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Abstract

We study the flows of curves in the pseudo-Galilean 3-space and its equiform geometry without any constraints. We find that the Frenet equations and intrinsic quantities of the inelastic flows 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.
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 dierential equations which characterize the
motionsofcurvesasafamily.
e motions of curves have been widely investigated by
many authors in dierent 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 identied 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 [411].
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 deni-
tions and theorems of the pseudo-Galilean 3-space. Next we
dene 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 ane 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−21.
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)dened 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 dierential form
=,givenas
()=,(),(),(6)
where (),() ∈ 3,thecurvature()and the torsion
()are dened 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

=2t,
t+n+
n+b+
b+n
=2t,
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-
bleowsinthepseudo-GalileanspaceG1
3.
Denition 4. Acurveevolutionr(,)and its ow r/in
the pseudo-Galilean space G1
3aresaidtobeinextensibleif

r

=0. (33)
According to Denition 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 coecient of proportionality 23.Otherline
elements (,,), which lie on an isotropic plane ( =
0), are matched into proportional ones with the coecient
12.So,alllinesegmentsarematchedintoproportionalones
with the same coecient 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.
Denition 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 dene 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 (/).eowofthecurve
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
satises 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,fromDenition 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 conict of interests
regarding the publication of this paper.
Acknowledgment
e authors would like to thank the referees for the helpful
suggestions.
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