The motion of whips and chains

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.Comment: 47 pages, 8 figure

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

“…This observation has been used by Preston to induce a Riemannian metric on the shape space of unparametrized curves, via metrics on the space of arclength parametrized curves; see [109,110]. A similar approach has been chosen in [74].…”

Stationary Strings in Air Flows and a Non-Linear Boundary Value Problem

“…The local existence and uniqueness of solution was proved for initial data near to a stationary state in [6] by using a hard implicit function theorem. The local existence and uniqueness was also proved in [5] by approximating with a discrete system. In those papers, however, the influence of air flows was not considered and the bending rigidity of string was neglected.…”

“…The motion of strings and chains was studied in [2], [6], [7] and [5]. The Bessel-Fourier expansion was e¤ectively used in [2] and [7].…”

Generalized solutions for inextensible string equations

“…Even though the analysis of the dynamics of inextensible strings subject to different kinds of boundary conditions is a notable problem which goes back to Galileo, Leibniz and Bernoulli (cf. [23,4,19]), and it has been investigated by many authors in various contexts (see e.g. [6,15,14,26,31]), there are still very few results about general wellposedness.…”

“…One of the existence results available is by Reeken [24,25] who proves well-posedness for an infinite string with gravity when the initial data is near the trivial (downwards vertical) stable stationary solution (close in H 26 ). Another one is due to Preston [19] who considers (1.1) in the absence of gravity with the whip boundary conditions (1.7). He obtains local existence and uniqueness in weighted Sobolev spaces for which the energy is bounded.…”