Intrinsic equations for a relaxed elastic line of second kind on an oriented surface (Q2802577)

Let \(\alpha(s)\) be an arc on an oriented surface \(S\) parametrized by arc length. The authors define the total square torsion of \(\alpha\) by \(H=\int_{0}^{1} \tau^2ds,\) where \(\tau\) is the torsion of \(\alpha.\) The arc \(\alpha\) is called a \textit{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 \(\alpha\). The aim of the paper is to obtain the differential equation and four boundary conditions for a relaxed elastic line of second kind.