#### Document Type

Article

#### Publication Date

2012

#### Abstract

We study geodesics of the H^{1 }Riemannian metric (see article for equation) on the space of inextensible curves (see article for equation). 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^{∞} in the Banach topology C^{1} ([0,1], R^{2}), and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one 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.

#### Recommended Citation

Preston, Stephen C. and Saxton, Ralph, "An H1 Model for Inextensible Strings" (2012). *Mathematics Faculty Publications.* Paper 19.

https://scholarworks.uno.edu/math_facpubs/19

## Comments

preprint

submitted to AIMS Journals