Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-12T19:42:48.388Z Has data issue: false hasContentIssue false
  • English
  • Français

GLOBAL SOLUTIONS OF THE EQUATION OF THE KIRCHHOFF ELASTIC ROD IN SPACE FORMS

Part of: Thin bodies, structures Variational problems in infinite-dimensional spaces Optimality conditions

Published online by Cambridge University Press:  27 September 2012

SATOSHI KAWAKUBO
SATOSHI KAWAKUBO*
Affiliation:
Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Kirchhoff elastic rod is one of the mathematical models of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler–Lagrange equations associated to the energy with the effect of bending and twisting. In this paper, we consider Kirchhoff elastic rods in a space form. In particular, we give the existence and uniqueness of global solutions of the initial-value problem for the Euler–Lagrange equations. This implies that an arbitrary Kirchhoff elastic rod of finite length extends to that of infinite length.

Keywords

Kirchhoff elastic rodscalculus of variationsordinary differential equationsinitial-value problemsglobal solutions

MSC classification

Secondary: 49K15: Problems involving ordinary differential equations 58E10: Applications to the theory of geodesics (problems in one independent variable) 74K10: Rods (beams, columns, shafts, arches, rings, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 1 , August 2013 , pp. 70 - 80
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

References

[1]Antman, S. S., ‘Ordinary differential equations of nonlinear elasticity. II. Existence and regularity theory for conservative boundary value problem’, Arch. Ration. Mech. Anal. 61(4) (1976), 353–393.CrossRefGoogle Scholar
[2]Antman, S. S., Nonlinear Problems of Elasticity, Applied Mathematical Sciences, 107 (Springer, New York, 1995).CrossRefGoogle Scholar
[3]Antman, S. S. & Jordan, K. B., ‘Qualitative aspects of the spatial deformation of non-linearly elastic rods’, Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 85–105.CrossRefGoogle Scholar
[4]Arroyo, J., Garay, O. J. & Mencía, J. J., ‘Closed generalized elastic curves in $S^2(1)$’, J. Geom. Phys. 48(2–3) (2003), 339–353.CrossRefGoogle Scholar
[5]Arroyo, J., Garay, O. J. & Mencía, J. J., ‘Extremals of curvature energy actions on spherical closed curves’, J. Geom. Phys. 51(1) (2004), 101–125.CrossRefGoogle Scholar
[6]Arroyo, J., Garay, O. J. & Mencía, J. J., ‘Quadratic curvature energies in the 2-sphere’, Bull. Aust. Math. Soc. 81(3) (2010), 496–506.CrossRefGoogle Scholar
[7]Barros, M., Garay, O. J. & Singer, D. A., ‘Elasticae with constant slant in the complex projective plane and new examples of Willmore tori in five spheres’, Tohoku Math. J. (2) 51(2) (1999), 177–192.CrossRefGoogle Scholar
[8]Bishop, R. L., ‘There is more than one way to frame a curve’, Amer. Math. Monthly 82 (1975), 246–251.CrossRefGoogle Scholar
[9]Bryant, R. & Griffiths, P. A., ‘Reduction for constrained variational problems and $\int {1\over 2}k^2 ds$’, Amer. J. Math. 108(3) (1986), 525–570.CrossRefGoogle Scholar
[10]Coddington, E. A. & Levinson, N., Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).Google Scholar
[11]Coleman, B. D., Dill, E. H., Lembo, M., Lu, Z. & Tobias, I., ‘On the dynamics of rods in the theory of Kirchhoff and Clebsch’, Arch. Ration. Mech. Anal. 121(4) (1992), 339–359.CrossRefGoogle Scholar
[12]Garay, O. J., ‘Extremals of the generalized Euler–Bernoulli energy and applications’, J. Geom. Symmetry Phys. 12 (2008), 27–61.Google Scholar
[13]Griffiths, P. A., Exterior Differential Systems and the Calculus of Variations, Progress in Mathematics, 25 (Birkhäuser, Boston, 1983).CrossRefGoogle Scholar
[14]Inoguchi, J. & Lee, J.-E., ‘Biminimal curves in 2-dimensional space forms’, Commun. Korean Math. Soc., to appear.Google Scholar
[15]Ivey, T. A. & Singer, D. A., ‘Knot types, homotopies and stability of closed elastic rods’, Proc. Lond. Math. Soc. (3) 79(2) (1999), 429–450.CrossRefGoogle Scholar
[16]Jurdjevic, V., ‘Non-Euclidean elastica’, Amer. J. Math. 117(1) (1995), 93–124.CrossRefGoogle Scholar
[17]Jurdjevic, V., Geometric Control Theory, Cambridge Studies in Advanced Mathematics, 52 (Cambridge University Press, Cambridge, 1997).Google Scholar
[18]Jurdjevic, V., ‘Integrable Hamiltonian systems on complex Lie groups’, Mem. Amer. Math. Soc. 178(838) (2005).Google Scholar
[19]Kawakubo, S., ‘Kirchhoff elastic rods in a Riemannian manifold’, Tohoku Math. J. (2) 54(2) (2002), 179–193.CrossRefGoogle Scholar
[20]Kawakubo, S., ‘Kirchhoff elastic rods in the three-sphere’, Tohoku Math. J. (2) 56(2) (2004), 205–235.CrossRefGoogle Scholar
[21]Kawakubo, S., ‘Kirchhoff elastic rods in three-dimensional space forms’, J. Math. Soc. Japan 60(2) (2008), 551–582.CrossRefGoogle Scholar
[22]Kawakubo, S., ‘First integrals and global solutions of the equation for Kirchhoff elastic rods’, Fukuoka Univ. Sci. Rep. 41(2) (2011), 147–153.Google Scholar
[23]Kawakubo, S., ‘Kirchhoff elastic rods in higher-dimensional space forms’, Proc. Japan Acad. Ser. A Math. Sci. 87(1) (2011), 5–9.CrossRefGoogle Scholar
[24]Kehrbaum, S. & Maddocks, J. H., ‘Elastic rods, rigid bodies, quaternions and the last quadrature’, Philos. Trans. R. Soc. Lond. Ser. A 355(1732) (1997), 2117–2136.CrossRefGoogle Scholar
[25]Kobayashi, S. & Nomizu, K., Foundations of Differential Geometry. Vol. I. Wiley Classics Library (Wiley, New York, 1996), Reprint of the 1963 original.Google Scholar
[26]Koiso, N., ‘Elasticae in a Riemannian submanifold’, Osaka J. Math. 29(3) (1992), 539–543.Google Scholar
[27]Koiso, N., ‘Convergence towards an elastica’, in: Geometry and Global Analysis (Sendai, 1993) (Tohoku Univ., Sendai, 1993), pp. 249–254.Google Scholar
[28]Koiso, N., ‘Convergence towards an elastica in a Riemannian manifold’, Osaka J. Math. 37(2) (2000), 467–487.Google Scholar
[29]Langer, J. & Singer, D. A., ‘The total squared curvature of closed curves’, J. Differential Geom. 20(1) (1984), 1–22.CrossRefGoogle Scholar
[30]Langer, J. & Singer, D. A., ‘Liouville integrability of geometric variational problems’, Comment. Math. Helv. 69(2) (1994), 272–280.CrossRefGoogle Scholar
[31]Langer, J. & Singer, D. A., ‘Lagrangian aspects of the Kirchhoff elastic rod’, SIAM Rev. 38(4) (1996), 605–618.CrossRefGoogle Scholar
[32]Linnér, A., ‘Curve-straightening in closed Euclidean submanifolds’, Comm. Math. Phys. 138(1) (1991), 33–49.CrossRefGoogle Scholar
[33]Loubeau, E. & Montaldo, S., ‘Biminimal immersions’, Proc. Edinb. Math. Soc. (2) 51(2) (2008), 421–437.CrossRefGoogle Scholar
[34]Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, 4th edn (Dover Publications, New York, 1944).Google Scholar
[35]Nizette, M. & Goriely, A., ‘Towards a classification of Euler–Kirchhoff filaments’, J. Math. Phys. 40(6) (1999), 2830–2866.CrossRefGoogle Scholar
[36]Pinkall, U., ‘Hopf tori in $S^3$’, Invent. Math. 81(2) (1985), 379–386.CrossRefGoogle Scholar
[37]Popiel, T. & Noakes, L., ‘Elastica in $\rm SO(3)$’, J. Aust. Math. Soc. 83(1) (2007), 105–124.CrossRefGoogle Scholar
[38]Shi, Y. & Hearst, J., ‘The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling’, J. Chem. Phys. 101 (1994), 5186–5200.CrossRefGoogle Scholar
[39]Tsuru, H., ‘Equilibrium shapes and vibrations of thin elastic rod’, J. Phys. Soc. Japan 56(7) (1987), 2309–2324.CrossRefGoogle Scholar