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

TL;DR

This paper derives differential equations and boundary conditions for relaxed elastic lines of the second kind on surfaces in Minkowski 3-space, focusing on minimizing total square torsion within specific arc families.

Contribution

It introduces the intrinsic equations governing relaxed elastic lines of the second kind in Minkowski 3-space, a novel extension in differential geometry.

Findings

Derived differential equations for relaxed elastic lines

Established boundary conditions for these lines

Extended elastic line theory to Minkowski 3-space

Abstract

Let $\alpha $ be an arc on a connected oriented surface $S$ in Minkowski 3-space, parameterized by arc length $s$, with torsion $\tau $ and length $l$. The total square torsion $H$ of $\alpha $ is defined by $% H=\int_{0}^{l}\tau ^{2}ds$. The arc 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 in Minkowski 3-space.