Intrinsic Equations for a Relaxed Elastic Line of Second Kind in Minkowski 3-Space


Ergin Bayram, Emin Kasap




Let $\alpha$ be an arc on a connected oriented surface $S$ in Minkowski 3-space, parameterized by arc length $s$, with torsion $\tau$ and length $l$. The total square torsion $H$ of $\alpha$ is defined by $H=\int_{0}^{l}\tau^2ds$. The arc $\alpha$ is called a relaxed elastic line of second kind if it is an extremal for the variational problem of minimizing the value of $H$ within the family of all arcs of length $l$ on $S$ having the same initial point and initial direction as $\alpha$. In this study, we obtain the differential equation and boundary conditions for a relaxed elastic line of second kind on an oriented surface in Minkowski 3-space. This formulation should give a more direct and more geometric approach to questions concerning relaxed elastic lines of second kind on a surface.