The Classical Elastic Curves in Lorentz-Minkowski Space

Filomat, 2015

Let ? be an arc on a connected oriented surface S in Minkowski 3-space, parameterized by arc length s, with torsion ? and length l. The total square torsion H of ? is defined by H = ?10 ?2ds. The arc ? is called a relaxed elastic line of second kind if it is an extremal for the variational problem of minimizing the value of H within the family of all arcs of length l on S having the same initial point and initial direction as ?. In this study, we obtain the differential equation and boundary conditions for a relaxed elastic line of second kind on an oriented surface in Minkowski 3-space. This formulation should give a more direct and more geometric approach to questions concerning relaxed elastic lines of second kind on a surface.

Physics Letters B, 1990

We give a complete description of timelike relativistic elastica, non-geodesic spacetime curves that solve the Euler-Lagrange equations for a lagrangian that depends on the square of the acceleration of the curve as well as its lorentzian length.

International Journal of Analysis and Applications, 2018

In this paper, we firstly introduce kinematics properties of the moving particle lying on a surface S. We assume that the particle corresponds to a different type of surface curves such that they are characterized by using the Darboux vector field W in Minkowski spacetime. Based on this result, we present geometrical understanding of the energy of the particle in each Darboux vector fields whether they lie on a spacelike surface or a timelike surface. Then, we also determine the bending elastic energy functional for the same particle on a surface S by assuming the particle has a bending feature of elastica. Finally, we prove that bending energy formula can be represented by the energy of the particle in each Darboux vector field W.

2011

Existence of acceleration pole points in special Frenet and Bishop motions for spacelike curve with a spacelike binormal in Minkowski 3-space E 1 are dependence into that, the curve α is not a general helix or planar. The ratio of torsion and curvature is by taking as a constant or non constant in our study. Then we show that, if the ratio of curvatures is constant, then there is not acceleration pole points of motion. AMS subject classifications. Primary 53A04; 53A17

International electronic journal of geometry, 2018

We present a variational study of finding null relaxed elastic lines which are extremals of a geometric energy functional, subject to suitable constraints and boundary conditions on a timelike surface in Minkowski 3-space. We derive an Euler-Lagrange equation for a null relaxed elastic line with regard to geodesic curvature, geodesic torsion and normal curvature of the curve on the timelike surface. Finally, we give some examples for null relaxed elastic lines on the pseudo-sphere and pseudo-cylinder.

Turkish Journal of Mathematics, 2000

We gived the intrinsic equations for a relaxed elastic line on an oriented surface in Ê 3 1 ([1],[2]). In this paper, we derived the intrinsic equations for a relaxed elastic line on an oriented time-like hypersurface and space-like hypersurface in the Minkowski space Ê n 1 and gived additional results about relaxed elastic lines on various timelike and spacelike hypersurface in the Minkowski space Ê n 1 .

Czechoslovak Journal of Physics - CZECH J PHYS, 2002

We show that the trajectory of a point charge in a uniform electromagnetic field is a helix if the Lorentz equation governs its motion. Our approach is totally relativistic, and it is based on the use of the Frenet-Serret formulae which describe the intrinsic geometry of world lines in Minkowski spacetime.

Mathematics and Statistics, 2020

In a theory of space curves, especially, a helix is the most elementary and interesting topic. A helix, moreover, pays attention to natural scientists as well as mathematicians because of its various applications, for example, DNA, carbon nanotube, screws, springs and so on. Also there are many applications of helix curve or helical structures in Science such as fractal geometry, in the fields of computer aided design and computer graphics. Helices can be used for the tool path description , the simulation of kinematic motion or the design of highways, etc. The problem of the determination of parametric representation of the position vector of an arbitrary space curve according to the intrinsic equations is still open in the Euclidean space E 3 and in the Minkowski space E 3 1. In this paper, we introduce some characterizations of a non-null slant helix which has a spacelike or timelike axis in E 3 1. We use vector differential equations established by means of Frenet equations in Minkowski space E 3 1. Also, we investigate some differential geometric properties of these curves according to these vector differential equations. Besides, we illustrate some examples to confirm our findings.

Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2018

We investigate the variational problem of the generalized relaxed elastic line defined as the problem of finding critical points of the functional obtained by adding the twisting energy to the bending energy functional, on a non-degenerate surface in Minkowski 3-space. There arise two different situations for the curve α given on any non-degenerate surface S in Minkowski 3-space according to the absolute value expression in the curvature and torsion formulas. We study the problem for both cases and as a result we characterize the generalized relaxed elastic line with an Euler-Lagrange equation and 3 boundary conditions in both cases. Finally, we search special solutions for the differential equation system obtained with regard to the geodesic curvature, geodesic torsion and normal curvature of the curve.

TURKISH JOURNAL OF MATHEMATICS, 2020

In this paper, the parametric expressions of spacelike and timelike curves with constant weighted curvature for some cases of a and b in Lorentz-Minkowski plane with density e ax+by are obtained.