Generalization of the catenary in the dual plane

TL;DR

This paper extends the classical catenary concept to the dual plane by defining and analyzing $\

Contribution

It introduces $\

Findings

Explicit equations for $\

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contribution":"It introduces $\

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 $\alpha$-catenaries in $\mathbb{D}^2$ as stationary points of a potential energy functional, where $\alpha\in \mathbb{R}$ is a real parameter. We derive the corresponding Euler-Lagrange equations and obtain explicit equations of these curves for specific values of $\alpha$. Furthermore, we establish a geometric characterization of $\alpha$-catenaries in terms of their curvature and unit normal vector field.