Abstract
Using the calculus of variations this paper derives the general equation for the "sliding catenary curve" — a hanging chain with terminal links free to slide along two poles, one tilted and one vertical. By applying physical assumptions along with the Euler-Lagrange equation, the Beltrami identity, the Legendre-Clebsch condition, the transversality condition, Lagrange multipliers, and the isoperimetric constraint, we derive the general equation for the sliding catenary curve through a functional that measures the potential energy of the hanging chain. This general equation is then compared to a real-life construction of a sliding catenary curve. Additionally the paper explores a more general case involving left-boundaries for the sliding catenary curve and provides Desmos activities as demonstrations for the reader.
Recommended Citation
Shade, Ethan
(2024)
"Derivation of the Sliding Catenary Curve via Calculus of Variations,"
Electronic Proceedings of Undergraduate Mathematics Day: Vol. 8, Article 1.
Available at:
https://ecommons.udayton.edu/epumd/vol8/iss1/1