Kirchhoff elastic rods in three-dimensional space forms

Journal of the Mathematical Society of Japan

Volumn 60, Issue 2, 2008, Pages 551-582

  Kawakubo, Satoshi ^a  

^a Fukuoka University   (Japan)

Author keywords

Calculus of variations; Elastic rod; Elastica

Indexed keywords

EID: 47949108125     PISSN: 00255645     EISSN: None     Source Type: Journal    
DOI: 10.2969/jmsj/06020551     Document Type: Article

References (28)

1. S. Antman, Nonlinear problems of elasticity, Springer, New York, 1995.
2. J. Arroyo, O. Garay and J. Mencía, Extremals of curvature energy actions on spherical closed curves, J. Geom. Phys., 51 (2004), 101-125.
3. 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.
4. 2/2 ds, Amer. J. Math., 108 (1986), 525-570.
5. P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and physicists, Springer, Berlin, 1954.
6. H. Hasiraoto and T. Kambe, Simulation of invariant shapes of a vortex filament with an elastic rod, J. Phys. Soc. Japan, 54 (1985), 5-7.
7. T. Ivey and D. Singer, Knot types, homotopies and stability of closed elastic rods, Proc. London Math. Soc. (3), 79 (1999), 429-450.
8. V. Jurdjevic, Non-Euclidean elastica, Amer. J. Math., 117 (1995), 93-124.
9. V. Jurdjevic, Geometric control theory, Cambridge University Press, Cambridge, 1997.
10. V. Jurdjevic, Integrable Hamiltonian systems on Lie groups: Kowalewski type, Ann. of Math. (2), 150 (1999), 605-644.
11. V. Jurdjevic, Integrable Hamiltonian systems on complex Lie groups, Mem. Amer. Math. Soc., 178 (2005), 838.
12. S. Kawakubo, Stability and bifurcation of circular Kirchhoff elastic rods, Osaka J. Math., 37 (2000), 93-137, Errata, Osaka J. Math., 37 (2000), 525.
13. S. Kawakubo, Kirchhoff elastic rods in the three-sphere, Tohoku Math. J., 56 (2004), 205-235.
14. S. Kida, A vortex filament moving without change of form, J. Fluid Mech., 112 (1981), 397-409.
15. G. Kirchhoff, Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. Reine Angew. Math., 56 (1859), 285-313.
16. N. Koiso, Elasticae in a Riemannian submanifold, Osaka J. Math., 29 (1992), 539-543.
17. J. Langer, Recursion in curve geometry, New York J. Math., 5 (1999), 25-51.
18. 3, J. London Math. Soc. (2), 30 (1984), 512-520.
19. 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.
20. J. Langer and D. Singer, The total squared curvature of closed curves, J. Differential Geom., 20 (1984), 1-22.
21. J. Langer and D. Singer, Liouville integrability of geometric variational problems, Comment. Math. Helv., 69 (1994), 272-280.
22. J. Langer and D. Singer, Lagrangian aspects of the Kirchhoff elastic rod, SIAM Rev., 38 (1996), 605-618.
23. A. E. H. Love, A treatise on the mathematical theory of elasticity, Fourth Ed. Dover Publications, New York, 1944.
24. G. Naber, The geometry of Minkowski spacetime, Springer, New York, 1992.
25. M. Nizette and A. Goriely, Towards a classification of Euler-Kirchhoff filaments, J. Math. Phys., 40 (1999), 2830-2866.
26. 3, Invent. Math., 81 (1985), 379-386.
27. Y. Shi and J. Hearst, The Kirchhoff elastic rod, the nonlinear Schrodinger equation, and DNA supercoiling, J. Chem. Phys., 101 (1994), 5186-5200.
28. H. Tsuru, Equilibrium shapes and vibrations of thin elastic rod, J. Phys. Soc. Japan, 56 (1987), 2309-2324.