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QUADRATIC CURVATURE ENERGIES IN THE 2-SPHERE

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

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: josujon.arroyo@ehu.es)
ÓSCAR J. GARAY*
Affiliation:
Department of Mathematics, Faculty of Science and Technology, University of the Basque Country, Aptdo 644, 48080 Bilbao, Spain (email: oscarj.garay@ehu.es)
JOSE MENCÍA
Affiliation:
Department of Mathematics, Faculty of Science and Technology, University of the Basque Country, Aptdo 644, 48080 Bilbao, Spain (email: jj.mencia@ehu.es)
*
✉For correspondence; e-mail: oscarj.garay@ehu.es
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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.

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