Intrinsic Equations for a Relaxed Elastic Line of Second Kind on an Oriented Surface

Abstract

Let α(s) be an arc on a connected oriented surface S in E3, parameterized by arc length s, with torsion t and length l. The total square torsion H of α is defined by H =?0lt2ds. 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 differential equation and boundary conditions for a relaxed elastic line of second kind on an oriented surface. © 2016 World Scientific Publishing Company.