TY  - JOUR
TT  - A Study on a Generalized Relaxed Curvature Energy Action
AU  - Özkan Tükel, Gözde
AU  - Yücesan, Ahmet
PY  - 2018
DA  - October
JF  - Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi
JO  - J. Nat. Appl. Sci.
PB  - Süleyman Demirel Üniversitesi
WT  - DergiPark
SN  - 1308-6529
SP  - 541
EP  - 546
VL  - 22
KW  - Generalized relaxed elastic line
KW  - Euler-Lagrange equations; Variational calculus
N2  - 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.
CR  - [1] Manning, G.S., 1987. Relaxed Elastic Line on a Curved Surface. Quart. Appl. Math., 45(3) 515-527.
CR  - [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.
CR  - [3] Yücesan, A., Özkan, G., 2012. Generalized Relaxed Elastic Line on an Oriented Surface. Ukranian Mathematical Journal, 64(8) 1121-1131.
CR  - [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.
CR  - [5] Lopez, R., 2014. Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space. International Electronic Journal of Geometry, 7(1) 44-107.
CR  - [6] O’Neill, B., 1993. Semi-Riemannian Geometry with Applications to Relativity. Academic Pres Inc., New York, 466p.
CR  - [7] Weinstock, R., 1952. Calculus of Variations with Application to Physics and Engineering. Dover Publications, Inc., New York, 326p.
CR  - [8] Özkan, G., Yücesan A., 2014. Relaxed Hyperelastic Curves on a Non-degenerate Surface. Mediterr. J. Math., 11 1241-1250.
UR  - https://dergipark.org.tr/tr/pub/sdufenbed/issue//470099
L1  - https://dergipark.org.tr/tr/download/article-file/553005
ER  -