Generalization of the catenary in the dual plane

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.