Đang chuẩn bị liên kết để tải về tài liệu:
Curves whose pseudo spherical indicatrices are elastic

The pseudo spherical indicatrix of a curve in Minkowski 3-space emerges as three types: The pseudo spherical tangent indicatrix, principal normal indicatrix, and binormal indicatrix of the curve. | Turk J Math (2018) 42: 3123 – 3132 © TÜBİTAK doi:10.3906/mat-1801-44 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Curves whose pseudo spherical indicatrices are elastic Ahmet YÜCESAN1 , Gözde ÖZKAN TÜKEL2 ,, Tunahan TURHAN3,∗, Department of Mathematics, Faculty of Arts and Science, Süleyman Demirel University, Isparta, Turkey 2 Department of Finance, Banking, and Insurance, Applied Sciences University of Isparta, Isparta, Turkey 3 Department of Electronics and Automation, Applied Sciences University of Isparta, Isparta, Turkey 1 Received: 15.01.2018 • Accepted/Published Online: 15.10.2018 • Final Version: 27.11.2018 Abstract: The pseudo spherical indicatrix of a curve in Minkowski 3 -space emerges as three types: the pseudo spherical tangent indicatrix, principal normal indicatrix, and binormal indicatrix of the curve. The pseudo spherical tangent, principal normal, and binormal indicatrix of a regular curve may be positioned on De Sitter 2 -space (pseudo sphere), pseudo hyperbolic 2-space, and two-dimensional null cone in terms of causal character of the curve. In this paper, we separately derive Euler–Lagrange equations of all pseudo spherical indicatrix elastic curves in terms of the causal character of a curve in Minkowski 3 -space. Then we give some results of solutions of these equations. Key words: Elastic curve, Euler–Lagrange equation, pseudo spherical indicatrix 1. Introduction An elastic curve γ minimizes the bending energy ∫ F(γ) = κ2 (s)ds (1) γ with fixed length and boundary conditions. Euler–Lagrange equations corresponding to critical points of the functional ( 1) are given by 2κ′′ + κ3 − 2κτ 2 − λκ = κτ ′ + 2κ′ τ = 0, 0, where κ and τ are curvature and torsion of γ , respectively. These equations can be solved by using Jacobi elliptic functions in Euclidean 3 -space [ 5, 12 ]. Elastic curves (spherical elastic curves) have been characterized on the sphere in Euclidean 3 -space. Brunnett and .