https://doi.org/10.5831/HMJ.2019.41.1.175
NEW VERSION OF THE MAGNETIC CURVES
ACCORDING TO THE BISHOP FRAME IN E3
Muhammed T. Sariaydin∗ and Talat K¨orpinar
Abstract. In this paper, it is investigated Lorentz force equations for N1 and N2-magnetic curves in 3-Dimensional Euclidean space.
We give the Lorentz force in the Bishop frame in E3. Then, we
obtain a new characterization for a magnetic field V . Also, we also give examples for each curve.
1. Introduction
A magnetic curve is called trajectory of a moving charged particle under the action of a magnetic field defined on a manifold. Let (M, g) be a Riemannian manifold of n−dimensional, then the magnetic F is defined as a closed 2−form. Also, for a one to one tensor field φ, Lorentz force of F is given by
g(φ(X), Y ) = F (X, Y ), ∀X, Y ∈ χ(M ).
The magnetic trajectories associated with a magnetic field F on (M, g) satisfy the following Lorentz force equation; ∇γ0γ0 = φ (γ0). Magnetic
curves generalize geodesics, which have the form of second order non-linear differential equation of motion ∇γ0γ0 = 0. Thus, geodesics can
be considered as the trajectory of a moving particle without any action on the magnetic field. In contrast to geodesics, the trajectories of the moving particle associated to a magnetic field corresponds to a flow line of its dynamically system.
Knowing the skew symmetrical (anti-symmetrical) property of the Lorentz force implies that magnetic curves have constant velocity. If magnetic curves are parametrized by arc length, then they are named
Received August 11, 2018. Accepted December 18, 2018. 2010 Mathematics Subject Classification. 78A35.
Key words and phrases. Magnetic curves, B¨acklund transformations, Killing magnetic field.
as a normal magnetic curve, [6, 13, 16]. Then, it was the subject of numerous studies. For example, Drut¸˘a-Romaniuc and Munteanu obtain Killing vector field in E31in [7]. Then, the author studies a short review of different approaches in [10]. Munteanu and Nistor study the trajectories of charged particles moving in a space modeled by the homogeneous 3-space in [12].
On the other hand, the classical purpose of the B¨acklund transfor-mation is to produce a new pseudo-spherical surface by considering the older one. vx− ux = 1 2((u + v) /2) , vy− uy = − 1 2((u − v) /2) .
A usefulness of this transformation appears on the relation of PDE and its associated solution. That is, knowing a family of the solution to the PDE implies that solution of the PDEs generated by the certain B¨acklund transformation can be estimated. In the case of the connected PDEs are exactly the same, a new type of transformation is defined as Auto-B¨acklund transformation. B¨acklund transformation is widely used to generate a new family of solutions on the integrable theory. Difficult PDEs can be connected to simpler ones having the obvious so-lution thanks to B¨acklund transformation. Also, non-trivial cases are generated if B¨acklund transformation is applied to the trivial solution, generally. Studies on the B¨acklund transformation at the classical level concentrate. Intensely on surface’s transformation having negative con-stant curvature in R3, [1, 5, 9]. Then, it was the subject of numerous studies. For example, In [4], the authors classified a class of nonlin-ear evolution equations. In [17], the authors investigate the B¨acklund transformations of integrable systems.
The rest of this paper is organized as follows. In Section 2, we give properties and the basic concepts of magnetic curves and B¨acklund transformations in E3. In Section 3, we give Killing magnetic vector fields according to Lorentz force equations in Euclidean 3-space.
2. Preliminaries
Given a spatial curve ξ : s → ξ(s), which is parameterized by arc-length parameter s. The derivative of the Frenet frame according to the
arc-length parameter is governed by the relations; ˙ T = κN, (2.1) ˙ N = −κT+τ B, ˙ B = −τ N,
where κ is the curvature and τ is torsion of the curve α. The Bishop frame is expressed as ˙ T = κ1N1+ κ2N2, (2.2) ˙ N1 = −κ1N1, ˙ N2 = −κ2N2, where θ = arctanτκ, τ = ˙θ and κ1 =
√
κ2+ τ2, [2].
Theorem 2.1. The following assertions hold; (i) [V, W ] = 0, (ii) V (v) = ∂v∂z (t, 0) = −ε1g (∇TV, e1) v, (iii) V (κ) =∂κ 2 ∂z (t, 0) =2ε2g ∇2e1V, ∇e1e1 + 4ε1κ 2g (∇ e1V, e1) + 2ε2g (R (V, e1) e1, ∇e1e1) , (iv) V (τ ) =∂τ 2 ∂z (t, 0) = − 2ε2g( 1 κ∇ 3 e1V − κ0 κ2∇ 2 e1V + ε1 ε2κ + c κ ∇e1V − ε1 cκ0 κ2V, τ e3). Here, ε1= g (e1, e1), ε2 = g (e2, e2), ε3= g (e3, e3), and κ0 = ∂κ∂t (t, 0), [11].
3. The Magnetic Curves in Euclidean 3-Space
In this chapter, we will study the magnetic curves of the B¨acklund transformation according to Bishop frame in the three-dimensional Eu-clidean space. Then, we will calculate the Killing magnetic vector fields of these curves.
Definition 3.1. Assume that α is any space curve in E3 and ω is the transformation of this curve. Then, it is show that
(3.1) ω = α + 2C tan γ
κ22+ C2(T cos γ + N1sin γ), where C = k2tanξ2 and γ is a solution of
dγ ds = κ ω 2 cos γ tan ξ 2− κ1 = −κ2cos γ tan ξ 2 − κ ω 1. Also, constant angle ω is ω 6= 0, and γ 6= π2, [9].
Definition 3.2. Assume that α : I ⊂ R → E3 is a curve and F is a magnetic field on E3, [8] Then, the curve α is call a Xi-magnetic curve if the Xi vector field of the curve satisfies the Lorentz force equation, that is,
(3.2) ∇TXi = Φ(Xi) = V × Xi,
where 1 ≤ i ≤ 3.
Theorem 3.3. Assume that ω is the transformation defined eq. (3.1) in the E3. Then, Bishop frame of ω can be written as
Tω = η1T + η2N1+ µη3κ2N2, Nω1 = (cos ξπ1− sin ξr1) T+η4N1
+ (cos ξπ2− sin ξr3) N2, Nω2 = (sin ξπ1+ cos ξr1) T+η5N1
+ (sin ξπ2+ cos ξr3) N2,
where η4= cos ξη1κ1π3− sin ξr2, η5= sin ξη1κ1π3+ cos ξr2. Proof. Eq. (3.1) can be written as
(3.3) ω = α + µ(T cos γ + N1sin γ),
where µ = 2C tan γk2 2+C2
. From [10], we have the frenet frame of eq. (3.1) as Tω = η1T + η2N1+ µη3κ2N2,
Nω = π1T + (η1κ1π3+ ˙η2π3)N1+ π2N2, Bω = r1T + r2N1+ r3N2.
Take into consideration the two equations together as (2.1) and (2.2), we can easily write the following equations;
Nω1 = (cos ξπ1− sin ξr1) T+η4N1 + (cos ξπ2− sin ξr3) N2, Nω2 = (sin ξπ1+ cos ξr1) T+η5N1
+ (sin ξπ2+ cos ξr3) N2. Thus, the proof is complete.
Theorem 3.4. Suppose that ω is a Tω-magnetic curve satisfies the Lorentz force equation. Then, the Lorentz force of B¨acklund transforma-tion in E3 is obtained as Φ(Tω) Φ(Nω 1) Φ(Nω2) = π1 η1κ1+ ˙η2 π2 λ1 0 Ω λ2 −Ω 0 T N1 N2 ,
where Ω = g(Φ(Nω1), Nω2) is a certain function,
λ1= −(π1(cos ξπ1− sin ξr1) + η4(η1κ1+ ˙η2) + π2(cos ξπ2− sin ξr3)), λ2= −(π1(sin ξπ1+ cos ξr1) + η5(η1κ1+ ˙η2) + π2(sin ξπ2+ cos ξr3)).
Proof. Let the ω be the Tω-magnetic curve, and the ω be transfor-mation satisfies the Lorentz force equation given by eq. (3.2). That is, it is clear that
Φ(Tω) = π1T + (η1κ1+ ˙η2)N1+π2N2. On the other hand, if the following equations are satisfied;
g(Φ(Nω1), Tω) = −g(Nω1, Φ(Tω)), g(Φ(Nω1), Nω1) = 0,
g(Φ(Nω1), Nω2) = Ω, then it can easily be written by
Φ(Nω1) = λ1= −(π1(cos ξπ1− sin ξr1) + η4(η1κ1+ ˙η2) +π2(cos ξπ2− sin ξr3))T + ΩN2.
In a similar way, Φ(Nω2) can be calculated as
Φ(Nω2) = −(π1(sin ξπ1+ cos ξr1) + η5(η1κ1+ ˙η2) +π2(sin ξπ2+ cos ξr3))T − ΩN1.
Lemma 3.5. Let ω be a Tω-magnetic curve in E3. If V is Killing magnetic field, then we write the following conditions as
π1η1+ (η1κ1+ ˙η2) η2+ π2µη3κ2 = 0, λ1(cos ξπ1− sin ξr1) + Ω (cos ξπ2− sin ξr3) = 0, λ2(sin ξπ1+ cos ξr1) − Ωη5 = 0, Proof. The proof is clear from the theorem 3.4.
Theorem 3.6. Suppose that ω is a Nω
1-magnetic curve satisfies the Lorentz force equation according to Bishop frame. Then, the Lorentz force of B¨acklund transformation in E3 is obtained as
Φ(Tω) Φ(Nω1) Φ(Nω2) = 0 λ3 −Λ η6 κ1(cos ξπ1− sin ξr1) + ˙η4 η7 Λ λ4 0 T N1 N2 ,
where Λ = g(Φ(Nω3), Tω) is a certain function,
λ3 = −(η1η6+ η2(κ1(cos ξπ1− sin ξr1) + ˙η4) + µη3η7κ2), λ4 = −(η6(sin ξπ1+ cos ξr1) + η5(κ1(cos ξπ1− sin ξr1) + ˙η4)
+ η7(sin ξπ2+ cos ξr3)),
η6 = (cos ξπ1− sin ξr1)0− η4κ1− κ2(cos ξπ2− sin ξr3) , η7 = κ2(cos ξπ1− sin ξr1) − (cos ξπ2− sin ξr3)0.
Proof. The proof is clear from the theorem 3.4
Lemma 3.7. Let ω be a Nω1-magnetic curve in E3. If V is Killing magnetic field, then we write the following conditions as
λ3η2− Λµη3κ2 = 0, η6(cos ξπ1− sin ξr1) + η4(κ1(cos ξπ1 − sin ξr1) + ˙η4) + η7(cos ξπ2− sin ξr3) = 0,
Λ (sin ξπ1+ cos ξr1) + λ4η5 = 0, Proof. The proof is clear from the theorem 3.4.
Theorem 3.8. Suppose that ω is a Nω2-magnetic curve satisfies the Lorentz force equation according to Bishop frame. Then, the Lorentz force of B¨acklund transformation in E3 is obtained as
Φ(Tω) Φ(Nω1) Φ(Nω2) = 0 Π λ5 −Π 0 λ6 η8 κ1(sin ξπ1+ cos ξr1) + ˙η5 η9 T N1 N2 ,
where Π = g(Φ(Tω), Nω1) is a certain function,
λ5 = −(η1η8+ η2(κ1(sin ξπ1+ cos ξr1) + ˙η5) + µη3η9κ2), λ6 = −(η8(cos ξπ1− sin ξr1) + η4(κ1(sin ξπ1+ cos ξr1) + ˙η5)
+ η9(cos ξπ2− sin ξr3)),
η8 = (sin ξπ1+ cos ξr1)0− η5κ1− κ2(sin ξπ2+ cos ξr3), η9 = κ2(sin ξπ1+ cos ξr1) + (sin ξπ2+ cos ξr3)0.
Proof. The proof is clear from the theorem 3.4 and theorem 3.6. Lemma 3.9. Let ω be a Nω2-magnetic curve in E3. If V is Killing magnetic field, then we write the following conditions as
Πη2+ λ5µη3κ2 = 0, −Π(cos ξπ1− sin ξr1) + λ6(cos ξπ2− sin ξr3) = 0,
η8(sin ξπ1+ cos ξr1) + η5(κ1(sin ξπ1 + cos ξr1) + ˙η5) + η9(sin ξπ2+ cos ξr3) = 0. Proof. The proof is clear from the theorem 3.4.
4. Geometrical and Physical Means of Nω1-Magnetic and Nω2 -Magnetic Curves
A charged particle moving along a curve in a magnetic field V ex-periences a sideways force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle. This force is known as the Lorentz force, and is given by
Φ (Tω) = V × Tω,
where T is velocity vector of the charged particle and V is the magnetic field.
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, and in which the speed of the particle remains constant. Because the magnetic force is always perpendicular to the motion, the magnetic field can do no work on an isolated charge. Also, if the charged particle moves parallel to magnetic field, the Lorentz force acts 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 (tangent vector), also the normal and binormal vector are expressed to the magnetic field V and experience a force given by
Φ (Nω1) = V × Nω1, and Φ (Nω2) = V × Nω2.
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Muhammed T. Sariaydin
Department of Mathematics, Sel¸cuk University, 42130, Konya, Turkey.
E-mail: talatsariaydin@gmail.com Talat K¨orpinar
Department of Mathematics, Mu¸s Alparslan University, 49250, Mu¸s, Turkey.
