科研之友
微信
新浪微博
Facebook
分享链接
摘要
We study geodesics of the H-1 Riemannian metric %26lt;br%26gt;%26lt;%26lt; u, v %26gt;%26gt; = integral(1)(0) %26lt; u(s), v(s)%26gt; + alpha(2) %26lt; u%26apos;(s), v%26apos;(s)%26gt; ds %26lt;br%26gt;on the space of inextensible curves gamma : [0, 1] -%26gt; R-2 with vertical bar gamma%26apos;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.
- 出版日期2013-5
