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Research Article

Motions of Curves in the Pseudo-Galilean Space G1

3

Suleyman Cengiz, Esra Betul Koc Ozturk, and Ufuk Ozturk

Department of Mathematics, Faculty of Science, C¸ankırı Karatekin University, 18100 C¸ankırı, Turkey

Correspondence should be addressed to Ufuk Ozturk; uuzturk@asu.edu

Received 28 November 2014; Revised 17 February 2015; Accepted 20 February 2015

Academic Editor: Qing-Wen Wang

Copyright © 2015 Suleyman Cengiz et al. is is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

cited.

We study the ows of curves in the pseudo-Galilean 3-space and its equiform geometry without any constraints. We nd that the

Frenet equations and intrinsic quantities of the inelastic ows of curves are independent of time. We show that the motions of

curves in the pseudo-Galilean 3-space and its equiform geometry are described by the inviscid and viscous Burgers’ equations.

1. Introduction

In mathematical modeling of many nonlinear events of the

natural and the applied sciences such as dynamics of vortex

laments, motions of interfaces, shape control of robot arms,

propagation of ame fronts, image processing, supercoiled

DNAs, magnetic uxes, deformation of membranes, and

dynamics of proteins, the motions of space curves are being

used. e evolutions of these nonlinear phenomena are

described by the dierential equations which characterize the

motionsofcurvesasafamily.

e motions of curves have been widely investigated by

many authors in dierent geometries. In 1992 Nakayama and

others explained that the close relation between the integrable

evolution equations and the motions of curves is based on

the equivalence of Frenet equations and the inverse scattering

problematzeroeigenvalue[1], so that they identied the

evolution equations that govern the 2D and 3D motions of

the curves. ey also studied the motions of the plane curves

in which the curvature obeys the mKDV equation and its

hierarchy [2]. Langer and Perline [3]gavethegeneralization

of the motions of curves to -dimensional Euclidean space.

Many well-known integrable equations or their hierarchies

relatedtothemotionsofspacecurvescanbefoundin

subsequent studies [4–11].

e subject of the curve ows in the pseudo-Galilean

space, which is a real Cayley-Klein space with projective sig-

nature, is a virgin area to be searched. Inelastic ows of curves

in the Galilean and the pseudo-Galilean spaces are studied at

[12,13]. Yoon [14] examined the inextensible ows of curves

in the equiform geometry of the Galilean 3-space. S¸ahin

[15] derived the intrinsic equations for a generalized relaxed

elastic line on an oriented surface in the Galilean space.

In this study we investigate the motions of curves in the

pseudo-Galilean 3-space and in its equiform geometry with-

out any constraints. e rst section gives the main deni-

tions and theorems of the pseudo-Galilean 3-space. Next we

dene the evolution of a one-parameter family of smooth

admissible curves in the pseudo-Galilean 3-space and nd the

ow equations of the curve evolution with use of the Frenet

equations. en we consider some particular cases where the

ow of the intrinsic quantities and induces the inviscid

Burgers’ equation. Finally we study the curve evolution in the

equiform geometry of the pseudo-Galilean 3-space regarding

the relations between the Frenet vectors of these spaces.

2. The Pseudo-Galilean Space G1

3

e pseudo-Galilean space G1

3is one of the real Cayley-

Klein spaces of projective signature (0,0,+,−)as explained

in [16]. e absolute gure of the pseudo-Galilean space G1

3

consists of an ordered triple {,,} where is the ideal

(absolute) plane, is the (absolute) line in ,andis the

xed hyperbolic involution of points of .ecurvesinG1

3

are described in [16,17].

Hindawi Publishing Corporation

Mathematical Problems in Engineering

Volume 2015, Article ID 150685, 6 pages

http://dx.doi.org/10.1155/2015/150685

2Mathematical Problems in Engineering

In the nonhomogeneous ane coordinates for points and

vectors (point pairs) the similarity group 8of G1

3has the

following form:

=11 +12

=21 +22+23cosh +23sinh

=31 +32+23sinh +23cosh , (1)

where 𝑖𝑗 and are real numbers. In particular, for 12 =23 =

1,thegroup(1) becomes the group 6⊂

8of isometries of

the pseudo-Galilean space G1

3as follows:

=11 +

=21 +22+cosh +sinh

=31 +32+sinh +cosh . (2)

Accordingtothemotiongroupinthepseudo-Galileanspace,

there are nonisotropic vectors x=(,,)(for which holds

=0) and four types of isotropic vectors: space-like ( =

0,2−2>0), time-like (=0,2−2<0),andtwotypesof

light-like vectors (=0,=±). A non-light-like isotropic

vector is a unit vector if 2−2=±1.

e scalar product of two vectors u=(1,2,3)and k=

(V1,V2,V3)can be written as

u,k=

1V1,if 1=0∨V1=0

2V2−3V3,if 1=0∧V1=0. (3)

is scalar product leaves invariant the pseudo-Galilean

norm of the vector u=(1,2,3)dened by

u=

1,if 1=0

2

2−2

3,if 1=0. (4)

Let be a spatial curve given rst by

:⊆R→ G1

3

→ ()=(),(),(), (5)

where (),(),() ∈ 3.enthecurve()is said to be

admissible if

() =0[16].Foranadmissiblecurvein G1

3

parameterized by the arc length =with dierential form

=,givenas

()=,(),(),(6)

where (),() ∈ 3,thecurvature()and the torsion

()are dened by

()= ()2− ()2(7)

()= () ()− () ()

2(),(8)

respectively. e pseudo-Galilean Frenet frame of the admis-

sible curve ()parameterizedbythearclengthhastheform

t()=()=1,(),(),

n()=1

() ()=1

()0, (), (),

b()=1

()0, (), (),

(9)

where t,n, and bare called the tangent vector, principal

normal vector, and binormal vector elds of the curve ,

respectively. Here =+1or −1is chosen by the criterion

det(t,n,b)=1.Ifnis a space-like or time-like vector, then the

curve ()given by (6) is time-like or space-like, respectively.

en the Frenet equations of the curve ()are given by

t()

n()

b()

𝑥

=

0

()0

00

()

0

()0

t()

n()

b()

,(10)

where t,n, and bare mutually orthogonal vectors [17,18].

3. Motions of Curves in

the Pseudo-Galilean Space G1

3

In this section we study the curve evolution in the pseudo-

Galilean3-spacebyusingtheFrenetframestructuretoobtain

somerelatedintegrableequations.

Let us consider a one-parameter family of smooth admis-

sible curves r(,) in the pseudo-Galilean space G1

3where

denotes the time or the scale and parameterizes each

curveofthefamily.Weassumethatthisfamilyr(,)evolves

according to the ow equation

r:= r

=(,)t+(,)n+(,)b,(11)

r(,0)=r(),(12)

where ,,are arbitrary functions.

Let

(,):=

r

=r

,r

(13)

denote the length along the curve. e arc length parameter

is given by

(,):=𝑢

0,.(14)

Mathematical Problems in Engineering 3

From (10) we can express the Frenet vectors and the intrinsic

quantities as

t:= r

=1

r

,

n:= 1

t

=1

t

,

b:= 1

n

=1

n

,

(15)

:=

t

=1

t

,

:=n,b

, (16)

respectively.

Now we will derive the ow equations for the Frenet

frame {t,n,b},themetric,thecurvature, and the torsion

for the curve evolution r(,)satisfying (12).Since2=

r/,r/taking the derivatives of both sides and using

(11) and (15) we can compute the ow of the metric as

2

=2r

,

r

=2t,

t+n+

n+b+

b+n

=2t,

t+

++n+

+b

=2

.(17)

So the ow of the metric equals

=

.(18)

It is important to notice that the variables and are

independent but and are not. As a consequence, we have

=

1

=−

+

.(19)

We can evaluate the ow equation of the unit tangent vector

tast

=

r

=−

r

+

r

=−

t+

t+n+

n+b+

b+n

=

++n+

+b.

(20)

Similarly for the ow of the unit normal vector nwe have

n

=1

++

−1

−

+

+n

+

+++1

+b.

(21)

Since n/,n=0we obtain

n

=

+++1

+b,(22)

=

++−

+

+. (23)

Also the ow of the binormal vector bbecomes

b

=

+++1

+n

+1

+++1

+

−1

−

b.(24)

From the equation b/,b=0we obtain

b

=

+++1

+n,(25)

=

+++1

+−

.

(26)

Since t,n=0and t,b=0we have

t

,n+t,n

=0,

t

,b+t,b

=0. (27)

en by (20),(22),and(25) we can write

++=0,

+=0. (28)

Hence the ow equations of the Frenet frame take the form

t

=0,

n

=0,

b

=0,

(29)

4Mathematical Problems in Engineering

and for the intrinsic quantities the ow equations become

=−

,

=−

.(30)

erefore, we have the following theorem.

eorem 1. Let r=r(,) be a one-parameter family of

smooth admissible curves in the pseudo-Galilean space G1

3.If

revolves according to (11), then, the Frenet frame {t,n,b}of r

is not time dependent and the intrinsic quantities and of r

satisfy the equations

=−

,

=−

,(31)

where is the arc length parameter of r.

Remark 2. Burgers’ equations describe various kinds of phe-

nomena such as a mathematical model of turbulence and the

approximate theory of ow through a shock wave traveling in

viscous uid. e inviscid Burgers’ equation is a model for the

nonlinear wave propagation, especially in uid mechanics. It

takes the form

+

=0, (32)

where (,)is a solution of the equation.

From Remark 2,ifwechoosethecurvature=or the

torsion =in (30), then we have that the intrinsic quantities

and evolve according to the inviscid Burgers’ equation. So,

we obtain the following corollary.

Corollary 3. Let r=r(,)be a curve evolution in the pseudo-

Galilean space G1

3with the intrinsic quantities and given by

(11).Ifonesets=or =, then the intrinsic quantities

and satisfy the inviscid Burgers’ equation.

3.1. Inextensible Curve Flows in the Pseudo-Galilean Space. In

this section, we investigate some properties of the inextensi-

bleowsinthepseudo-GalileanspaceG1

3.

Denition 4. Acurveevolutionr(,)and its ow r/in

the pseudo-Galilean space G1

3aresaidtobeinextensibleif

r

=0. (33)

According to Denition 4 and (11),incasethefamilyof

curves r(,)is inextensible, from (18) we get

=0, (,)=()(34)

forsomesinglevariablefunction. erefore, we have the

following corollary.

Corollary 5. e curve evolution r(,)which is given by (11)

isinextensibleifandonlyif/=0.

If we now restrict ourselves to the arc length parame-

terized admissible curves that undergo purely inextensible

deformations, that is, (,) = () = 1and / = 0,

then the local coordinate corresponds to the arc length

parameter . us the ow of the curve is expressed as

r:= r

=(,)t+(,)n+(,)b(35)

and the ow of the Frenet frame {t,n,b}with the intrinsic

quantities and is given by

t

=0,

n

=0,

b

=0,

=0,

=0.

(36)

So, we get the following corollary.

Corollary 6. Let r=r(,)beacurveevolutioninthepseudo-

Galilean space G1

3with its ow r/given by (11).Ifthecurve

ow r(,)is inextensible, then the Frenet vectors {t,n,b},the

curvature , and the torsion are not time dependent.

4. Motions of Curves in

the Equiform Geometry of G1

3

Similarity group (1) matches an ordinary (formal) line ele-

ment (=0,,)in a pseudo-Euclidean plane (i.e., =

const.) into a segment of length proportional to the original

one with the coecient of proportionality 23.Otherline

elements (,,), which lie on an isotropic plane ( =

0), are matched into proportional ones with the coecient

12.So,alllinesegmentsarematchedintoproportionalones

with the same coecient of proportionality if and only if

12 =

23.enweobtainasubgroup7⊂

8which

preserves length ratio of segments and angles between planes

and lines, respectively. is group is called the group of

equiform transformations of the pseudo-Galilean space.

Denition 7. Geometry of the pseudo-Galilean space G1

3

induced by the 7-parameter equiform group 7is called the

equiform geometry of the space G1

3.

Let :→G3be an admissible curve with the arc length

parameter . We dene the equiform invariant parameter of

by

=1

, (37)

Mathematical Problems in Engineering 5

where =1/is the radius of the curvature of the curve .It

follows that

=1

.(38)

We then have the new equiform invariant Frenet equations as

T()

N()

B()

𝜎

=

10

0

0

T()

N()

B()

,(39)

where

is called the equiform curvature and

is called the

equiform torsion of the curve [12]. ese are related to the

curvature and torsion by the equations

=−𝑠

2,

=

.(40)

Also the equiformly invariant Frenet vectors T,N, and Bare

related to the pseudo-Galilean Frenet vectors t,n, and bas

T=t

=t,

N=n

=n,

B=b

=b.

(41)

e equiformly invariant arc length parameter of the curve

evolution r(,)canbedenedasafunctionofby

()=𝑢

01

,.(42)

So the operator /is equal to (/).eowofthecurve

evolution r(,)canbeexpressedintheform

r

=T+N+B,(43)

where ,,andare arbitrary functions. e preceding

ow of r(,) is related to ow (11) in the pseudo-Galilean

space G1

3as r

=t+n+b,(44)

with =,=,and=.enusingtheformulas

in Section 3 we obtain the ow of the metric

=

=

=

+

(45)

or

=

=

+

. (46)

e partial derivatives / and / do not commute in

general while the partials /and /commute:

=−

+

− 1

+

.(47)

Using (41) and (29) the ow equation of the equiformly

invariant tangent vector eld Tis calculated as

T

=

t

=−1

T

=−

+

T.

(48)

Similarly, we can write the ows of the equiformly invariant

principal normal and binormal vector elds, the equiform

curvature, and the equiform torsion, respectively, as follows:

N

=

n=−

+

N,(49)

B

=

n=−

+

B,(50)

=−2

+

−

+

, (51)

=0. (52)

erefore, we obtain the following theorem.

eorem 8. Let r=r(,) be an admissible curve in the

equiform geometry of G1

3with the equiform invariant Frenet

frame (39).Ifrevolves according to (43), then the ows of

(i) the equiform invariant Frenet vectors T,N,and Bof r

are, respectively, given as

T

=−

+

T,

N

=−

+

N,

B

=−

+

B,

(53)

(ii) the equiform curvature

and the equiform torsion

of r

are, respectively, given as

=−2

+

−

+

,

=0, (54)

where istheequiforminvariantparameterandis an

arbitrary function.

Remark 9. Viscous Burgers’ equation can be regarded as

a one-dimensional analog of the Navier-Stokes equations

which model the behavior of viscous uids. It is given by the

equation

+

=V2

2,(55)

where (,)is a solution of the equation.

6Mathematical Problems in Engineering

From Remark 9,ifwechoose(/) +

= (1/

2)(

/)in (51),thenweseethattheintrinsicquantity

evolves according to the viscous Burgers’ equation. So, we

have the following corollary.

Corollary 10. Let r=r(,)be an equiform invariant curve

evolutionintheequiformgeometryofG1

3with the intrinsic

quantity

given by (39).Iftheequality/ +

=

(1/2)(

/)holds, then the intrinsic quantity

satises the

viscous Burgers’ equation.

4.1. Inextensible Curve Flows in the Equiform Geometry of G1

3.

In this section, we investigate some properties of the inex-

tensible ows in the equiform geometry of G1

3.

Let r(,)be an inextensible curve evolution in the equi-

form geometry of G1

3given by (43).en,fromDenition 4,

we have

+

=0 (56)

and from this equation we get

=

,(57)

where is an integration constant. So, we get the following

corollary.

Corollary 11. e curve evolution r(,),whichisgivenby

(43), is inextensible if and only if =/for some integra-

tion constant .

From eorem 8 and Corollary 11,wehavethefollowing

corollary.

Corollary 12. If the curve evolution r(,),whichisgiven

by (43), is inextensible, then the Frenet vectors {T,N,B},the

curvature

, and the torsion

of rare not time dependent.

Conflict of Interests

e authors declare that there is no conict of interests

regarding the publication of this paper.

Acknowledgment

e authors would like to thank the referees for the helpful

suggestions.

References

[1] K. Nakayama, H. Segur, and M. Wadati, “Integrability and the

motion of curves,” Physical Review Letters,vol.69,no.18,pp.

2603–2606, 1992.

[2] K. Nakayama and M. Wadati, “Motion of curves in the plane,”

JournalofthePhysicalSocietyofJapan,vol.62,no.2,pp.473–

479, 1993.

[3] J. Langer and R. Perline, “Curve motion inducing modied

Korteweg-de Vries systems,” Physics Letters A,vol.239,no.1-2,

pp. 36–40, 1998.

[4] R. E. Goldstein and D. M. Petrich, “e Korteweg-de Vries

hierarchy as dynamics of closed curves in the plane,” Physical

Review Letters,vol.67,no.23,pp.3203–3206,1991.

[5] M. G ¨

urses, “Motion of curves on two-dimensional surfaces and

soliton equations,” Physics Letters A,vol.241,no.6,pp.329–334,

1998.

[6] K. Nakayama, “Motion of curves in hyperboloids in the Mink-

owski space II,” JournalofthePhysicalSocietyofJapan,vol.68,

no.10,pp.3214–3218,1999.

[7] W. K. Schief and C. Rogers, “Binormal motion of curves of con-

stant curvature and torsion. Generation of soliton surfaces,” e

Royal Society of London. Proceedings. Series A. Mathematical,

Physical and Engineering Sciences, vol. 455, no. 1988, pp. 3163–

3188, 1999.

[8] K.-S. Chou and C. Qu, “e KdV equation and motion of plane

curves,” JournalofthePhysicalSocietyofJapan,vol.70,no.7,pp.

1912–1916, 2001.

[9] K.-S. Chou and C. Qu, “Integrable equations arising from

motions of plane curves,” Physica D: Nonlinear Phenomena,vol.

162,no.1-2,pp.9–33,2002.

[10] K.-S. Chou and C. Qu, “Integrable motions of space curves in

ane geometry,” Chaos, Solitons & Fractals,vol.14,no.1,pp.

29–44, 2002.

[11] K.-S. Chou and C. Qu, “Motions of curves in similarity geom-

etries and Burgers-mKdV hierarchies,” Chaos, Solitons and

Fractals,vol.19,no.1,pp.47–53,2004.

[12] Z. Erjavec and B. Divjak, “e equiform dierential geometry

of curves in the pseudo-Galilean space,” Mathematical Commu-

nications,vol.13,no.2,pp.321–332,2008.

[13] A. O. ¨

O˘

grenmis¸, M. Yenero˘

glu, and M. K¨

ulahcı, “Inelastic

admissible curves in the pseudo-Galilean space 1

3,” Interna-

tionalJournalofOpenProblemsinComputerScienceandMath-

ematics,vol.4,no.3,pp.199–207,2011.

[14] D. W. Yoon, “Inelastic ows of curves according to equiform in

Galilean space,” Journal of the Chungcheong Mathematical Soci-

ety,vol.24,no.4,pp.665–673,2011.

[15] T. S¸ahin, “Intrinsic equations for a generalized relaxed elastic

line on an oriented surface in the Galilean space,” Acta Mathe-

matica Scientia. Series B. English Edition,vol.33,no.3,pp.701–

711, 2013.

[16] B. Divjak and ˇ

Z. Milin-ˇ

Sipuˇ

s, “Special curves on ruled surfaces

in Galilean and pseudo-Galilean spaces,” Acta Mathematica

Hungarica,vol.98,no.3,pp.203–215,2003.

[17] B. Divjak, “Curves in pseudo-Galilean geometry,” Annales

Universitatis Scientiarum Budapestinensis de Rolando E¨

otv¨

os

Nominatae. Sectio Mathematica,vol.41,pp.117–128,1998.

[18] B. Divjak and ˇ

Z. Milin-ˇ

sipuˇ

s, “Minding isometries of ruled

surfaces in pseudo-Galilean space,” Journal of Geometry,vol.77,

no. 1-2, pp. 35–47, 2003.

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