The motion of whips and chains

TL;DR

This paper analyzes the mathematical motion of a whip modeled as an inextensible string, proving local existence, uniqueness, and persistence of smooth solutions using a discrete approximation approach.

Contribution

It establishes the local well-posedness and smoothness persistence for the whip's motion equations, employing a novel method of lines with a chain of pendula approximation.

Findings

Proved local existence and uniqueness of solutions.

Demonstrated persistence of smooth solutions over time.

Validated the method of lines approach with discrete pendula model.

Abstract

We study the motion of an inextensible string (a whip) fixed at one point in the absence of gravity, satisfying the equations $$ \eta_{tt} = \partial_s(\sigma \eta_s), \qquad \sigma_{ss}-\lvert \eta_{ss}\rvert^2 = -\lvert \eta_{st}\rvert^2, \qquad \lvert \eta_s\rvert^2 \equiv 1 $$ with boundary conditions $\eta(t,1)=0$ and $\sigma(t,0)=0$. We prove local existence and uniqueness in the space defined by the weighted Sobolev energy $$ \sum_{\ell=0}^m \int_0^1 s^{\ell} \lvert \partial_s^{\ell}\eta_t\rvert^2 \, ds + \int_0^1 s^{\ell+1} \lvert \partial_s^{\ell+1}\eta\rvert^2 \, ds, $$ when $m\ge 3$. In addition we show persistence of smooth solutions as long as the energy for $m=3$ remains bounded. We do this via the method of lines, approximating with a discrete system of coupled pendula (a chain) for which the same estimates hold.