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Automation and Remote Control

, Volume 79, Issue 7, pp 1191–1206 | Cite as

Multiple Solutions in Euler’s Elastic Problem

  • A. A. Ardentov
Nonlinear Systems
  • 14 Downloads

Abstract

The paper is devoted to multiple solutions of the classical problem on stationary configurations of an elastic rod on a plane; we describe boundary values for which there are more than two optimal configurations of a rod (optimal elasticae). We define sets of points where three or four optimal elasticae come together with the same value of elastic energy. We study all configurations that can be translated into each other by symmetries, i.e., reflections at the center of the elastica chord and reflections at the middle perpendicular to the elastica chord. For the first symmetry, the ends of the rod are directed in opposite directions, and the corresponding boundary values lie on a disk. For the second symmetry, the boundary values lie on a Möbius strip. As a result, we study both sets numerically and in some cases analytically; in each case, we find sets of points with several optimal configurations of the rod. These points form the currently known part of the reachability set where elasticae lose global optimality.

Keywords

Euler’s elastica optimal control Maxwell stratum symmetries elasticity theory elliptic integral 

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References

  1. 1.
    Ardentov, A.A. and Sachkov, Yu.L., Solution to Euler’s Elastic Problem, Autom. Remote Control, 2009, vol. 70, no. 4, pp. 633–643.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Cambridge: Univ. Press, 1892.zbMATHGoogle Scholar
  3. 3.
    Bernoulli, J., Curvatura Laminae Elasticae. Ejus Identitas cum Curvatura Lintei a Pondere Inclusi Fluidi Expansi. Radii Circulorum Osculantium in Terminis Simplicissimis Exhibiti; Una cum Novis Quibusdam Theorematis huc Pertinentibus, &c, Acta Eruditorum, June 1694, pp. 262–276.Google Scholar
  4. 4.
    Bernoulli, D., The 26th Letter to Euler (October 1742), Correspondence Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIème Siècle, vol. 2, Fuss, P.H., Ed., St. Petersburg: Academia Imperiale des Sciences, 1843.Google Scholar
  5. 5.
    Euler, L., Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, sive Solutio Problematis Isoperimitrici Latissimo Sensu Accepti, Lausanne: Bousquet, 1744.zbMATHGoogle Scholar
  6. 6.
    Saalschütz, L., Der belastete Stab unter Einwirkung einer seitlichen Kraft, Leipzig: Teubner, 1880.zbMATHGoogle Scholar
  7. 7.
    Born, M., Stabilität der elastischen Linie in Ebene und Raum, Göttingen: Dieterich, 1906.zbMATHGoogle Scholar
  8. 8.
    Levien, R., The Elastica: A Mathematical History, Technical Report no. UCB/EECS-2008-103, 2008.Google Scholar
  9. 9.
    Sachkov, Yu.L., Maxwell Strata in the Euler Elastic Problem, J. Dynam. Control Syst., 2008, vol. 14, no. 2, pp. 169–234.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Sachkov, Yu.L., Conjugate Points in Euler’s Elastic Problem, J. Dynam. Control Syst., 2008, vol. 14, no. 3, pp. 409–439.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sachkov, Yu.L., Closed Euler Elasticae, Differ. Equat. Dynam. Syst. Collected Papers, Tr. Mat. Inst. Steklova, 2012, vol. 278, pp. 227–241.Google Scholar
  12. 12.
    Sachkov, Yu.L. and Sachkova, E.F., Exponential Mapping in Euler’s Elastic Problem, J. Dynam. Control Syst., 2014, vol. 20, no. 4, pp. 443–464.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kushner, A.G., Lychagin, V.V., and Rubtsov, V.N., Contact Geometry and Nonlinear Differential Equations, Cambridge: Cambridge Univ. Press, 2007.zbMATHGoogle Scholar
  14. 14.
    Agrachev, A.A. and Sachkov, Yu.L., Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences (Book 87), Berlin: Springer, 2005.Google Scholar
  15. 15.
    Arnold, V.I., Gusein-Zade, S.M., and Varchenko, A.N., Singularities of Differentiable Maps, Basel: Birkhäuser, 2012.CrossRefzbMATHGoogle Scholar
  16. 16.
    Akhiezer, N.I., Elements of the Theory of Elliptic Functions, Translations of Mathematical Monographs, Providence: American Mathematical Society, 1990.Google Scholar
  17. 17.
    Krantz, S.G. and Parks, H.R., The Implicit Function Theorem: History, Theory, and Applications, Basel: Birkauser, 2003.CrossRefzbMATHGoogle Scholar
  18. 18.
    Arnold, V.I., Mathematical Methods of Classical Mechanics, New York: Springer, 1989.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Ailamazyan Program Systems InstituteRussian Academy of SciencesPereslavl-ZalesskyRussia

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