Intrinsic equations for a relaxed elastic line of second kind on an oriented surface

Let alpha(s) be an arc on a connected oriented surface S in E-3, parameterized by arc length s, with torsion tau and length l. The total square torsion H of alpha is defined by H = integral(l)(0) tau(2) ds. 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 differential equation and boundary conditions for a relaxed elastic line of second kind on an oriented surface.