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

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Volume 278, Pages 227–241 (Mi tm3409)

This article is cited in 18 scientific papers (total in 18 papers)

Closed Euler elasticae

Yu. L. Sachkov

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

Full-text PDF (275 kB) Citations (18)

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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.

Received in February 2011

English version:
Proceedings of the Steklov Institute of Mathematics, 2012, Volume 278, Pages 218–232
DOI: https://doi.org/10.1134/S0081543812060211

Bibliographic databases:

Document Type: Article

UDC: 517.977

Language: English

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

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tm3409

https://www.mathnet.ru/eng/tm/v278/p227

This publication is cited in the following 18 articles:

Tatsuya Miura, Kensuke Yoshizawa, “General rigidity principles for stable and minimal elastic curves”, Journal für die reine und angewandte Mathematik (Crelles Journal), 2024
David Krejčiřík, Vladimir Lotoreichik, “Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain”, Calc. Var., 63:9 (2024)
Yu. L. Sachkov, “Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions”, Russian Math. Surveys, 78:1 (2023), 65–163
Matsutani Sh., Previato E., “An Algebro-Geometric Model For the Shape of Supercoiled Dna”, Physica D, 430 (2022), 133073
Kensuke Yoshizawa, “The critical points of the elastic energy among curves pinned at endpoints”, DCDS, 42:1 (2022), 403
Bartels S., Reiter Ph., “Numerical Solution of a Bending-Torsion Model For Elastic Rods”, Numer. Math., 146:4 (2020), 661–697
Miura T., “Elastic Curves and Phase Transitions”, Math. Ann., 376:3-4 (2020), 1629–1674
F. Dayrens, S. Masnou, M. Novaga, “Existence, regularity and structure of confined elasticae”, ESAIM-Control OPtim. Calc. Var., 24:1 (2018), 25–43
A. A. Ardentov, “Multiple solutions in Euler's elastic problem”, Autom. Remote Control, 79:7 (2018), 1191–1206
D. Bucur, A. Henrot, “A new isoperimetric inequality for elasticae”, J. Eur. Math. Soc., 19:11 (2017), 3355–3376
T. Kemmochi, “Energy dissipative numerical schemes for gradient flows of planar curves”, Bit, 57:4 (2017), 991–1017
B. Kawohl, “Two dimensions are easier”, Arch. Math., 107:4 (2016), 423–428
V. Ferone, B. Kawohl, C. Nitsch, “The elastica problem under area constraint”, Math. Ann., 365:3-4 (2016), 987–1015
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
Yuliy Baryshnikov, Cheng Chen, 2016 IEEE 55th Conference on Decision and Control (CDC), 2016, 2561
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
Yu. L. Sachkov, E. F. Sachkova, “Exponential mapping in Euler's elastic problem”, J. Dyn. Control Syst., 20:4 (2014), 443–464
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

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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