AN H1 MODEL FOR INEXTENSIBLE STRINGS

We study geodesics of the H-1 Riemannian metric << u, v >> = integral(1)(0) < u(s), v(s)> + alpha(2) < u'(s), v'(s)> ds on the space of inextensible curves gamma : [0, 1] -> R-2 with vertical bar gamma'vertical bar 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-infinity in the Banach topology C-1 ([0, 1], 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.