On the Cauchy problem for a dynamical Euler's elastica
Abstract
The dynamics for a thin, closed loop inextensible Euler's elastica moving in three dimensions are considered. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives and a second order elliptic equation for its tension. A Hasimoto transformation is used to rewrite the equations in convenient coordinates for the space and time derivatives of the tangent vector. A feature of this elastica is that it exhibits time-dependent monodromy. A frame parallel-transported along the elastica is in general only quasi-periodic, resulting in time-dependent boundary conditions for the coordinates. This complication is addressed by a gauge transformation, after which a contraction mapping argument can be applied. Local existence and uniqueness of elastica solutions are established for initial data in suitable Sobolev spaces.
Cite
@article{arxiv.math/0202278,
title = {On the Cauchy problem for a dynamical Euler's elastica},
author = {Almut Burchard and Lawrence E. Thomas},
journal= {arXiv preprint arXiv:math/0202278},
year = {2007}
}
Comments
31 pages. Revised introduction