## Journal of the Mathematical Society of Japan

### Kirchhoff elastic rods in three-dimensional space forms

Satoshi KAWAKUBO

#### Abstract

The Kirchhoff elastic rod is one of the mathematical models of thin elastic rods, and is characterized as a critical point of the energy functional obtained by adding the effect of twisting to the bending energy. In this paper, we investigate Kirchhoff elastic rods in three-dimensional space forms. In particular, we give explicit formulas of Kirchhoff elastic rods in the three-sphere and in the three-dimensional hyperbolic space in terms of Jacobi sn function and the elliptic integrals.

#### Article information

Source
J. Math. Soc. Japan Volume 60, Number 2 (2008), 551-582.

Dates
First available: 30 May 2008

http://projecteuclid.org/euclid.jmsj/1212156662

Digital Object Identifier
doi:10.2969/jmsj/06020551

Mathematical Reviews number (MathSciNet)
MR2421988

Zentralblatt MATH identifier
1142.58012

#### Citation

KAWAKUBO, Satoshi. Kirchhoff elastic rods in three-dimensional space forms. Journal of the Mathematical Society of Japan 60 (2008), no. 2, 551--582. doi:10.2969/jmsj/06020551. http://projecteuclid.org/euclid.jmsj/1212156662.

#### References

• S. Antman, Nonlinear problems of elasticity, Springer, New York, 1995.
• J. Arroyo, O. Garay and J. Mencí a, Extremals of curvature energy actions on spherical closed curves, J. Geom. Phys., 51 (2004), 101–125.
• M. Barros, O. Garay and D. Singer, Elasticae with constant slant in the complex projective plane and new examples of Willmore tori in five spheres, Tohoku Math. J., 51 (1999), 177–192.
• R. Bryant and P. Griffiths, Reduction for constrained variational problems and $\int{k^2/2}\,ds$, Amer. J. Math., 108 (1986), 525–570.
• P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and physicists, Springer, Berlin, 1954.
• H. Hasimoto and T. Kambe, Simulation of invariant shapes of a vortex filament with an elastic rod, J. Phys. Soc. Japan, 54 (1985), 5–7.
• T. Ivey and D. Singer, Knot types, homotopies and stability of closed elastic rods, Proc. London Math. Soc. (3), 79 (1999), 429–450.
• V. Jurdjevic, Non-Euclidean elastica, Amer. J. Math., 117 (1995), 93–124.
• V. Jurdjevic, Geometric control theory, Cambridge University Press, Cambridge, 1997.
• V. Jurdjevic, Integrable Hamiltonian systems on Lie groups: Kowalewski type, Ann. of Math. (2), 150 (1999), 605–644.
• V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups, Mem. Amer. Math. Soc., 178 (2005), 838.
• S. Kawakubo, Stability and bifurcation of circular Kirchhoff elastic rods, Osaka J. Math., 37 (2000), 93–137, Errata, Osaka J. Math., 37 (2000), 525.
• S. Kawakubo, Kirchhoff elastic rods in the three-sphere, Tohoku Math. J., 56 (2004), 205–235.
• S. Kida, A vortex filament moving without change of form, J. Fluid Mech., 112 (1981), 397–409.
• G. Kirchhoff, Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. Reine Angew. Math., 56 (1859), 285–313.
• N. Koiso, Elasticae in a Riemannian submanifold, Osaka J. Math., 29 (1992), 539–543.
• J. Langer, Recursion in curve geometry, New York J. Math., 5 (1999), 25–51.
• J. Langer and D. Singer, Knotted elastic curves in $\RR^3$, J. London Math. Soc. (2), 30 (1984), 512–520.
• J. Langer and D. Singer, Curves in the hyperbolic plane and mean curvature of tori in $3$-space, Bull. London Math. Soc., 16 (1984), 531–534.
• J. Langer and D. Singer, The total squared curvature of closed curves, J. Differential Geom., 20 (1984), 1–22.
• J. Langer and D. Singer, Liouville integrability of geometric variational problems, Comment. Math. Helv., 69 (1994), 272–280.
• J. Langer and D. Singer, Lagrangian aspects of the Kirchhoff elastic rod, SIAM Rev., 38 (1996), 605–618.
• A. E. H. Love, A treatise on the mathematical theory of elasticity, Fourth Ed. Dover Publications, New York, 1944.
• G. Naber, The geometry of Minkowski spacetime, Springer, New York, 1992.
• M. Nizette and A. Goriely, Towards a classification of Euler-Kirchhoff filaments, J. Math. Phys., 40 (1999), 2830–2866.
• U. Pinkall, Hopf tori in $S^3$, Invent. Math., 81 (1985), 379–386.
• Y. Shi and J. Hearst, The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling, J. Chem. Phys., 101 (1994), 5186–5200.
• H. Tsuru, Equilibrium shapes and vibrations of thin elastic rod, J. Phys. Soc. Japan, 56 (1987), 2309–2324.