A Convex Optimization Approach to the Discrete Hanging Chain Problem

TL;DR

This paper models the discrete hanging chain problem with variable link properties as a convex optimization problem, simplifying it to a single nonlinear equation under symmetry, offering a new computational approach.

Contribution

It introduces a convex optimization framework for the discrete hanging chain problem with arbitrary link properties, reducing the problem to a single nonlinear equation in symmetric cases.

Findings

Shape of the chain derived from convex optimization

Reduction to a single nonlinear equation for symmetric links

Provides a computational method for chain shape analysis

Abstract

In this paper we investigate the discrete version of the classical hanging chain problem. We generalize the problem, by allowing for arbitrary mass and length of each link. We show that the shape of the chain can be obtained by solving a convex optimization problem. Then we use optimality conditions to show that the problem can be further reduced to solving a single non-linear equation, when the links of the chain have symmetric mass and length.