A Study on a Generalized Relaxed Curvature Energy Action

764 255


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 $\alpha $ 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.

Anahtar kelimeler

Generalized relaxed elastic line; Euler-Lagrange equations; Variational calculus

Tam metin:



[1] Manning, G.S., 1987. Relaxed Elastic Line on a Curved Surface. Quart. Appl. Math., 45(3) 515-527.

[2] Nickerson, H.K., Manning G.S., 1988. Intrinsic Equations for a Relaxed Elastic Line on an Oriented Surface. Geom. Dedicata, 27(2) 127-136.

[3] Yücesan, A., Özkan, G., 2012. Generalized Relaxed Elastic Line on an Oriented Surface. Ukranian Mathematical Journal, 64(8) 1121-1131.

[4] Akutagawa, K., Nishikawa S., 1990. The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space. Tohoku Math. J., 42 67-82.

[5] Lopez, R., 2014. Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. International Electronic Journal of Geometry, 7(1) 44-107.

[6] O’Neill, B., 1993. Semi-Riemannian Geometry with Applications to Relativity. Academic Pres Inc., New York, 466p.

[7] Weinstock, R., 1952. Calculus of Variations with Application to Physics and Engineering. Dover Publications, Inc., New York, 326p.

[8] Özkan, G., Yücesan A., 2014. Relaxed Hyperelastic Curves on a Non-degenerate Surface. Mediterr. J. Math., 11 1241-1250.