abstract - Universidade Estadual de Maringá · PDF fileCharacterization Of Inextensible...
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Bol. Soc. Paran. Mat. (3s.) v. 31 2 (2013): 47–53. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v31i2.15957 Characterization Of Inextensible Flows Of Spacelike Curves With Sabban Frame In S 2 1 Mahmut Ergüt, Essin Turhan and Talat Körpınar abstract: In this paper, we study inextensible flows of spacelike curves on S 2 1 . We obtain partial differential equations in terms of inextensible flows of spacelike curves according to Sabban frame on S 2 1 . Key Words:Inextensible flows, Sabban Frame. Contents 1 Introduction 47 2 Preliminaries 47 3 Inextensible Flows of Curves According to Sabban Frame in S 2 1 48 1. Introduction Physically, inextensible curve and surface flows give rise to motions in which no strain energy is induced. The swinging motion of a cord of fixed length, for example, or of a piece of paper carried by the wind, can be described by inextensible curve and surface flows. Such motions arise quite naturally in a wide range of physical applications. This study is organised as follows: Firstly, we study inextensible flows of space- like curves on S 2 1 . Secondly, we obtain partial differential equations in terms of inextensible flows of spacelike curves according to Sabban frame on S 2 1 . 2. Preliminaries The Minkowski 3-space E 3 provided with the standard flat metric given by 〈, 〉 = −dx 2 1 + dx 2 2 + dx 2 3 , where (x 1 ,x 2 ,x 3 ) is a rectangular coordinate system of E 3 1 . Recall that, the norm of an arbitrary vector a ∈ E 3 1 is given by ‖a‖ = 〈a,a〉.γ is called a unit speed curve if velocity vector v of γ satisfies ‖a‖ =1. Denote by {T, N, B} the moving Frenet–Serret frame along the spacelike curve γ in the space E 3 1 . For an arbitrary spacelike curve γ with first and second curvature, 2000 Mathematics Subject Classification: 53A35 47 Typeset by B S P M style. c Soc. Paran. de Mat.
Transcript of abstract - Universidade Estadual de Maringá · PDF fileCharacterization Of Inextensible...
Bol. Soc. Paran. Mat. (3s.) v. 31 2 (2013): 47–53.c©SPM –ISSN-2175-1188 on line ISSN-00378712 in press
SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v31i2.15957
Characterization Of Inextensible Flows Of Spacelike Curves With
Sabban Frame In S2
1
Mahmut Ergüt, Essin Turhan and Talat Körpınar
abstract: In this paper, we study inextensible flows of spacelike curves on S2
1.
We obtain partial differential equations in terms of inextensible flows of spacelikecurves according to Sabban frame on S2
1.
Key Words: Inextensible flows, Sabban Frame.
Contents
1 Introduction 47
2 Preliminaries 47
3 Inextensible Flows of Curves According to Sabban Frame in S2
148
1. Introduction
Physically, inextensible curve and surface flows give rise to motions in which nostrain energy is induced. The swinging motion of a cord of fixed length, for example,or of a piece of paper carried by the wind, can be described by inextensible curveand surface flows. Such motions arise quite naturally in a wide range of physicalapplications.
This study is organised as follows: Firstly, we study inextensible flows of space-like curves on S
2
1. Secondly, we obtain partial differential equations in terms of
inextensible flows of spacelike curves according to Sabban frame on S2
1.
2. Preliminaries
The Minkowski 3-space E3 provided with the standard flat metric given by
〈, 〉 = −dx2
1+ dx2
2+ dx2
3,
where (x1, x2, x3) is a rectangular coordinate system of E3
1. Recall that, the norm
of an arbitrary vector a ∈ E3
1is given by ‖a‖ =
√
〈a, a〉. γ is called a unit speedcurve if velocity vector v of γ satisfies ‖a‖ = 1.
Denote by {T,N,B} the moving Frenet–Serret frame along the spacelike curveγ in the space E
3
1. For an arbitrary spacelike curve γ with first and second curvature,
2000 Mathematics Subject Classification: 53A35
47Typeset by B
SPM
style.c© Soc. Paran. de Mat.
48 Mahmut Ergüt, Essin Turhan and Talat Körpınar
κ and τ in the space E3
1, the following Frenet–Serret formulae is given
T′ = κN
N′ = −κT + τB (2.1)
B′ = τN,
where
〈T,T〉 = 1, 〈N,N〉 = 1, 〈B,B〉 = −1,
〈T,N〉 = 〈T,B〉 = 〈N,B〉 = 0.
Here, curvature functions are defined by κ = κ(s) = ‖T′(s)‖ and τ(s) =−
⟨
N,B′⟩
.Torsion of the spacelike curve γ is given by the aid of the mixed product
τ =[γ′, γ′′, γ′′′]
κ2.
Now we give a new frame different from Frenet frame. Let α : I −→ S2
1be
unit speed spherical curve. We denote σ as the arc-length parameter of α . Let usdenote t (σ) = α′ (σ) , and we call t (σ) a unit tangent vector of α. We now set avector s (σ) = α (σ)× t (σ) along α. This frame is called the Sabban frame of α onS
2
1. Then we have the following spherical Frenet-Serret formulae of α :
α′ = t,
t′ = −α− κgs, (2.2)
s′ = −κgt,
where κg is the geodesic curvature of the spacelike curve α on the S2
1and
g (t, t) = 1, g (α, α) = 1, g (s, s) = −1,
g (t, α) = g (t, s) = g (α, s) = 0.
3. Inextensible Flows of Curves According to Sabban Frame in S2
1
Let α (u, t) is a one parameter family of smooth spacelike curves in S2
1.
The arclength of α is given by
σ(u) =
u∫
0
∣
∣
∣
∣
∂α
∂u
∣
∣
∣
∣
du, (3.1)
where∣
∣
∣
∣
∂α
∂u
∣
∣
∣
∣
=
∣
∣
∣
∣
⟨
∂α
∂u,∂α
∂u
⟩∣
∣
∣
∣
1
2. (3.2)
Characterization Of Inextensible Flows 49
The operator∂
∂σis given in terms of u by
∂
∂σ=
1
ν
∂
∂u,
where v =
∣
∣
∣
∣
∂α
∂u
∣
∣
∣
∣
and the arclength parameter is dσ = vdu.
Any flow of α can be represented as
∂α
∂t= fS
1α+ fS
2t + fS
3s. (3.3)
Letting the arclength variation be
σ(u, t) =
u∫
0
vdu.
In the S2
1the requirement that the curve not be subject to any elongation or
compression can be expressed by the condition
∂
∂tσ(u, t) =
u∫
0
∂v
∂tdu = 0, (3.4)
for all u ∈ [0, l] .
Definition 3.1. The flow∂α
∂tin S
2
1are said to be inextensible if
∂
∂t
∣
∣
∣
∣
∂α
∂u
∣
∣
∣
∣
= 0.
Lemma 3.2. Let∂α
∂t= fS
1α+ fS
2t + fS
3s be a smooth flow of the spacelike curve α.
The flow is inextensible if and only if
∂v
∂t−∂fS
2
∂u= fS
1v − fS
3vkg. (3.5)
Proof: Suppose that∂α
∂tbe a smooth flow of the spacelike curve α.
From (3.3), we obtain
v∂v
∂t=
⟨
∂α
∂u,∂
∂u
(
fS1α+ fS
2t + fS
3s)
⟩
.
By the formula of the Sabban, we have
∂v
∂t=
⟨
t,
(
∂fS1
∂u− fS
2v
)
α+
(
fS1v +
∂fS2
∂u− fS
3vκg
)
t +
(
∂fS3
∂u− fS
2vκg
)
s
⟩
.
50 Mahmut Ergüt, Essin Turhan and Talat Körpınar
Making necessary calculations from above equation, we have (3.5), which provesthe lemma.
2
Theorem 3.3. Let∂α
∂t= fS
1α + fS
2t + fS
3s be a smooth flow of the spacelike curve
α. The flow is inextensible if and only if
∂fS2
∂u= fS
3vκg − fS
1v. (3.7)
Proof: From (3.4), we have
∂
∂tσ(u, t) =
u∫
0
∂v
∂tdu =
u∫
0
(
fS1v +
∂fS2
∂u− fS
3vκg
)
du = 0. (3.8)
Substituting (3.5) in (3.8) complete the proof of the theorem.2
We now restrict ourselves to arc length parametrized curves. That is, v = 1and the local coordinate u corresponds to the curve arc length σ. We require thefollowing lemma.
Lemma 3.4. Let∂α
∂t= fS
1α+ fS
2t + fS
3s be a smooth flow of the spacelike curve α.
Then,
∂t
∂t=
(
∂fS1
∂σ− fS
2
)
α+
(
∂fS3
∂σ− fS
2κg
)
s, (3.9)
∂α
∂t= −
(
∂fS1
∂σ− fS
2
)
t + ψs, (3.10)
∂s
∂t=
(
∂fS3
∂σ− fS
2κg
)
t − ψα, (3.11)
where ψ =
⟨
∂α
∂t, s
⟩
.
Proof: Using definition of α, we have
∂t
∂t=
∂
∂t
∂α
∂σ=
∂
∂σ(fS
1α+ fS
2t + fS
3s).
Using the Sabban equations, we have
∂t
∂t=
(
∂fS1
∂σ− fS
2
)
α+
(
fS1
+∂fS
2
∂σ− fS
3κg
)
t +
(
∂fS3
∂σ− fS
2κg
)
s. (3.12)
Characterization Of Inextensible Flows 51
On the other hand, substituting (3.7) in (3.12), we get
∂t
∂t=
(
∂fS1
∂σ− fS
2
)
α+
(
∂fS3
∂σ− fS
2κg
)
s.
Since, we express :
∂fS1
∂σ− fS
2+
⟨
t,∂α
∂t
⟩
= 0,
−∂fS
3
∂σ+ fS
2κg +
⟨
t,∂s
∂t
⟩
= 0,
ψ +
⟨
α,∂s
∂t
⟩
= 0.
Then, a straight forward computation using above system gives
∂α
∂t= −
(
∂fS1
∂σ− fS
2
)
t + ψs,
∂s
∂t=
(
∂fS3
∂σ− fS
2κg
)
t − ψα,
where ψ =
⟨
∂α
∂t, s
⟩
. Thus, we obtain the theorem.
2
The following theorem states the conditions on the curvature and torsion forthe flow to be inextensible.
Theorem 3.5. Let∂α
∂tis inextensible. Then, the following system of partial dif-
ferential equations holds:
∂κg
∂σ− ψ =
∂
∂σ(fS
2κg) −
∂2fS3
∂σ2,
κgψ =∂2fS
1
∂σ2−∂fS
2
∂σ.
Proof:
Using (3.9), we have
∂
∂σ
∂t
∂t=
∂
∂σ
[(
∂fS1
∂σ− fS
2
)
α+
(
∂fS3
∂σ− fS
2κg
)
s
]
=
(
∂2fS1
∂σ2−∂fS
2
∂σ
)
α+ [
(
∂fS1
∂σ− fS
2
)
−κg
(
∂fS3
∂σ+ fS
2κg
)
]t
+
(
∂2fS3
∂σ2−
∂
∂σ(fS
2κg)
)
s.
52 Mahmut Ergüt, Essin Turhan and Talat Körpınar
On the other hand, from Sabban frame we have
∂
∂t
∂t
∂σ=
∂
∂t(−α−κgs)
= (ψ −∂κg
∂σ)s + [
(
∂fS1
∂σ− fS
2
)
+ κg
(
∂fS3
∂σ− fS
2κg
)
]t + κgψα.
Hence we see that∂κg
∂σ− ψ =
∂
∂σ(fS
2κg) −
∂2fS3
∂σ2.
and
κgψ =∂2fS
1
∂σ2−∂fS
2
∂σ.
Thus, we obtain the theorem. 2
Theorem 3.6. Let∂α
∂t= fS
1α + fS
2t + fS
3s be a smooth flow of the spacelike curve
α. Then,
κg
(
∂fS1
∂σ− fS
2
)
=
(
∂fS3
∂σ+ fS
2κg
)
+∂ψ
∂σ.
Proof:
Similarly, we have
∂
∂σ
∂s
∂t=
∂
∂σ
[(
∂fS3
∂σ− fS
2κg
)
t − ψα
]
= [
(
∂2fS3
∂σ−
∂
∂σ
(
fS2κg
)
− ψ
)
t−κg
(
∂fS3
∂σ− fS
2κg
)
s
+[−
(
∂fS3
∂σ+ fS
2κg
)
−∂ψ
∂σ]α].
On the other hand, a straightforward computation gives
∂
∂t
∂s
∂σ=
∂
∂t(−κgt)
= −∂κg
∂tt − κg
[(
∂fS1
∂σ− fS
2
)
α+
(
∂fS3
∂σ− fS
2κg
)
s
]
.
Combining these we obtain the corollary. 2
In the light of Theorem 3.6, we express the following corollary without proof:
Corollary 3.7.
ψ =∂2fS
3
∂σ−
∂
∂σ
(
fS2κg
)
+∂κg
∂t.
Characterization Of Inextensible Flows 53
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Mahmut Ergüt
Fırat University,
Department of Mathematics,
23119 Elazığ, Turkey
E-mail address: [email protected]
and
Essin Turhan
Fırat University,
Department of Mathematics,
23119 Elazığ, Turkey
E-mail address: [email protected]
and
Talat Körpınar
Fırat University,
Department of Mathematics,
23119 Elazığ, Turkey
E-mail address: [email protected]
