The Hanging Rope: A Convex Optimization Problem in the Calculus of Variations

Abstract: We study the problem first introduced by Verma and Keller in 1984 of how to taper a heavy rope such that its elongation is minimized. The problem is stated as an optimization problem of a functional J[w]. Specifically we provide a proof of optimality for the solution using traditional convex optimization techniques. We also utilize the Legendre transformation when studying the Euler–Lagrange equation — this is nice because it sheds some light on the structure of the solution in a natural way. In the last section we consider a similar problem but where the functional J is a function of itself; J = J[w,J]. This problem is unfortunately not solved but might be subject to future research.