Sagging ropes demonstrate transversality conditions of variational problems

We present an account of transversality conditions of variational problems and show how they give rise to essential results in the analysis of different physical phenomena. We find such conditions to be a powerful and elegant approach to a diverse number of variational problems with free endpoints. In this work, we illustrate such condition with the study of a heavy inextensible rope sagging both symmetrically and asymmetrically between two variously shaped, steering-guide wires without friction. In such a situation, the transversality conditions lead to the orthogonality of the rope to the wires at endpoints of the rope, which we confirm experimentally. Freeing the endpoints of the rope yields exact analytical equations that predict the tension of the rope. Heavy ropes whose endpoints are free to slip between arbitrarily shaped, steering wires are discussed.