Elastic Strips with Spacelike Directrix - Springer Link

Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-018-0622-0

Elastic Strips with Spacelike Directrix Gözde Özkan Tükel1 · Ahmet Yücesan1

Received: 20 November 2017 / Revised: 27 February 2018 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Abstract The elasticity of a rectifying strip with a spacelike directrix (base curve) is studied using the differential geometry of developable surfaces and variational techniques. Two conservation laws are derived in Minkowski 3-space R13 by constructing two different variations including only Lorentz translations or rotations. By means of these laws two new types of elastic strips are defined: momentum strips and forcefree strips. There exists a connection between elastic curves on de Sitter 2-space and pseudohyperbolic space and elastic strips with spacelike directrix. Keywords Variational calculus · Elastic strips · Conservation laws Mathematics Subject Classification 53A35 · 53B30 · 35A15

1 Introduction An elastic strip is a special ruled surface that may be completely characterized using its directrix. The surface is expressed as a solution to a variational problem. These thin rods are completely determined (up to Euclidean motions) by curvature and torsion of its centerline. Sadowsky (1930) introduced a model for elastic strips by using the techniques of differential geometry, and Wunderlich (1962) improved on this [9, 10]. Wunderlich shows that Willmore functional S H 2 d A of an infinitely narrow inextensible strip is proportional to the Sadowsky functional

Communicated by Young Jin Suh.

B 1

Gözde Özkan Tükel [email protected] Süleyman Demirel University, Isparta, Turkey

123

G. Özkan Tükel, A. Yücesan

S=

κ γ

2

τ2 1+ 2 κ

2 ds.

(1)

This implies equilibrium configurations of elastic strips can be obtained in terms of critical points of the Sadowsky functional when it is applied to all variations having fixed end points and fixed length. Hangan [5] and Chubelaschwili and Pinkall [4] derive two Euler–Lagrange equations of the functional (1) at different times in Euclidean 3-space R 3 . The characterization of the centerline of an elastic strip presented by Chubelaschwili and Pinkall corresponds to conservation laws generated by the symmetry group of Euclidean motions. The idea that the balanced form of the Euler–Lagrange equations of the functional (1) may be useful for certain problems involving non-Euclidean symmetry groups. The present paper is devoted to the study of elastic strips with a spacelike directrix. This article is organized as follows: In Sect. 2 we derive two Euler–Lagrange equations for characterization of elastic strips with a spacelike directrix in Minkowski 3-space R13 . We also show that planar critical points of the modified Sadowsky functional are just planar elastic curves in Minkowski 3-space R13 . In Sect. 3, we obtain two conservation laws to derive two new classes of integrable elastic strips with spacelike directrix. Using these laws, we give a different characterization for elastic strips with spacelike directrix. In Sect. 4 we define elastic momentum strips and force-free strips with a spacelike directrix and present some relations between elastic curves on hyperquadrics and elastic strips with spacelike directrix.

2 Euler–Lagrange Equations In Euclidean or Minkowski 3-space, the Euler–Lagrange equation which characterizes an elastic surface is given by the fourth-order equation H + 2H H 2 − K = 0, where denotes the Laplace–Beltrami operator and K is the Gaussian curvature [3]. As in Euclidean 3-space R 3 , the Willmore functional, defined on developable ruled surfaces’ directrix using the Darboux vector, is proportional to the Sadowsky functional in Minkowski 3-space R13 . In this section we introduce Minkowski rectifying strips, build the variational problem for elastic strips with spacelike directrix and then derive two Euler–Lagrange equations for the critical points of modified Sadowsky functional. We provide an example by solving this system of equations. Recall that Minkowski 3-space R13 is a three-dimensional real vector space equipped with the metric < x, y >= x1 y1 + x2 y2 − x3 y3 , x = (x1 , x2 , x3 ) , y = (y1 , y2 , y3 ) ∈ R13 , which is of a non-degenerate, symmetric and bilinear form. A smooth curve in R13 is a spacelike (resp., a timelike and a lightlike), if its tangent vector is a spacelike (resp., a timelike and a lightlike) [6,7].

123

Elastic Strips with Spacelike Directrix

Let γ : [0, ] → R13 be a regular spacelike curve with velocity υ = γ in Minkowski 3-space R13 . At a point γ (t) of γ , let T (t), N (t) and B(t), respectively, denote the unit tangent vector, the unit normal vector and the binormal vector. For the Frenet frame {T, N , B} along γ , we have < T, T >= 1, < N , N >= ε

< B, B >= −ε, ε = ∓1

and T × N = −ε B,

N × B = T,

B × T = εN .

The curvature, the torsion and the modified torsion are denoted by

γ × γ det γ , γ , γ τ κ= , τ = −ε and λ = . γ × γ κ γ 3 The Frenet formulas are given by T = ευκ N N = −υκ T − ευλκ B B = −ευλκ N .

(2)

An elastic strip is a surface with minimum bending energy. To get a detailed description of these strips, we introduce a developable ruled surface whose directrix is a spacelike curve. Since a developable surface is a special ruled surface, parametrization of a thin, inextensible and developable strip derives from Frenet components of a curve γ called directrix of the strip, i.e., if γ (s) is a parametrization of a curve, then Fγ : [0, ] × [− , ] → R13 → Fγ (t, δ) = γ (t) + δ D (t) (t, δ)

(3)

is a parametrization of a strip with directrix γ and of length and width 2 , where D (t) = λT (t) + B (t) is the modified Darboux vector, T is the unit tangent vector, B is the unit binormal, and τ (s) λ (s) = κ (s) is the modified torsion of γ such that κ is the curvature and τ is the torsion of γ . Because

det γ (t) , D (t) , D (t) = 0 in (3), Fγ is a developable surface. Definition 1 Any developable ruled surface with spacelike directrix defined by (3) is called Minkowski rectifying strip. We next study infinitely narrow Minkowski rectifying strips constructed by using critical points of the Sadowsky functional (1) within all space curves with fixed end points and

123

G. Özkan Tükel, A. Yücesan ·

:=

∂ (γt ) = 0. ∂δ δ=0

(4)

The results obtained by Wunderlich imply that there is a 1-dimensional variational problem involving the critical base curve, as opposed to a 2-dimensional variational problem involving the surface itself. We therefore look for elastic strips with spacelike directrix defined as follows. Definition 2 Let Fγ be a Minkowski rectifying strip. If the spacelike directrix γ of Fγ is an extremal for the modified Sadowsky functional 2 2 2 κ 1+λ − μ υds, Sμ (γ ) =

(5)

0

then Fγ is an elastic strip with spacelike directrix. Here μ is a Lagrange multiplier due to the length constraint. In the above definition, we observe that a spacelike curve γ : (0, ) → R13 defines elastic strip, and so does γ : [0, ] → R13 . Lemma 1 If γ : [0, ] → R13 is an arclength parametrized spacelike curve and γ : [0, ] × [− , ] → R13 · → γ (s, δ) = γδ (s) = γ (s) + δ γ (s) (s, δ) is a variation of γ with variational vector field ∂ · γ (s) = γδ (s) = u 1 (s) T (s) + u 2 (s) N (s) + u 3 (s) B (s) , ∂δ δ=0

(6)

where ·

·

·

u 1 =< γ (s) , T (s) >, u 2 = ε < γ (s) , N (s) >, u 3 = −ε < γ (s) , B (s) >, then we have ·

υ = u 1 − κu 2 ,

κ = u 1 κ + u 2 κ 1 + ελ2 − 2u 3 λκ − u 3 (λκ) + εu 2 ·

and

2

(λκ) (λκ) κ 3 λ = u1λ + u2 ε 2 − ε − ελ κ + λκ κ κ3 λ λ (λκ) + εu 2 − u 3 λλ + u 3 −ε + λ2 + u 2 2ε + ε 2 κ κ κ κ 1 + u 3 3 − u 3 2. κ κ ·

123

(7) (8)

(9)


· ·

Proof By using the fact γ = γ , we obtain the equality ·

· υT + T = u 1 − u 2 κ T + εu 1 κ + u 2 − εu 3 λκ N + −εu 2 λκ + u 3 B,

from which we derive (7) and ·

T = εu 1 κ + u 2 − εu 3 λκ N + −εu 2 λκ + u 3 B.

Thus, we conclude the following equalities · ·

· · T = ευκ N + εκ N + εκ N , ·

T = − εu 1 κ + u 2 − εu 3 λκ N + εu 1 κ + u 2 − εu 3 λκ (−κ T − ελκ B)

+ −εu 2 λκ + u 3 B − −εu 2 λκ + u 3 ελκ N . · ·

Equation (8) directly follows from Eq. (7) and the fact T = T , both of which imply that ·

N = − u 1 κ + εu 2 − u 3 λκ T

ε −εu 2 λκ + u 3 − ελκ εu 1 κ + u 2 − εu 3 λκ B. + κ

By using (7) and (8), we obtain the following equations · ·

< N , B >= −λκ 2 u 2 + ε2 κu 3 + λκu 1 − λκ 2 u 2 + λκ + u 1 λκ + 1 + ελ2 λκ 2 u 2 − 2u 3 λ2 κ − u 3 λ (λk) + ελu 2 ε ·

−εu 2 λκ + u 3 − ελκ εu 1 κ + u 2 − εu 3 λκ < N , B >= −ε . κ

Employing the equality

· ·

N = N , we obtain (9). Assume that an arclength

parametrized spacelike curve γ : [0, ] → R13 is the directrix of an elastic strip. Consider a variation of γ having the variational vector field (6). Calculate the first variation of the Sadowsky functional

Sμ (γδ ) =

0

2 κδ2 1 + λ2δ − μ υδ dt

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as 2 ∂ ∂ 2 2 κδ 1 + λδ − μ υδ Sμ (γδ ) (γt ) = ∂δ ∂δ δ=0 δ=0 2 ∂ κδ2 1 + λ2δ − μ υδ + dt, 0 ∂δ δ=0 and then by taking condition (4) into consideration, we obtain the following equality 2 ∂ ∂ κδ2 1 + λ2δ − μ υδ = dt. Sμ (γδ ) ∂δ 0 ∂δ δ=0 δ=0 Now from equalities (7), (8) and (9), we conclude that 1 2

∂ ∂δ

0

2 κδ2 1 + λ2δ − μ υδ

δ=0

dt =

u 2 f 1 + u 3 f 2 + b dt,

0

where 2 2 2 + 2κ 1 + λ λλ f1 : = ε κ 1 + λ 2 κ 2 κ 1 + λ2 (1 + (5 + 4ε) λ2 ) + μ) + λκ(−εκ 2 1 + λ2 λ 2 κ 1 + λ2 λ + 2ε 1 + λ2 λ ), + 2ε κ 2 f 2 : = −(κ 2 1 + λ2 λ + (−ε − 1) 2κ 2 λ 1 + λ2 κ 1 + λ2 λ + −2 1 + λ2 λ )) + −2 κ 2 + λκ(κ 1 + λ2 + 2κ 1 + λ2 λλ ) +

and

2 1 2 κ 1 + λ2 − μ 2 + εu 2 6λλ κ + 2λ2 κ 1 + λ2 − κ 3λ2 + 1 1 + λ2 + εu 2 κ 3λ2 + 1 1 + λ2

κ 1 + λ2 + u 3 (−ε − 1) 2κ 2 λ 1 + λ2 + −2λ κ − u 3 2λ 1 + λ2

b := u 1

123

(10)


κ 1 + λ2 + −2λ 1 + λ2 − u 3 −2λ κ . − u 3 2λ 1 + λ2

(11)

Theorem 1 If Fγ is an elastic strip with spacelike directrix γ , then γ satisfies the Euler–Lagrange equations (12) f 1 = f 2 = 0. In addition, for each variation of γ which leaves the integrand of the modified Sadowsky functional invariant, we obtain b = 0, when γ is an extremal of Sμ . Proof Suppose that a spacelike curve γ is an extremal for Sμ in which γ is parametrized with respect to the arclength. From (10), we obtain ∂ Sμ (γδ ) = (u 2 (s) f 1 (s) + u 3 (s) f 2 (s)) ds + b () − b (0) = 0. ∂δ 0 δ=0 On the other hand, since b () = b (0) = 0 for a suitable variation, the claimed Euler–Lagrange equations hold. Furthermore, when γ is critical for Sμ it satisfies Euler–Lagrange equations stated in (12) so that 2 1 ∂ 2 2 κ 1 + λδ − μ υδ = u 2 f 1 + u 3 f 2 + b = b = 0. 2 ∂δ δ=0 δ The Euler–Lagrange equations (12) show that planar critical points of functional (5) correspond to planar elastic curves in Minkowski 3-space R13 [2]. Moreover, a spacelike helix with non-null normal vector satisfies the Euler–Lagrange equations (12). We give an example for elastic strips with spacelike helix directrix. √ √ √ 3 cosh 13s, sinh 13s, 23 13s be an unit-speed spaceExample 1 Let ξ (s) = 13 like helix [1]. The helix has the curvature and the modified torsion κ = 3 and λ =

2 . 3

(13)

By using values (13) and Euler–Lagrange equations (12), we see that the Minkowski rectifying strip Fξ (s, δ) =

√ √ 3 3 2 cosh 13s, sinh 13s, √ s + 13 13 13

√ 13 δ 3

with directrix ξ is an elastic strip with spacelike directrix chosen μ ≈ − 35, 469.

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3 Conservation Laws In this section we set up two different variations including only Lorentzian translations or rotations. We then state the first and second conservation laws of elastic strips with 3 spacelike directrix Minkowski space R1 . in three-dimensional

2 We recall that κ 2 1 + λ2 − μ is invariant under Lorentzian motions. Now we choose variations involving only translations and rotations, respectively. First choose a variation involving translations γδ (s) = γ (s) + δ with the variation vector field ·

γ δ (s) = = u 1 T + u 2 N + u 3 B, where is an arbitrary vector such that u 1 =< , T >, u 2 = ε < , N > and u 3 = −ε < , B > .The derivatives u 2 , u 3 , u 3 are calculated as

and

u 2 = −εκ < , T > −λκ < , B >, u 3 = λκ < , N >

(14) (15)

u 3 = (λκ) < , N > −λκ 2 < , T > −ελ2 κ 2 < , B > .

(16)

Combine Eq. (11) with derivatives (14), (15) and (16), and we obtain b =< , W0 >, where 2 2 1 2 2 2 2 W0 := κ 1+λ N +μ T + κ 1+λ + 2κλλ 1 + λ 2 2 B − −ελκ 2 1 + λ2 + 2ε (ε + 1) κ 2 λ 1 + λ2

κ 2ε λ 1 + λ2 − + 2ελ 1 + λ2 B. κ When the spacelike curve γ is critical for the Sadowsky functional, b is a constant. Thus, W0 is a constant for any ∈ R13 . Now, we investigate the elastic strips with spacelike directrix for a variation consisting only of Lorentzian rotations. Taking into account that ∂ Aδ γ (s) = ˜ × γ (s) = u 1 T + u 2 N + u 3 B ∂δ δ=0

123


for ˜ ∈ R13 , Aδ ∈ S O1 (3), u 1 =< ˜ × γ , T >, u 2 = ε < ˜ × γ , N > and u 3 = −ε < ˜ × γ , B > . By using similar method, we get ˜ W1 >, b =< , where

1 2κλ 1 + λ2 N W1 :=2λκ 1 + λ2 T + κ − εκ 1 + λ2 1 − λ2 B − γ × W0 .

Therefore, W1 is constant when the spacelike curve γ is critical for the Sadowsky functional, and W0 is constant for any ∈ R13 In the following theorems we show that elastic strips with a spacelike directrix are determined by W0 and W1 . Theorem 2 (The first conservation law) A Minkowski rectifying strip Fγ is an elastic strip with spacelike directrix if and only if the force vector W0 = a1 T + a2 N + a3 B is constant, where 2 1 2 2 κ 1+λ a1 := +μ , (17) 2 2 (18) a2 := κ 1 + λ2 + 2κλλ 1 + λ2 , and

2 a3 := − −ελκ 2 1 + λ2 + 2ε (ε + 1) κ 2 λ 1 + λ2

κ 2 2 2ε λ 1 + λ − + 2ελ 1 + λ . κ

Proof It suffices to show

(19)

W0 = ε f 1 N − ε f 2 B,

(20)

since the force vector W0 is a constant if and only if f 1 = f 2 = 0 in Eq. (20). We use Frenet equations (2) to obtain

W0 = (a1 − εκa2 )T + a2 + εκa1 − ελκa3 N + a3 + ελκa2 B.

(21)

On the other hand, from (17) and (18), we find a2 =

ε a . κ 1

It then follows that the coefficient of T in Eq. (12) vanishes. Now, by using (17), (18) and (19), the coefficients of N and B can be stated as a2 + εκa1 − ελκa3 = ε f 1 , a3 + ελκa2 = −ε f 2 ,

(22)

123


respectively. Equation (22) shows that γ defines an elastic strip with spacelike directrix if and only if f 1 = f 2 = 0. Theorem 3 (The second conservation law) Fγ is an elastic strip with spacelike directrix if and only if the torque vector W1 = s1 T + s2 N + s3 B − γ × W0 is constant, where s1 := 2λκ 1 + λ2 , 1 s2 := 2κλ 1 + λ2 , κ

(23) (24)

s3 := −εκ 1 + λ2 1 − λ2 .

and

(25)

Moreover, if W1 is a constant but γ does not define an elastic strip with spacelike directrix, then γ is conserved. Proof Using the Frenet equations (2), we obtain the first derivative of W1 as follows ⎞

⎛

⎟ ⎜ W1 = s1 − κs2 T + ⎝εκs1 + s2 − ελκs3 − ε < T × W0 , N >⎠ N ⎛

−a3

0

⎞

−a3

⎜ ⎟ + ⎝s3 − εs2 λκ + ε < T × W0 , B >⎠ B − γ × W0 .

(26)

a2

a2

Substituting (23), (24), (25) and the derivatives s1 , s2 and s3 in (26), we see that coefficients of T, N and B vanish. Then Eq. (26) reduces to W1 = −γ × W0 .

(27)

Equation (27) implies that W1 is zero if W0 is a constant. From Theorem 2, we know W0 is a constant if and only if γ defines an elastic strip with spacelike directrix. On the contrary, γ does not define an elastic strip with spacelike directrix while W1 = 0. Then, substituting (20) in Eq. (27), we get 0 = W1 = −γ × (ε f 1 N − ε f 2 B) . Hence, γ ∈ Span{N , B}. So we have < γ , γ > = 2 < γ , T >= 0.

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4 Some Relations Between Elastic Curves on Hyperquadrics and Elastic Strips with Spacelike Directrix We begin this section by recalling hyperquadrics of Minkowski 3-space R13 and elastic curves on hyperquadrics. De Sitter 2-space of radius r > 0 in R13 is the hyperquadric S12 = { p ∈ R13 < p, p >= r 2 } of dimension 2 and index 1 [7]. As it is well known, de Sitter 2-space S12 is a timelike surface [6]. Note also that a spacelike and a timelike elastic curves with geodesic curvature λ on S12 satisfy the following equalities 2 1 4 λ − λ + 1+ 4 2 1 4 λ − λ + −1 + 4

σ 2 λ = A, 2 σ λ2 = A, 2

A = const A = const,

(28)

respectively, where σ is a Lagrange multiplier [8]. The pseudohyperbolic space of radius r > 0 in R13 is the hyperquadric H 2 = H02 = { p ∈ R13 < p, p >= −r 2 } with dimension 2 and index 1 [7]. In this case the pseudohyperbolic space H 2 is a spacelike surface. A spacelike elastic curve with geodesic curvature λ on H 2 satisfies 2 1 4 σ 2 λ + λ − 1+ λ = A, 4 2

A = const.,

(29)

where σ is a Lagrange multiplier [8]. Now, we define elastic momentum strips with spacelike directrix. Definition 3 Any spacelike curve γ defines an momentum strip in three-dimensional Minkowski space R13 , if W1 + γ × W0 , T is a constant nonzero function. Our next result provides a relation between momentum strips with spacelike directrix in R13 and spacelike elastic curves on pseudohyperbolic space H 2 . Theorem 4 Let a regular spacelike curve γ : [0, ] → R13 define a momentum strip with a spacelike directrix and Lagrange multiplier μ. (i) If the principle normal vector N of γ is spacelike, then the binormal vector B of γ corresponds a spacelike elastic curve with Lagrange multiplier −μ on pseudohyperbolic space H 2 . Conversely for each such arclength parametrized

123


˜ → H 2 with nonvanishing, non-constant geodesic curvature λ and B : [0, ] T := B × B , the spacelike curve γ (t) =

1 2

t 1+ 0

1 T (s) ds λ2 (s)

corresponds to a momentum strip with spacelike directrix with 1 + λ2 dt − μ (γ ) . Sμ (γ (t)) = 2 0

(ii) If the principle normal vector N of γ is timelike, then the binormal vector B of γ corresponds a timelike elastic curve with Lagrange multiplier −μ on de Sitter ˜ → S2 2-space S12 . Conversely for each such arclength parametrized B : [0, ] 1 with nonvanishing, non-constant geodesic curvature λ and T := B × B , the spacelike curve t 1 1+ 2 T (s) ds γ (t) = λ (s) 0

corresponds to a momentum strip with spacelike directrix with

Sμ (γ (t)) =

1 + λ2 dt − μ (γ ) .

0

Proof (i) Let a regular spacelike curve γ with spacelike principle normal vector N define a momentum strip with Lagrange multiplier μ. Then binormal vector B of γ is a timelike vector field. Because of the fact that < B , B >= λ2 κ 2 > 0, B is a spacelike curve on pseudohyperbolic space H 2 . Once we choose < W1 + γ × W0 , T >= s1 = 4, we obtain κ=

2

λ 1 + λ2

(30)

and

2 2 λ (s) 1 1 1 2 1 < W0 , W0 > − μ = 1 + λ (s)2 + 4 4 16 4 4 λ (s) λ (s) 1 4 1 . μ + 1 − + λ (s)2 2 1 + λ (s)2

123

(31)


With a change in parameter λ (t) = λ (s (t)) , t (s) = (31) as

2 1+λ2

in (31), we may write

1 4 1 2 1 1 1 2 (−μ) < W0 , W0 > − μ . + − 1+ = 4 2 16 4 λ (t) λ (t) (32) By using Frenet equations (2) we can easily see that t is the arclength parameter of

1 B and λ1 is its curvature. In addition, we have 16 < W0 , W0 > − 41 μ2 = const and Eq. (32) has at least a nonzero solution. The binormal B of the spacelike curve γ corresponds to the spacelike elastic curves with Lagrange multiplier −μ on pseudo hyperbolic space H 2 with radius 1 by (29).

1 λ (t)

2

Conversely, let B be such an arclength parametrized spacelike elastic curve on pseudohyperbolic space H 2 with nonvanishing, non-constant geodesic curvature λ and let B be the binormal of a spacelike curve γ with Frenet frame {T, N , B} in R13 . Then the Darboux trihedron of B is {−N , −B, T } and derivative formulas of the Darboux frame are given by −B = − N , −N = − λT + B, T =λN .

(33)

If we consider curve B having a reparametrization with an arclength s the spacelike ds 1 1 with dt = 2 1 + λ2 there is a spacelike curve 1 γ (t) = 2

t 1+ 0

1 T (s) ds. λ2 (s)

Now from the stated equations in (33), the curvature and the modified torsion of γ can be written as 1 τ 2 andλ = = − . (34) κ= 1 1 κ λ 1 + λ λ2 From Definition 1 and (34), we conclude that the spacelike curve denotes a momentum strip with spacelike directrix. Substituting 1˜ for λ in Eq. (32) yields λ

2 1 4 μ 2 1 1 λ = < W0 , W0 > − μ2 . λ + λ − 1− 4 2 16 4 We conclude that < W0 , W0 > and < W0 , W1 > are conserved, since B is a spacelike elastic curve with Lagrange multiplier −μ on pseudohyperbolic space H 2 . So, we can easily check from (20) and (27), f 1 = f 2 = 0. (ii) One can easily prove by a similar calculation as part i of this theorem.

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Definition 4 An elastic strip with a spacelike directrix in Minkowski 3-space R13 is called a force-free strip with spacelike directrix if W0 = 0. Now we suppose that γ defines a force-free strip with spacelike directrix. By Theorem 2, we have W0 = a1 T + a2 N + a3 B = 0, where T, N and B are components of the Frenet frame. This implies that 2 1 2 2 a1 = κ 1+λ + μ = 0 ⇒ μ < 0. 2 We can assume that μ = −1. If we take the first derivative of a1 , we get 2 0 = κ 1 + λ2 + 2κλλ 1 + λ2 = a2 and

1

. κ= 1 + λ2

This allows one to prove the following lemma.

(35)

Lemma 2 If γ is a spacelike curve with non-constant modified torsion λ = κτ , then the following conditions are equivalent: (i) γ defines a force-free strip with spacelike

directrix, (ii) W1 = 2λT + 2λ 1 + λ2 N − ε 1 − λ2 B is a constant, (iii) a1 = 0 and < J, J > is conserved, where J = s1 T + s2 N + s3 B. Theorem 5 Let a spacelike curve γ define a force-free strip with spacelike directrix. Then, the tangent vector of γ corresponds to a spacelike elastic curve with Lagrange multiplier 1 on de Sitter 2-space S12 . Conversely for each such arclength parametrized curve T : [0, ] → S12 with geodesic curvature λ, the spacelike curve γ (t) =

t

1 + λ2 (s) T (s) ds

0

defines a force-free strip with spacelike directrix with S−1 (γ ) = 2 1 + λ2 dt = 2 (γ ) . 0

123


Proof Suppose that a spacelike curve γ defines a force-free strip with arclength parameter s. From (iii) of Lemma 2 and (35), we have 2 < W1 , W1 >= 1 + λ2 4λ2 + 1 − 6λ2 + λ4 .

(36)

1 On the other hand, if we chose the arclength t with t (s) = 1+λ 2 for the tangent vector T of γ and we write λ (t) = λ (s (t)) in (36), then by Lemma 2, we obtain

2 < W1 , W1 > +ε =ε λ (t) − 4

ε 4 ε 2 λ (t) + 1 + λ . 4 2

(37)

By means of Frenet equations (2), we see that λ is the geodesic curvature of T . In order to characterize an elastic curve of Eq. (37) on de Sitter 2-space S12 , the coefficient of λ4 must be − 41 . It means ε = 1, that is, the binormal vector of γ is a timelike vector. Then (37) reduces to 1 4 1 < W1 , W1 > +1 2 = λ (t) − λ (t) + 1 + λ2 . 4 4 2 From (28), we see the tangent vector T of γ corresponds to a spacelike elastic curve with Lagrange multiplier 1 on de Sitter 2-space S12 with radius 1. Conversely, let T be such an arclength parametrized spacelike elastic curve with geodesic curvature λ on de Sitter 2−space S12 and let T be the tangent vector of a spacelike curve γ with Frenet frame {T, N , B} in R13 . Then the Darboux trihedron of T is {N , T, −B} and derivative formulas of the Darboux frame are given by T =N , N = − T + λB,

(38)

−B = − λN . Taking into consideration a spacelike curve T which has a reparametrization with an arclength s, we have a spacelike curve t 1 + λ2 (s) T (s) ds, γ (t) = 0

where γ (t) is the arclength reparametrized curve γ˜ (s) := γ (t (s)) with ds dt = 2 1 + λ (t (s)) . By using Eq. (38), the curvature and the modified torsion of γ are found as 1 . κ= 1 + λ2 Then we find 2 1 κ 2 1 + λ2 − 1 = 0. a1 = 2

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On the other hand, for the curve γ (s) = γ (t (s)) , we obtain < J, J > +1 2 1 4 1 = λ − λ + 1+ λ2 . 4 4 2

(39)

Equation (39) shows that < J, J > is conserved, since T is a spacelike elastic curve with Lagrange multiplier 1 on S12 . Then we see from Lemma 2, the spacelike curve γ defines a force-free strip with spacelike directrix. Acknowledgements The authors want to express her/his thanks to the referees for her/his valuable comments and suggestions. This work was partially supported by the unit of Scientific Research Project Coordination of Süleyman Demirel University under Project 3356 − D1 − 12.

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