Sagging Ropes Demonstrate the Transversality Conditions of Variational Problems

TL;DR

This paper demonstrates how the transversality conditions in variational problems explain the behavior of sagging ropes, including their orthogonality to guide wires and tension predictions, validated through experiments.

Contribution

It applies transversality conditions to analyze sagging ropes with free endpoints, providing new analytical equations and experimental validation.

Findings

Rope is orthogonal to guide wires at endpoints.

Transversality conditions predict rope tension accurately.

Experimental results confirm theoretical predictions.

Abstract

An account of the transversality conditions of variational problems gives rise to essential results in the analysis of different physical phenomena. This powerful and elegant approach has proven to be fruitful in a diversity of variational problems with free endpoints, when the endpoints are free to slip along preset curves. We illustrate the transversality condition by the study of a heavy inextensible rope sagging both symmetrically and asymmetrically between two steering variously-shaped guide wires without friction. In this case, the transversality conditions lead to the orthogonality of the rope to the wires at endpoints of the rope, which is confirmed experimentally. Freeing the endpoints of the rope yields exact and simple analytical equations predicting the tension of the rope. Heavy ropes whose endpoints are free to slip between variously-shaped wires are discussed.