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Generalization of the catenary in the dual plane

Muhittin Evren Aydin, Rafael López

Abstract

In this paper, we study a dual analogue of the classical catenary within the class of admissible curves in the dual plane $\mathbb{D}^2$. We introduce $α$-catenaries in $\mathbb{D}^2$ as stationary points of a potential energy functional, where $α\in \mathbb{R}$ is a real parameter. We derive the corresponding Euler-Lagrange equations and obtain explicit equations of these curves for specific values of $α$. Furthermore, we establish a geometric characterization of $α$-catenaries in terms of their curvature and unit normal vector field.

Generalization of the catenary in the dual plane

Abstract

In this paper, we study a dual analogue of the classical catenary within the class of admissible curves in the dual plane . We introduce -catenaries in as stationary points of a potential energy functional, where is a real parameter. We derive the corresponding Euler-Lagrange equations and obtain explicit equations of these curves for specific values of . Furthermore, we establish a geometric characterization of -catenaries in terms of their curvature and unit normal vector field.
Paper Structure (3 sections, 9 theorems, 66 equations)

This paper contains 3 sections, 9 theorems, 66 equations.

Table of Contents

  1. Introduction and Formulation of The Problem
  2. The Euler-Lagrange equation and examples
  3. A characterization of catenaries in $\mathbb D^2$

Key Result

Proposition 2.1

Let $\alpha\in \mathbb R$, $\alpha\neq0$. An admissible curve $\gamma:[a,b]\to\mathbb D^2$, $\gamma=\psi+\varepsilon \phi$, is an $\alpha$-catenary in $\mathbb D^2$ if and only if $\psi$ is an $\alpha$-catenary in $\mathbb R^2$ and $\mathcal{E}_1'[\psi,\phi]=0$.

Theorems & Definitions (20)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 10 more