Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
5/2001

Global curvature and self-contact of nonlinearly elastic curves and rods

Oscar Gonzalez, John H. Maddocks, Friedemann Schuricht and Heiko von der Mosel

Abstract

Many different physical systems, e.g. super-coiled DNA molecules, have been successfully modelled as elastic curves, ribbons or rods. We will describe all such systems as framed curves, and will consider problems in which a three dimensional framed curve has an associated energy that is to be minimized subject to the constraint of there being no self-intersection. For closed curves the knot type may therefore be specified a priori. Depending on the precise form of the energy and imposed boundary conditions, local minima of both open and closed framed curves often appear to involve regions of self-contact, that is, regions in which points that are distant along the curve are close in space. While this phenomenon of self-contact is familiar through every day experience with string, rope and wire, the idea is surprisingly difficult to define in a way that is simultaneously physically reasonable, mathematically precise, and analytically tractable. Here we use the notion of global radius of curvature of a space curve in a new formulation of the self-contact constraint, and exploit our formulation to derive existence results for minimizers, in the presence of self-contact, of a range of elastic energies that define various framed curve models. As a special case we establish the existence of ideal shapes of knots.

Received:
May 18, 2001
Published:
May 18, 2001
MSC Codes:
49J99, 53A04, 57M25, 74B20, 92C40

Related publications

inJournal
2002 Repository Open Access
Oscar Gonzalez, John H. Maddocks, Friedemann Schuricht and Heiko von der Mosel

Global curvature and self-contact of nonlinearly elastic curves and rods

In: Calculus of variations and partial differential equations, 14 (2002) 1, pp. 29-68