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

TL;DR

This paper derives differential equations and boundary conditions for relaxed elastic lines of the second kind on oriented surfaces, focusing on minimizing total torsion within specific arc families.

Contribution

It introduces the differential equations and boundary conditions characterizing relaxed elastic lines of second kind on oriented surfaces.

Findings

Derived the differential equation for relaxed elastic lines of second kind.

Established boundary conditions for these elastic lines.

Provided a mathematical framework for analyzing torsion-minimizing curves.

Abstract

Let {\alpha}(s) be an arc on a connected oriented surface S in E3, parameterized by arc length s, with torsion {\tau} and length l. The total square torsion F of {\alpha} is defined by T=\int_{0}^{l}\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 F 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.