Abstract
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.
Highlights
A relaxed elastic line is a solution of a variational problem introduced by Manning [1] in 1987 to examine mechanical features of DNA molecule
We define the generalized relaxed elastic line as a critical point of the functional which consists of the addition of twisting energy to bending energy
We study the problem of generalized relaxed elastic line on a connected oriented non-degenerate surface S in Minkowski 3-space R31
Summary
Keywords Generalized relaxed elastic line, Euler-Lagrange equations, Variational calculus. Abstract: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. Minkowski 3-uzayda bulunan herhangi bir dejenere olmamıs S yüzeyi üzerinde verilen α egrisi için egrilik ve burulma formüllerinde yer alan mutlak deger ifadesine göre iki farklı durum ortaya çıktı. Her iki durum için problemi inceledik ve sonuç olarak her iki durumda da genellestirilmisgevsetilmiselastik çizgiyi bir Euler-Lagrange denklemi ve 3 sınır sart ile karakterize ettik. Son olarak egrinin jeodezik egriligine, jeodezik burulmasına ve normal egriligine baglı olarak elde edilen diferansiyel denklem sistemi için özel çözümler arastırdık
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