An $H^1$ model for inextensible strings
Stephen C. Preston  Department of Mathematics, University of Colorado, Boulder, CO 803090395, United States (email) Abstract: We study geodesics of the $H^1$ Riemannian metric $$ « u,v » = ∫_0^1 ‹ u(s), v(s)› + α^2 ‹ u'(s), v'(s)› ds$$ on the space of inextensible curves $\gamma\colon [0,1]\to\mathbb{R}^2$ with $ γ'≡ 1$. This metric is a regularization of the usual $L^2$ metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The $H^1$ geodesic equation represents a limiting case of the PochhammerChree equation from elasticity theory. We show the geodesic equation is $C^{\infty}$ in the Banach topology $C^1([0,1], \mathbb{R}^2)$, and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one endpoint of the curves fixed, we have globalintime solutions. We conclude with some surprising features in the periodic case, along with an analogy to the equations of incompressible fluid mechanics.
Keywords: Inextensible strings, Riemannian geometry, infinitedimensional manifolds.
Received: November 2011; Revised: February 2012; Available Online: December 2012. 
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