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Relaxed elastic line in a Riemannian manifold

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We obtain a differential equation with 2 boundary conditions for a relaxed elastic line in a Riemannian manifold. This differential equation, which is found with respect to constant sectional curvature G, geodesic curvature κ, and 2 boundary conditions, gives a more direct and more geometric approach to questions concerning a relaxed elastic line in a Riemannian manifold. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 746 – 752 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1303-38 Relaxed elastic line in a Riemannian manifold ∗ ¨ ¨ G¨ ozde OZKAN , Ahmet YUCESAN Department of Mathematics, S¨ uleyman Demirel University, 32260, Isparta, Turkey Received: 19.03.2013 • • Accepted: 23.04.2013 Published Online: 25.04.2014 • Printed: 23.05.2014 Abstract: We obtain a differential equation with 2 boundary conditions for a relaxed elastic line in a Riemannian manifold. This differential equation, which is found with respect to constant sectional curvature G , geodesic curvature κ , and 2 boundary conditions, gives a more direct and more geometric approach to questions concerning a relaxed elastic line in a Riemannian manifold. We give various theorems and results in terms of a relaxed elastic line. Consequently, we examine the concept of a relaxed elastic line in 2− and 3− dimensional space forms. Key words: Relaxed elastic line, Riemannian manifold, geodesic curvature, space forms 1. Introduction An elastic curve (or elastica), as proposed by Daniel Bernoulli to Leonhard Euler in 1744 , is the solution to a variational problem of minimizing the integral of the squared curvature ∫ ℓ κ2 (s) ds (1.1) 0 for curves of a fixed length ℓ satisfying given first-order boundary conditions, where s, 0 ≤ s ≤ ℓ, is arc length. The elastic curve was studied by David Singer in 3−dimensional Euclidean space in [7] . He used the classical techniques of the calculus of variations to derive the equations of the elastic curve. He also formulated a generalized variational problem, that of the elastic curve in a Riemannian manifold. If no boundary conditions are imposed at s = ℓ , and if no external forces act at any s , the elastic curve is relaxed. Thus, a relaxed elastic line (or curve) with fewer boundary conditions than an elastic curve is a more general solution to variational problem of elastic