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 $${\eta }_{tt}={\partial }_s(\sigma {\eta }_s),{\sigma }_{ss}-|{\eta }_{ss}{|}^2=-|{\eta }_{st}{|}^2,|{\eta }_s{|}^2\equiv 1$$ 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 $$\sum _{\ell =0}^m{\int }_0^1{s}^{\ell }|{\partial }_s^{\ell }{\eta }_t{|}^{2}ds+{\int }_0^1{s}^{\ell +1}|{\partial }_s^{\ell +1}\eta {|}^{2}ds,$$ 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.

47 pages, 8 figures