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 timedependent monodromy. A frame paralleltransported along the elastica is in general only quasiperiodic, resulting in timedependent 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2002
 DOI:
 10.48550/arXiv.math/0202278
 arXiv:
 arXiv:math/0202278
 Bibcode:
 2002math......2278B
 Keywords:

 Analysis of PDEs;
 Mathematical Physics;
 35Q72 (74K10)
 EPrint:
 31 pages. Revised introduction