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Trudy MIAN, 2012, Volume 278, Pages 227–241 (Mi tm3409)  

This article is cited in 12 scientific papers (total in 12 papers)

Closed Euler elasticae

Yu. L. Sachkov

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

Abstract: Euler's classical problem on stationary configurations of an elastic rod in a plane is studied as an optimal control problem on the group of motions of a plane. We show complete integrability of the Hamiltonian system of the Pontryagin maximum principle. We prove that a closed elastica is either a circle or a figure-of-eight elastica, wrapped around itself several times. Finally, we study local and global optimality of closed elasticae: the figure-of-eight elastica is optimal only locally, while the circle is optimal globally.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2012, 278, 218–232

Bibliographic databases:

UDC: 517.977
Received in February 2011

Citation: Yu. L. Sachkov, “Closed Euler elasticae”, Differential equations and dynamical systems, Collected papers, Trudy MIAN, 278, MAIK Nauka/Interperiodica, Moscow, 2012, 227–241; Proc. Steklov Inst. Math., 278 (2012), 218–232

Citation in format AMSBIB
\by Yu.~L.~Sachkov
\paper Closed Euler elasticae
\inbook Differential equations and dynamical systems
\bookinfo Collected papers
\serial Trudy MIAN
\yr 2012
\vol 278
\pages 227--241
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2012
\vol 278
\pages 218--232

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    This publication is cited in the following articles:
    1. Bartels S., Reiter Ph., “Numerical Solution of a Bending-Torsion Model For Elastic Rods”, Numer. Math.  crossref  mathscinet  isi
    2. S. Avvakumov, O. Karpenkov, A. Sossinsky, “Euler elasticae in the plane and the Whitney-Graustein theorem”, Russ. J. Math. Phys., 20:3 (2013), 257–267  crossref  mathscinet  zmath  isi  elib  scopus
    3. Yu. L. Sachkov, E. F. Sachkova, “Exponential mapping in Euler's elastic problem”, J. Dyn. Control Syst., 20:4 (2014), 443–464  crossref  mathscinet  zmath  isi  scopus
    4. Ya. A. Butt, A. I. Bhatti, Yu. L. Sachkov, “Integrability by quadratures in optimal control of a unicycle on hyperbolic plane”, 2015 American Control Conference (ACC) (Chicago, IL, USA), IEEE, 2015, 4251–4256  crossref  mathscinet  isi  scopus
    5. B. Kawohl, “Two dimensions are easier”, Arch. Math., 107:4 (2016), 423–428  crossref  mathscinet  zmath  isi  elib  scopus
    6. V. Ferone, B. Kawohl, C. Nitsch, “The elastica problem under area constraint”, Math. Ann., 365:3-4 (2016), 987–1015  crossref  mathscinet  zmath  isi  elib  scopus
    7. Yu. Baryshnikov, Ch. Chen, “Shapes of Cyclic Pursuit and Their Evolution”, 2016 IEEE 55Th Conference on Decision and Control (CDC), IEEE Conference on Decision and Control, IEEE, 2016, 2561–2566  isi
    8. D. Bucur, A. Henrot, “A new isoperimetric inequality for elasticae”, J. Eur. Math. Soc., 19:11 (2017), 3355–3376  crossref  mathscinet  zmath  isi  scopus
    9. T. Kemmochi, “Energy dissipative numerical schemes for gradient flows of planar curves”, Bit, 57:4 (2017), 991–1017  crossref  mathscinet  zmath  isi  scopus
    10. F. Dayrens, S. Masnou, M. Novaga, “Existence, regularity and structure of confined elasticae”, ESAIM-Control OPtim. Calc. Var., 24:1 (2018), 25–43  crossref  mathscinet  zmath  isi  scopus
    11. A. A. Ardentov, “Multiple solutions in Euler's elastic problem”, Autom. Remote Control, 79:7 (2018), 1191–1206  mathnet  crossref  isi  elib
    12. Miura T., “Elastic Curves and Phase Transitions”, Math. Ann., 376:3-4 (2020), 1629–1674  crossref  mathscinet  isi
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