [ s √ 2 ] ,−cos [ s √ 2 ] , 0 ) , BT (sT ) = (0, 0, 1) . Moreover, the natural representation and the curvature of γT are respectively, sT = 1 2 s, f = 1, σ = 0, κT = √ 2, In addition, the torsion and the certain function of γT are respectively, τT = 0 and Ψ2 = 0, it means that γT is NT-magnetic as well as BT-magnetic curve. (a) (b) Figure 7. The circular helix γ and its spherical image γT . 5. 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. 16 Int. J. Anal. Appl. (2022), 20:18 Acknowledgment The researchers wish to extend their sincere gratitude to the Deanship of Scientific Research at the Islamic University of Madinah for the support provided to the Post-Publishing Program 1. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, 1976. [2] R. Talman, Geometric Mechanics: Toward a Unification of Classical Physics, second ed., Wiley-VCH, New York, 2007. [3] A. Comtet, On the Landau Levels on the Hyperbolic Plane, Ann. Phys. 173 (1987), 185–209. https://doi.org/ 10.1016/0003-4916(87)90098-4. [4] T. 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Romero, Magnetic Vortex Filament Flows, J. Math. Phys. 48 (2007), 082904. https://doi.org/10.1063/1.2767535. https://doi.org/10.1016/0003-4916(87)90098-4 https://doi.org/10.1016/0003-4916(87)90098-4 https://doi.org/10.3792/pjaa.70.12 https://doi.org/10.3836/tjm/1270043477 https://doi.org/10.1088/1751-8113/42/19/195201 https://doi.org/10.1007/s10474-009-9005-1 https://doi.org/10.1063/1.3659498 https://doi.org/10.1016/j.geomphys.2011.10.002 https://doi.org/10.1209/0295-5075/77/34002 https://doi.org/10.1209/0295-5075/77/34002 https://doi.org/10.1063/1.2767535 1. Introduction 2. Basic concepts 3. Magnetic curves of the tangent indicatrix 3.1. TT-magnetic curve 3.2. NT-magnetic curve 3.3. BT-magnetic curve 4. Applications 5. Conclusion Acknowledgment References