Tohoku Mathematical Journal

Kirchhoff elastic rods in a Riemannian manifold

Satoshi Kawakubo

Full-text: Open access

Abstract

Imagine a thin elastic rod like a piano wire. We consider the situation that the elastic rod is bent and twisted and both ends are welded together to form a smooth loop. Then, does there exist a stable equilibrium? In this paper, we generalize the energy of uniform symmetric Kirchhoff elastic rods in the $3$-dimensional Euclidean space to consider such a variational problem in a Riemannian manifold. We give the existence and regularity of minimizers of the energy in a compact or homogeneous Riemannian manifold.

Article information

Source
Tohoku Math. J. (2), Volume 54, Number 2 (2002), 179-193.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113247562

Digital Object Identifier
doi:10.2748/tmj/1113247562

Mathematical Reviews number (MathSciNet)
MR1904948

Zentralblatt MATH identifier
1011.58008

Subjects
Primary: 58E10: Applications to the theory of geodesics (problems in one independent variable)
Secondary: 74G60: Bifurcation and buckling 74K10: Rods (beams, columns, shafts, arches, rings, etc.)

Citation

Kawakubo, Satoshi. Kirchhoff elastic rods in a Riemannian manifold. Tohoku Math. J. (2) 54 (2002), no. 2, 179--193. doi:10.2748/tmj/1113247562. https://projecteuclid.org/euclid.tmj/1113247562


Export citation

References

  • S. S. Antman, Ordinary differential equations of nonlinear elasticity, II, Existence and regularity theory for conservative boundary value problem, Arch. Rational Mech. Anal. 61 (1976), 353--393.
  • N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1958.
  • I. M. Gel'fand and S. V. Fomin, Calculus of variations, Prentice-Hall, 1963.
  • T. Ivey and D. Singer, Knot types, homotopies and stability of closed elastic rods, Proc. London Math. Soc. (3) 79 (1999), 429--450.
  • S. Kawakubo, Stability and bifurcation of circular Kirchhoff elastic rods, Osaka J. Math. 37 (2000), 93--137.
  • S. Kawakubo, Errata to ``Stability and bifurcation of circular Kirchhoff elastic rods by Satoshi Kawakubo, Osaka J. Math. 37 (2000), 93--137", Osaka J. Math. 37 (2000), 525.
  • J. L. Kelly, General Topology, Van Nostrand, 1955.
  • S. Kobayashi and K. Nomizu, Foundations of differential geometry, I, II, Interscience, 1963, 1969.
  • N. Koiso, Elasticae in a Riemannian submanifold, Osaka J. Math. 29 (1992), 539--543.
  • J. Langer and D. Singer, The total squared curvature of closed curves, J. Differential Geom. 20 (1984), 1--22.
  • J. Langer and D. Singer, Curve-straightening in Riemannian manifolds, Ann. Global Anal. Geom. 5 (1987), 133--150.
  • J. Langer and D. Singer, Lagrangian aspects of the Kirchhoff elastic rod, SIAM Rev. 38 (1996), 605--618.
  • C. B. Morrey, Jr., Multiple integrals in the calculus of variations, Springer-Verlag, New York, 1966.