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 geodesicequations 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.
Highlights
To ensure that η satisfies the inextensibility constraint |ηs| ≡ 1, where α > 0 is some given parameter
|ηt(t, s)|2 + α2|ηts(t, s)|2 ds dt under the constraint |ηs(t, s)| ≡ 1, and they trace out geodesics in the infinitedimensional manifold A of inextensible curves with the corresponding Riemannian metric
We will show that a weak formulation of equations (1)–(2) give a C∞ ordinary differential equation on A, and there is a C∞ Riemannian exponential map which maps initial conditions (γ0, v0) ∈ T A to time-one solutions (η(1), ηt(1)). (We demonstrate that if the data is only piecewise C1, solutions will remain piecewise C1 with jumps at the same locations, and that if the initial data is C2 or smoother we obtain classical solutions of (1)–(2).) As a consequence we obtain existence of minimizing geodesics between sufficiently close curves, and we can use curvature computations rigorously to understand stability of solutions
Summary
Follow this and additional works at: https://scholarworks.uno.edu/math_facpubs Part of the Applied Mathematics Commons. Recommended Citation Preston, Stephen C. and Saxton, Ralph, "An H1 Model for Inextensible Strings" (2012).
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