Intrinsic equations for a relaxed elastic line of second kind in Minkowski 3-space

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 $$. The arc 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 differential equation and boundary conditions for a relaxed elastic line of second kind on an oriented surface in Minkowski 3-space.