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QUADRATIC CURVATURE ENERGIES IN THE 2-SPHERE

Part of: Global differential geometry Hamiltonian and Lagrangian mechanics Elastic materials

Published online by Cambridge University Press:  02 March 2010

JOSU ARROYO ,
ÓSCAR J. GARAY  and
JOSE MENCÍA
JOSU ARROYO
Affiliation:
Department of Mathematics, Faculty of Science and Technology, University of the Basque Country, Aptdo 644, 48080 Bilbao, Spain (email: [email protected])
ÓSCAR J. GARAY*
Affiliation:
Department of Mathematics, Faculty of Science and Technology, University of the Basque Country, Aptdo 644, 48080 Bilbao, Spain (email: [email protected])
JOSE MENCÍA
Affiliation:
Department of Mathematics, Faculty of Science and Technology, University of the Basque Country, Aptdo 644, 48080 Bilbao, Spain (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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The classical variational analysis of curvature energy functionals, acting on spaces of curves of a Riemannian manifold, is extremely complicated, and the procedure usually can not be completely developed under such a degree of generality. Sometimes this difficulty may be overcome by focusing on specific actions in real space forms. In this note, we restrict ourselves to quadratic Lagrangian energies acting on the space of closed curves of the 2-sphere. We solve the Euler–Lagrange equation and show that there exists a two-parameter family of closed critical curves. We also discuss the stability of the circular critical points. Since, even for this class of energies, the complete variational analysis is quite involved, we use instead a numerical approach to provide a useful method of visualization of relevant aspects concerning uniqueness, stability and explicit representation of the closed critical curves.

Keywords

elastic curvesEuler–Lagrange equationscurvature energy functionals

MSC classification

Secondary: 53C20: Global Riemannian geometry, including pinching 70H03: Lagrange's equations 74B05: Classical linear elasticity
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 3 , June 2010 , pp. 496 - 506
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This research has been partially supported by grants GIC07/58-IT-256-07 of Gobierno Vasco and MTM2007-61990 of Ministerio de Ciencia e Innovación. Spain.

References

[1]Arroyo, J., Barros, M. and Garay, O. J., ‘Some examples of critical points for the total mean curvature functional’, Proc. Edinb. Math. Soc. 43 (2000), 587–603.Google Scholar
[2]Arroyo, J., Garay, O. J. and Mencía, J. J., ‘Closed generalized elastic curves in 𝕊2(1)’, J. Geom. Phys. 48 (2003), 339–353.CrossRefGoogle Scholar
[3]Arroyo, J., Garay, O. J. and Mencía, J. J., ‘Extremals of curvature energy actions on spherical closed curves’, J. Geom. Phys. 51 (2004), 101–125.CrossRefGoogle Scholar
[4]Arroyo, J., Garay, O. J. and Mencía, J. J., ‘Elastic circles in 2-spheres’, J. Phys. A.: Math. Gen. 39 (2006), 2307–2324.CrossRefGoogle Scholar
[5]Barros, M., ‘Simple geometric models with applications to physics’, in: Curvature and Variational Modeling in Physics and Biophysics, AIP Conference Proceedings, 1002, (eds. Garay, O. J., García-Río, E. and Vázquez-Lorenzo, R.) (AIP, Melville, NY, 2008), pp. 71–113.Google Scholar
[6]Bryant, R. and Griffiths, P., ‘Reduction of order for constrained variational problems and ’, Amer. J. Math. 108 (1986), 525–570.CrossRefGoogle Scholar
[7]Gradsteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products (Academic Press, New York, 1980).Google Scholar
[8]Jurdjevic, V., ‘Non-Euclidean elastica’, Amer. J. Math. 117 (1995), 93–124.CrossRefGoogle Scholar
[9]Kapustnikov, A. A., Pashnev, A. and Pichugin, A., ‘Canonical quantization of the kink model beyond the static solution’, Phys. Rev. D 55 (1977), 2257–2264.CrossRefGoogle Scholar
[10]Kawakubo, S., ‘Kirchhoff elastic rods in the three-dimensional space forms’, J. Math. Soc. Japan 60 (2008), 551–582.CrossRefGoogle Scholar
[11]Koiso, N., ‘Elasticae in a Riemannian manifold’, Osaka J. Math. 29 (1992), 539–543.Google Scholar
[12]Landau, L. and Lifshitz, E., Theory of Elasticity, Course of Theoretical Physics, 7 (Butterworth-Heinemann, Boston, MA, 1986).Google Scholar
[13]Langer, J. and Singer, D. A., ‘The total squared curvature of closed curves’, J. Differential Geom. 20 (1984), 1–22.Google Scholar
[14]Langer, J. and Singer, D. A., ‘Knotted elastic curves in ℝ3’, J. Lond. Math. Soc. 16 (1984), 512–520.CrossRefGoogle Scholar
[15]Langer, J. and Singer, D. A., ‘Curve-straightening in Riemannian manifolds’, Ann. Global Anal. Geom. 5(2) (1987), 133–150.CrossRefGoogle Scholar
[16]Linner, A., ‘Curve straightening and the Palais–Smale condition’, Trans. Amer. Math. Soc. 350 (1998), 3743–3765.Google Scholar
[17]Noakes, L. and Popiel, T., ‘Elastica in ’, J. Aust. Math. Soc. 83(1) (2007), 105–124.Google Scholar
[18]Plyushchay, M. S., ‘Massless particle with rigidity as a model for description of bosons and fermions’, Phys. Lett. B 243 (1990), 383–388.CrossRefGoogle Scholar