Geometry of whips and chains

TL;DR

This paper explores the geometric properties of whips and chains, demonstrating how discrete chains approximate continuous whips and their configuration spaces, with implications for modeling fluids.

Contribution

It establishes the convergence of chain motions and curvature to those of whips, providing a geometric framework for discrete approximations of continuous systems.

Findings

Chain motions converge to whip motions

Curvature of chain configuration space converges to that of whip

Speculation on fluid approximation by discrete systems

Abstract

We study the geometry of the inextensible string (the whip) and its discrete approximation (the chain). In the absence of gravity, both motions represent geodesic motions on certain manifolds. We show how the motion of the chain converges to that of a whip, and how the curvature of the chain's configuration space converges to that of the whip's configuration space. Finally we speculate on the analogous approximation of an incompressible fluid by a discrete system.