We study a relaxed elastic line in the general case on an oriented surface. In particular, we obtain a differential equation with three boundary conditions for a generalized relaxed elastic line. Then we analyze the results in a plane, on a sphere, on a cylinder, and on the geodesics of these surfaces.
Similar content being viewed by others
References
M. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ (1976).
G. S. Manning, “Relaxed elastic line on a curved surface,” Quart. Appl. Math., 45, No. 3, 515–527 (1987).
H. K. Nickerson and G. S. Manning, “Intrinsic equations for a relaxed elastic line on an oriented surface,” Geom. Dedicata, 27, No. 2, 127–136 (1988).
D. A. Singer, “Lectures on elastic curves and rods,” AIP Conf. Proc., 1002, 3–32 (2008).
R. Weinstock, Calculus of Variations: with Applications to Physics and Engineering, McGraw-Hill, New York (1952).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 8, pp. 1121–1131, August, 2012.
Rights and permissions
About this article
Cite this article
Yücesan, A., Özkan, G. Generalized relaxed elastic line on an oriented surface. Ukr Math J 64, 1277–1289 (2013). https://doi.org/10.1007/s11253-013-0715-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0715-8