Calculus of Variations and Partial Differential Equations

, Volume 14, Issue 1, pp 29–68

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

  • O. Gonzalez
  • J.H. Maddocks
  • F. Schuricht
  • H. von der Mosel
Original article

DOI: 10.1007/s005260100089

Cite this article as:
Gonzalez, O., Maddocks, J., Schuricht, F. et al. Calc Var (2002) 14: 29. doi:10.1007/s005260100089


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.

Mathematics Subject Classification (2000): 49J99, 53A04, 57M25, 74B20, 92C40

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • O. Gonzalez
    • 1
  • J.H. Maddocks
    • 2
  • F. Schuricht
    • 3
  • H. von der Mosel
    • 4
  1. 1.Department of Mathematics, The University of Texas, Austin, TX 78712, USA (e-mail:
  2. 2.Département de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (e-mail:
  3. 3.Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22–26, 04103 Leipzig, Germany (e-mail:
  4. 4.Mathematics Institute, University of Bonn, Beringstrasse 4, 53115 Bonn, Germany (e-mail: