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Maxwell strata in the Euler elastic problem

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Abstract

The classical Euler problem on stationary configurations of elastic rod in the plane is studied in detail by geometric control techniques as a left-invariant optimal control problem on the group of motions of a two-dimensional plane E(2).

The attainable set is described, the existence and boundedness of optimal controls are proved. Extremals are parametrized by the Jacobi elliptic functions of natural coordinates induced by the flow of the mathematical pendulum on fibers of the cotangent bundle of E(2).

The group of discrete symmetries of the Euler problem generated by reflections in the phase space of the pendulum is studied. The corresponding Maxwell points are completely described via the study of fixed points of this group. As a consequence, an upper bound on cut points in the Euler problem is obtained.

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References

  1. A. Agrachev, B. Bonnard, M. Chyba, and I. Kupka, Sub-Riemannian sphere in Martinet flat case. ESAIM Control Optim. Calc. Var. 2 (1997). 377–448.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. A. Agrachev and Yu. L. Sachkov, Geometric control theory. Fizmatlit, Moscow (2004); English translation: Control theory from the geometric viewpoint. Springer-Verlag, Berlin (2004).

    Google Scholar 

  3. _____, An intrinsic approach to the control of rolling bodies. Proc. 38th IEEE Conf. Decision and Control, Phoenix, Arizona, USA, December 7–10 (1999), Vol. 1, pp. 431–435.

  4. S. S. Antman, The influence of elasticity on analysis: Modern developments, Bull. Amer. Math. Soc. 9 (1983), No. 3, 267–291.

    Article  MathSciNet  MATH  Google Scholar 

  5. V. I. Arnold, Singularities of caustics and wave fronts. Kluwer (1990).

  6. D. Bernoulli, 26th letter to L. Euler (October, 1742). In: Correspondance mathématique et physique, Vol. 2, St. Petersburg (1843).

  7. J. Bernoulli, Véritable hypothèse de la résistance des solides, avec la demonstration de la corbure des corps qui font ressort. In: Collected works, Vo. 2, Geneva (1744).

  8. G. Birkhoff and C. R. de Boor, Piecewise polynomial interpolation and approximation. In: Approximation of Functions (Proc. Symp. General Motors Res. Lab., 1964), Elsevier, Amsterdam (1965), pp. 164–190.

    Google Scholar 

  9. M. Born, Stabilität der elastischen Linie in Ebene und Raum. Preisschrift und Dissertation, Göttingen, Dieterichsche Universitäts-Buchdruckerei Göttingen (1906). Reprinted in: Ausgewählte Abhandlungen, Göttingen, Vanderhoeck & Ruppert (1963), Vol. 1, pp. 5–101.

  10. U. Boscain and F. Rossi, Invariant Carno–Caratheodory metrics on S 3, SO(3), SL(2), and lens spaces. SIAM J. Control (2008), accepted.

  11. R. Brockett and L. Dai, Non-holonomic kinematics and the role of elliptic functions in constructive controllability. In: Nonholonomic Motion Planning (Z. Li and J. Canny, eds.), Kluwer, Boston (1993), pp. 1–21.

    Google Scholar 

  12. L. Cesari, Optimization. Theory and applications. Problems with ordinary differential equations. Springer-Verlag, New York–Heidelberg–Berlin (1983).

    MATH  Google Scholar 

  13. L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimitrici latissimo sensu accepti. Lausanne, Geneva (1744).

    Google Scholar 

  14. M. Golumb and J. Jerome, Equilibria of the curvature functional and manifolds of nonlinear interpolating spline curves. SIAM J. Math. Anal. 13 (1982), 421–458.

    Article  MathSciNet  Google Scholar 

  15. S. Jacquet, Regularity of sub-Riemannian distance and cut locus. Preprint No. 35, May 1999, Universita degli Studi di Firenze, Dipartimento di Matematica Applicata “G. Sansone,” Italy.

  16. C. G. J. Jacobi, Volresungen über Dynamik. G. Reimer, Berlin (1891).

    Google Scholar 

  17. J. W. Jerome, Minimization problems and linear and nonlinear spline functions, I: Existence. SIAM J. Numer. Anal. 10 (1973), 808–819.

    Article  MathSciNet  MATH  Google Scholar 

  18. ____, Smooth interpolating curves of prescribed length and minimum curvature. Proc. Amer. Math. Soc. 51 (1975), 62–66.

    Article  MathSciNet  MATH  Google Scholar 

  19. V. Jurdjevic, The geometry of the ball-plate problem. Arch. Rat. Mech. Anal. 124 (1993), 305–328.

    Article  MathSciNet  MATH  Google Scholar 

  20. _____, Non-Euclidean elastica. Amer. J. Math. 117 (1995), 93–125.

    Article  MathSciNet  MATH  Google Scholar 

  21. _____, Geometric control theory. Cambridge Univ. Press (1997).

  22. D. F. Lawden, Elliptic functions and applications. Springer-Verlag (1989).

  23. A. Linnér, Unified representations of non-linear splines. J. Approx. Theory 84 (1996), 315–350.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. E. H. Love, A treatise on the mathematical theory of elasticity. Dover, New York (1927).

    MATH  Google Scholar 

  25. R. S. Manning, J. H. Maddocks, and J. D. Kahn, A continuum rod model of sequence-dependent DNA structure. J. Chem. Phys. 105 (1996), 5626–5646.

    Article  Google Scholar 

  26. R. S. Manning, K. A. Rogers, and J. H.Maddocks, Isoperimetric conjugate points with application to the stability of DNA minicircles. Proc. Roy. Soc. London A 454 (1998), 3047–3074.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Mumford, Elastica and computer vision. In: Algebraic geometry and its applications (C. L. Bajaj, Ed.), Springer-Verlag, New York (1994), pp. 491–506.

    Google Scholar 

  28. O. Myasnichenko, Nilpotent (3,6) sub-Riemannian problem. J. Dynam. Control Systems 8 (2002), No. 4, 573–597.

    Article  MathSciNet  MATH  Google Scholar 

  29. L. Saalschütz, Der belastete Stab. Leipzig (1880).

  30. Yu. L. Sachkov, Exponential mapping in generalized Dido’s problem. Mat. Sb. 194 (2003), No. 9, 63–90.

    MathSciNet  Google Scholar 

  31. _____, Discrete symmetries in the generalized Dido problem. Mat. Sb. 197 (2006), No. 2, 95–116. 235–257.

    MathSciNet  Google Scholar 

  32. _____, The Maxwell set in the generalized Dido problem. Mat. Sb. 197 (2006), No. 4, 123–150. 595–621.

    MathSciNet  Google Scholar 

  33. _____, Complete description of the Maxwell strata in the generalized Dido problem. Mat. Sb. 197 (2006), No. 6, 111–160.

    MathSciNet  Google Scholar 

  34. _____, Maxwell strata in Euler’s elastic problem. Preprint SISSA 04/2007/M (January 15th 2007), Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy; available at: arXiv:0705.0614 [math.OC], 3 May 2007.

  35. _____, Conjugate points in Euler’s elastic problem. J. Dynam. Control Systems (accepted), available at: arXiv:0705.1003 [math.OC], 7 May 2007.

  36. _____, Exponential mapping in Euler’s elastic problem. In preparation.

  37. A. V. Sarychev and D. F. M. Torres, Lipschitzian regularity of minimizers for optimal control problems with control-affine dynamics. Appl. Math. Optim. 41 (2000), 237–254.

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Timoshenko, History of strength of materials. McGraw-Hill, New York (1953).

    Google Scholar 

  39. C. Truesdell, The influence of elasticity on analysis: The classic heritage. Bull. Amer. Math. Soc. 9 (1983), No. 3, 293–310.

    Article  MathSciNet  MATH  Google Scholar 

  40. E. T. Whittaker and G. N. Watson, 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 Univ. Press, Cambridge (1996).

    MATH  Google Scholar 

  41. S. Wolfram, Mathematica: a system for doing mathematics by computer. Addison-Wesley, Reading, MA (1991).

    Google Scholar 

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Authors and Affiliations

  1. Program Systems Institute, Russian Academy of Sciences, Pereslavl-Zalessky, 152020, Russia

    Yu. L. Sachkov

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Correspondence to Yu. L. Sachkov.

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This work was partially supported by the Russian Foundation for Basic Research, project No. 05-01-00703-a.

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Sachkov, Y.L. Maxwell strata in the Euler elastic problem. J Dyn Control Syst 14, 169–234 (2008). https://doi.org/10.1007/s10883-008-9039-7

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  • DOI: https://doi.org/10.1007/s10883-008-9039-7

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