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

Abstract

Let α be an arc on a connected oriented surface S in Minkowski 3-space, parameterized by arc length s, with torsion τ and length l. The total square torsion H of α is defined by H=∫₀^{l}τ²ds. The arc α 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 α. 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.