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The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold



Nemat Abazari and Yusuf Yayli
 
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ABSTRACT

In this study the mathematical idealization of the classical variational problem in 3-dimensional indefinite-Riemannian Manifolds is studied for the curve α which is timelike and spacelike, parameterized by the arc-length. The geodesic curvature and torsion of an elastic curve are evaluated if they exist as the solutions of the differential equations for all different cases. Due to elastic curve definition, the minimum principle theorem is applied to elastic energy function which is defined as the integral of the squared geodesic curvature of the curve.

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  How to cite this article:

Nemat Abazari and Yusuf Yayli, 2012. The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold. Journal of Applied Sciences, 12: 1303-1307.

DOI: 10.3923/jas.2012.1303.1307

URL: https://scialert.net/abstract/?doi=jas.2012.1303.1307
 
Received: November 15, 2011; Accepted: April 24, 2012; Published: June 30, 2012



INTRODUCTION

The elastic curve is one of the important topics in geometry that appear in computation of the solution of a variational problems which for the first time was considered by Daniel Bernoulli and Leonhard Euler in 1744. The main goal of their study was minimizing the bending energy of a thin inextensible wire. Some additional information about their works is available in work of Love (1927), Hsu (2007), Yanti and Mahlia (2009), Akanmu and Gambo (2007, 2008). The major idea of this variational problem is minimizing the energy function defined as the integral of the squared curvature for a curve of a fixed length subject to boundary conditions (Barros et al., 1999).

According to Koiso (1992), for a curve in a Riemannian manifold, we can define two quantities: the length and the total square curvature of curve. Also a curve is called as an elastica if it is a critical point of the functional total square curvature restricted to the space of curves of a fixed length.

One of essential work in elastica is classified the all closed elastic curves in the Euclidean space, (Langer and Singer, 1984). Also, Koiso (1992) found the unique long time solution of an initial value problem in Euclidean space and studied the elastic curves restricted in a submanifold. Barros et al. (1999) studied the complete classification of elastic curves in complex projective plane.

Singer (2007), studied the elastica in Euclidean space and classified the elastic curves in a Riemannian manifold with constant sectional curvature G. The cross sectional curvature G = 0 has been studied by Sager et al. (2011). The aim of this paper was to study the classical variational problem in the 3-dimensional indefinite-Riemannian manifold.

PRELIMINARIES

Let Mγ be a 3-dimensional indefinite-Riemannian manifold of index γ (0≤γ≤3) isometrically immersed into an m-dimensional indefinite-Riemannian manifold Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold of index i for m≥3. Then is Mγ called 3-dimensional indefinite-Riemannian submanifold of Mi. Especially if γ = 1, then M1 is called a Lorentzian submanifold of Mi (Lopez, 2008). We denote the metrics of Mγ and Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold by the symbol 〈,〉 and the covariant differentiation of Mγ (resp. Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold) by ∇ (resp.Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold). Then we have the Gauss formula:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(1)

where, X and Y are tangent vector fields of Mγ and B is the second fundamental form of Mγ.

Let α(t) be a regular curve on a 3-dimensional indefinite-Riemannian manifold Mγ. We denote the tangent vector field α’(t) = X. When 〈X, Y〉 = ±1, α is called a unit speed curve.

THE ELASTICA IN A 3-DIMENSIONAL INDEFINITE-RIEMANNIAN MANIFOLD

This study has formulated a generalized variational problem, that of the elastica in 3-dimensional indefinite-Riemannian manifold. By this we mean a curve which is an extremal for the integral of the squared (geodesic) curvature among curves with specified boundary conditions. Here, we summarize the machinery needed for calculations.

In what follows, M is a smooth 3-dimensional indefinite-Riemannian manifold, with indefinite-Riemannian metric g (X, Y) = 〈X, Y〉 = x1y1+x2y2-x3y3, where X = (x1, x2, x3) and Y = (y1, y2, y3), that is, a symmetric bilinear form on tangent vectors X and Y at each point. The covariant derivative ∇XY, measures the derivative of a vector field Y in the direction of a vector X.

Definition: A vector field V is called spacelike if 〈V,V〉>0 or V = 0, timelike if 〈V,V〉<0 and lightlike if 〈V,V〉 = 0 and V≠0.

For vector fields X and Y the equality of mixed partial derivatives is replaced by the bracket formula:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Let α(t) be an immersed curve in M, then it has velocity vector V = vT and squared geodesic curvature:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Set the Frenet frame for a family of curves αw(t) = g(w,t) by (T, N, B), therefore we can write:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

where, V is velocity, Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold is speed, W represents an infinitesimal variation of the curve and s is the arc-length parameter along a curve.

The basic formulas needed in calculating the Euler equations are as follows:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(2)

So:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(3)

So:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(4)

Here, the curvature tensor R is given by:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

The proof of Eq. 6, is mentioned in Singer (2007).

In what follows, α[0,1]→M is a curve of length L. Now for fixed constant λ let:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

For a variation αW with variation field W we compute:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

One of the symmetries of the curvature tensor allows us to replace 〈R(W, T)T, ∇TT〉 with 〈R(∇TT, T)T, W〉. Now integrate by parts, using g = -〈∇TW, T〉, we get:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

where:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

and:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

THE FRENET EQUATIONS

Let α be a curve in 3-dimensional indefinite-Riemannian manifold M with speed v(t) = |α’(t)|, curvature k, torsion τ and Frenet frame {T, N, B}. The Frenet equations are written down as follows (Fernandez et al., 2006):

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

where, ε1 is 〈T, T〉, ε2 is 〈N, N〉 and ε3 is 〈B, B〉.

The timelike case: Let α(t) be a timelike curve in 3-dimensional indefinite-Riemannian manifold M. If the normal vector field N and the binormal vector field B are spacelike, then we have the following Frenet formulas along α(t):

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(5)

where, κ and τ are curvature and torsion of α, respectively.

Theorem 1: Let α(t) be a timelike curve in the smooth 3-dimensional indefinite-Riemannian manifold M, then α(t) is the elastic curve if and only if the curvature κ, torsion τ and sectional curvature G of α being as follows:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

where, c1, c2 are constants and:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Proof: Since M is a manifold of sectional curvature G, the formula for E can be simplified to:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Then:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(6)

The equations E = 0 for the elastica become:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Solving above system, we get:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

where, c1, c2 are constants and:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Conversely, by substituting the κ, τ and G in the Eq. 6, we can get E = 0, therefore, α is an elastic curve.

The spacelike case: Let α(t) be a spacelike curve in M. There are three possibilities depending on the causal character of ∇TT, which are given, respectively, in the following theorems:

Theorem 2: Let α(t) be a spacelike elastic curve in the smooth 3-dimensional indefinite-Riemannian manifold M with a constant sectional curvature G and covariant differentiation ∇. Suppose that the vector field ∇TT be a spacelike, then the curvature is Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold and the torsion is Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold where c, m, p, q and r are constants and s is arc-length.

Proof: The Frenet equations for α are as follows:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(7)

where, κ and τ are curvature and torsion of α, respectively. In this case, the formula for E can be simplified to:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

where, s is the arc-length.

The equations E = 0 for the elastica become:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

The second equation integrates to:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

where, c is constant. Multiplication of the first equation by 2κs and integration yields:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(8)

where, A is undetermined constant. Letting u = κ2, this becomes:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(9)

Since, this equation is of the form u2s = p (u) and P a third degree polynomial, it can be solved by standard techniques in terms of elliptic functions (Jing-Lei and Zhi-Jian, 2011; Abazari, 2011; Koklu, 2002). The cubic polynomial P (u) satisfies P (0) = 4c2≥0 and limu→±∞ P (u) = ±∞. Furthermore, if u = κ2 is a nonconstant solution to (9), it must obviously take on values at which P (u)>0. It follows that we may assume P(u) has three real roots, given in two cases:

Case I : 1,-α23, satisfying -α1≤α2≤0≤α3
Case II : α123 satisfying 0≤α1≤α2≤α3

We can now write Eq. 9 in the form:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(10)

The solution of Eq. 10 is given by:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(11)

where:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(12)

For background on the solution of such equations, (Davis, 1962). Also, α1, α2 and α3 are related to the coefficients of P(u) by:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(13)

where, the up and down symbol are related to case I and case II, respectively.

Theorem 3: Let α(t) be a spacelike curve in the smooth 3-dimensional indefinite-Riemannian manifold M with a covariant differentiation ∇. Suppose that the vector field ∇TT be a timelike, then α(t) is the elastic curve if and only if the curvature κ, torsion τ and sectional curvature G of α is as follows:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

where, c1, c2 are constants and:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Proof: The Frenet equations for α are as follows:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(14)

where, κ and τ are curvature and torsion of α, respectively. In this case, the formula for E can be simplified to:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(15)

The equations E= 0 for the elastica become:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Integrates of first and third equation yields:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Therefore, G = c where:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Conversely, by substituting the κ, τ and G in the Eq. 15, we can get E = 0, therefore, α is an elastic curve.

Theorem 4: Let α(t) be a spacelike curve in the smooth 3-dimensional indefinite-Riemannian manifold M with a constant sectional curvature G and covariant differentiation ∇. Suppose that the vector field ∇TT be a lightlike, then the curvature is κ = 1 and the torsion is:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(16)

where:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

and c is integration constant.

Proof: The vector field ∇TT is lightlike. Then, similar on previous case, the Frenet equations for α are as follows:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(17)

where, κ = 1 and τ is torsion of α. In this case, the formula for E can be simplified to:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

The equations E = 0 for the elastica become:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

or:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold
(18)

where:

Image for - The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

Since, d is a constant then the general solution of Eq. 18 is obtained as Eq. 16.

ACKNOWLEDGMENTS

The authors would like to thank Prof. Dr. H.H. Hacisalihoglu for his careful reading of the first draft and many helpful suggestions. Also the first author would like to thanks the Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran, for its financial support.

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11:  Sager, I., N. Abazari, E. Ekmekci and Y. Yayli, 2011. The classical elastic curves in lorentz-minkowski space. Int. J. Contemp. Math. Sci., 62: 309-320.
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12:  Lopez, R., 2008. Differential geometry of curves and surfaces in Lorentz-Minkowski space. Department of Geometry and Topology, University of Granada, Granada, Spain. http://arxiv.org/pdf/0810.3351.pdf.

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14:  Davis, H.T., 1962. Introduction to Nonlinear Differential and Integral Equations. Dover Publication Inc., New York, USA., ISBN-13: 9780486609713, Pages: 566.

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16:  Abazari, R., 2011. Numerical simulation of coupled nonlinear schrodinger equation by RDTM and comparison with DTM. J. Applied Sci., 11: 3454-3463.
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