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 η t t = ∂ s ( σ η s ) , σ s s − | η s s | 2 σ = − | η s t | 2 , | η s | 2 ≡ 1 with boundary conditions η ( t , 1 ) = 0 and σ ( t , 0 ) = 0 . We prove local existence and uniqueness in the space defined by the weighted Sobolev energy ∑ ℓ = 0 m ∫ 0 1 s ℓ | ∂ s ℓ η t | 2 d s + ∫ 0 1 s ℓ + 1 | ∂ s ℓ + 1 η | 2 d s , when m ⩾ 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.