Journal of Dynamical and Control Systems

, Volume 20, Issue 4, pp 443–464 | Cite as

Exponential Mapping in Euler’s Elastic Problem



The Euler problem on optimal configurations of elastic rod in the plane with fixed endpoints and tangents at the endpoints is considered. The global structure of the exponential mapping that parameterises extremal trajectories is described. It is proved that open domains cut out by the Maxwell strata in the preimage and image of the exponential mapping are mapped diffeomorphically. As a consequence, computation of globally optimal elasticae with given boundary conditions is reduced to solving systems of algebraic equations having unique solutions in the open domains. For certain special boundary conditions, optimal elasticae are presented.


Euler elastica Optimal control Exponential mapping 

Mathematics Subject Classifications (2010)

49J15 93B29 93C10 74B20 74K10 65D07 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agrachev AA, Barilari D. Sub-Riemannian structures on 3D Lie groups. J Dynam Control Syst. 2012;18(1):21–44.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Agrachev AA, Sachkov YuL. Geometric control theory. Moscow: Fizmatlit; 2004. English transl. Control theory from the geometric view- point. 2004.Google Scholar
  3. 3.
    Ardentov AA, Sachkov YuL. Extremal trajectories in nilpotent sub-Riemannian problem on Engel group. Sbornik Mathematics 2011;202(11):31–54. English translation: Sbornik Mathematics 2011;202(11):1593–1616.CrossRefMathSciNetGoogle Scholar
  4. 4.
    Ardentov AA, Sachkov YuL. Solution of Euler’s elastic problem. Avtomatika i Telemekhanika 2009;4:78–88. (in Russian, English translation in automation and remote control).MathSciNetGoogle Scholar
  5. 5.
    Euler L. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive Solutio problematis isoperimitrici latissimo sensu accepti. Lausanne: 1744.Google Scholar
  6. 6.
    Krantz SG, Parks HR. The implicit function theorem: history theory, and applications. Boston: Birkauser; 2001.Google Scholar
  7. 7.
    Love AEH. A treatise on the mathematical theory of elasticity, 4th ed. New York: Dover; 1927.MATHGoogle Scholar
  8. 8.
    Mashtakov A, Sachkov YuL. Extremal trajectories and Maxwell points in the plate-ball problem. Sbornik Mathematics 2011;202(9):97–120.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Sachkov YuL. Complete description of the Maxwell strata in the generalized Dido problem (in Russian). Matem Sbornik 2006;197(6):111–160. English translation in: Sbornik Mathematics 2006;197(6):901–50.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Sachkov YuL. Maxwell strata in the Euler elastic problem. J Dynam Control Syst. 2008;14(2):169–234.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Sachkov YuL. Conjugate points in Euler’s elastic problem. J Dynam Control Syst. 2008;14(3):409–439.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Sachkov YuL. Closed Euler elasticae. Proc Steklov Inst Math. 2012;278:218–232.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Whittaker ET, Watson GN. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of principal transcendental functions. Cambridge: Cambridge University Press; 1996.MATHGoogle Scholar
  14. 14.
    Wolfram S. Mathematica: a system for doing mathematics by computer. Reading: Addison-Wesley; 1991.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

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

Personalised recommendations