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Cengiz, Suleyman; Koc Ozturk, Esra Betul; Ozturk, Ufuk

doi:https://doi.org/10.1155/2015/150685

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2015 | Article ID 150685 | https://doi.org/10.1155/2015/150685

Motions of Curves in the Pseudo-Galilean Space

Suleyman Cengiz,¹Esra Betul Koc Ozturk,¹and Ufuk Ozturk¹

Academic Editor: Qing-Wen Wang

Received28 Nov 2014

Revised17 Feb 2015

Accepted20 Feb 2015

Published26 Feb 2015

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.

1. Introduction

In mathematical modeling of many nonlinear events of the natural and the applied sciences such as dynamics of vortex filaments, motions of interfaces, shape control of robot arms, propagation of flame fronts, image processing, supercoiled DNAs, magnetic fluxes, deformation of membranes, and dynamics of proteins, the motions of space curves are being used. The evolutions of these nonlinear phenomena are described by the differential equations which characterize the motions of curves as a family.

The motions of curves have been widely investigated by many authors in different 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 problem at zero eigenvalue [1], so that they identified the evolution equations that govern the 2D and 3D motions of the curves. They also studied the motions of the plane curves in which the curvature obeys the mKDV equation and its hierarchy [2]. Langer and Perline [3] gave the generalization of the motions of curves to -dimensional Euclidean space. Many well-known integrable equations or their hierarchies related to the motions of space curves can be found in subsequent studies [4–11].

The subject of the curve flows in the pseudo-Galilean space, which is a real Cayley-Klein space with projective signature, is a virgin area to be searched. Inelastic flows of curves in the Galilean and the pseudo-Galilean spaces are studied at [12, 13]. Yoon [14] examined the inextensible flows of curves in the equiform geometry of the Galilean 3-space. Şahin [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 without any constraints. The first section gives the main definitions and theorems of the pseudo-Galilean 3-space. Next we define the evolution of a one-parameter family of smooth admissible curves in the pseudo-Galilean 3-space and find the flow equations of the curve evolution with use of the Frenet equations. Then we consider some particular cases where the flow 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

The pseudo-Galilean space is one of the real Cayley-Klein spaces of projective signature as explained in [16]. The absolute figure of the pseudo-Galilean space consists of an ordered triple where is the ideal (absolute) plane, is the (absolute) line in , and is the fixed hyperbolic involution of points of . The curves in are described in [16, 17].

In the nonhomogeneous affine coordinates for points and vectors (point pairs) the similarity group of has the following form: where and are real numbers. In particular, for , the group (1) becomes the group of isometries of the pseudo-Galilean space as follows: According to the motion group in the pseudo-Galilean space, there are nonisotropic vectors (for which holds ) and four types of isotropic vectors: space-like , time-like , and two types of light-like vectors . A non-light-like isotropic vector is a unit vector if .

The scalar product of two vectors and can be written as This scalar product leaves invariant the pseudo-Galilean norm of the vector defined by Let be a spatial curve given first by where . Then the curve is said to be admissible if [16]. For an admissible curve in parameterized by the arc length with differential form , given as where , the curvature and the torsion are defined by respectively. The pseudo-Galilean Frenet frame of the admissible curve parameterized by the arc length has the form where , and are called the tangent vector, principal normal vector, and binormal vector fields of the curve , respectively. Here or is chosen by the criterion . If is a space-like or time-like vector, then the curve given by (6) is time-like or space-like, respectively. Then the Frenet equations of the curve are given by where , and are mutually orthogonal vectors [17, 18].

3. Motions of Curves in the Pseudo-Galilean Space

In this section we study the curve evolution in the pseudo-Galilean 3-space by using the Frenet frame structure to obtain some related integrable equations.

Let us consider a one-parameter family of smooth admissible curves in the pseudo-Galilean space where denotes the time or the scale and parameterizes each curve of the family. We assume that this family evolves according to the flow equation where are arbitrary functions.

Let denote the length along the curve. The arc length parameter is given by From (10) we can express the Frenet vectors and the intrinsic quantities as respectively.

Now we will derive the flow equations for the Frenet frame , the metric , the curvature , and the torsion for the curve evolution satisfying (12). Since taking the derivatives of both sides and using (11) and (15) we can compute the flow of the metric as So the flow of the metric equals It is important to notice that the variables and are independent but and are not. As a consequence, we have We can evaluate the flow equation of the unit tangent vector as Similarly for the flow of the unit normal vector we have Since we obtain Also the flow of the binormal vector becomes From the equation we obtain Since and we have Then by (20), (22), and (25) we can write Hence the flow equations of the Frenet frame take the form and for the intrinsic quantities the flow equations become Therefore, we have the following theorem.

Theorem 1. Let be a one-parameter family of smooth admissible curves in the pseudo-Galilean space . If evolves according to (11), then, the Frenet frame of is not time dependent and the intrinsic quantities and of satisfy the equations where is the arc length parameter of .

Remark 2. Burgers’ equations describe various kinds of phenomena such as a mathematical model of turbulence and the approximate theory of flow through a shock wave traveling in viscous fluid. The inviscid Burgers’ equation is a model for the nonlinear wave propagation, especially in fluid mechanics. It takes the form where is a solution of the equation.

From Remark 2, if we choose the curvature 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 be a curve evolution in the pseudo-Galilean space with the intrinsic quantities and given by (11). If one sets 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 inextensible flows in the pseudo-Galilean space .

Definition 4. A curve evolution and its flow in the pseudo-Galilean space are said to be inextensible if

According to Definition 4 and (11), in case the family of curves is inextensible, from (18) we get for some single variable function . Therefore, we have the following corollary.

Corollary 5. The curve evolution which is given by (11) is inextensible if and only if .

If we now restrict ourselves to the arc length parameterized admissible curves that undergo purely inextensible deformations, that is, and , then the local coordinate corresponds to the arc length parameter . Thus the flow of the curve is expressed as and the flow of the Frenet frame with the intrinsic quantities and is given by So, we get the following corollary.

Corollary 6. Let be a curve evolution in the pseudo-Galilean space with its flow given by (11). If the curve flow is inextensible, then the Frenet vectors , the curvature , and the torsion are not time dependent.

4. Motions of Curves in the Equiform Geometry of

Similarity group (1) matches an ordinary (formal) line element in a pseudo-Euclidean plane (i.e., .) into a segment of length proportional to the original one with the coefficient of proportionality . Other line elements , which lie on an isotropic plane , are matched into proportional ones with the coefficient . So, all line segments are matched into proportional ones with the same coefficient of proportionality if and only if . Then we obtain a subgroup which preserves length ratio of segments and angles between planes and lines, respectively. This group is called the group of equiform transformations of the pseudo-Galilean space.

Definition 7. Geometry of the pseudo-Galilean space induced by the 7-parameter equiform group is called the equiform geometry of the space .

Let be an admissible curve with the arc length parameter . We define the equiform invariant parameter of by where is the radius of the curvature of the curve . It follows that We then have the new equiform invariant Frenet equations as where is called the equiform curvature and is called the equiform torsion of the curve [12]. These are related to the curvature and torsion by the equations Also the equiformly invariant Frenet vectors , and are related to the pseudo-Galilean Frenet vectors , and as The equiformly invariant arc length parameter of the curve evolution can be defined as a function of by So the operator is equal to . The flow of the curve evolution can be expressed in the form where , and are arbitrary functions. The preceding flow of is related to flow (11) in the pseudo-Galilean space as with , , and . Then using the formulas in Section 3 we obtain the flow of the metric or The partial derivatives and do not commute in general while the partials and commute: Using (41) and (29) the flow equation of the equiformly invariant tangent vector field is calculated as Similarly, we can write the flows of the equiformly invariant principal normal and binormal vector fields, the equiform curvature, and the equiform torsion, respectively, as follows: Therefore, we obtain the following theorem.

Theorem 8. Let be an admissible curve in the equiform geometry of with the equiform invariant Frenet frame (39). If evolves according to (43), then the flows of
(i) the equiform invariant Frenet vectors , and of are, respectively, given as (ii) the equiform curvature and the equiform torsion of are, respectively, given as where is the equiform invariant parameter and is 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 fluids. It is given by the equation where is a solution of the equation.

From Remark 9, if we choose in (51), then we see that the intrinsic quantity evolves according to the viscous Burgers’ equation. So, we have the following corollary.

Corollary 10. Let be an equiform invariant curve evolution in the equiform geometry of with the intrinsic quantity given by (39). If the equality holds, then the intrinsic quantity satisfies the viscous Burgers’ equation.

4.1. Inextensible Curve Flows in the Equiform Geometry of

In this section, we investigate some properties of the inextensible flows in the equiform geometry of .

Let be an inextensible curve evolution in the equiform geometry of given by (43). Then, from Definition 4, we have and from this equation we get where is an integration constant. So, we get the following corollary.

Corollary 11. The curve evolution , which is given by (43), is inextensible if and only if for some integration constant .

From Theorem 8 and Corollary 11, we have the following corollary.

Corollary 12. If the curve evolution , which is given by (43), is inextensible, then the Frenet vectors , the curvature , and the torsion of are not time dependent.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for the helpful suggestions.

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Copyright © 2015 Suleyman Cengiz et al. This 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.

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