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An $H^1$ model for inextensible strings

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  • 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 Pochhammer-Chree 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 global-in-time solutions. We conclude with some surprising features in the periodic case, along with an analogy to the equations of incompressible fluid mechanics.
    Mathematics Subject Classification: 35G31, 35B44, 35B45, 35B65, 35Q74, 58J47, 58J90.


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