Multiple Solutions in Euler’s Elastic Problem

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 elastic, optimal control, Maxwell stratum, symmetry, theory of elasticity, elliptic integral

Received

28.09.2018

Publication date

29.09.2018

Number of characters

1003

Cite Download pdf To download PDF you should sign in

GOST	Ardentov A. Multiple Solutions in Euler’s Elastic Problem // Avtomatika i Telemekhanika – 2018. – Issue 7 C. 22-40 [Electronic resource]. URL: http://ras.jes.su/ait/s000523100000265-5-1-en (circulation date: 24.04.2024). DOI: 10.31857/S000523100000265-5
MLA	Ardentov, Andrey "Multiple Solutions in Euler’s Elastic Problem." Avtomatika i Telemekhanika 7 (2018):22-40. DOI: 10.31857/S000523100000265-5
APA	Ardentov A. (2018). Multiple Solutions in Euler’s Elastic Problem. Avtomatika i Telemekhanika (7), pp.22-40 DOI: 10.31857/S000523100000265-5

Размещенный ниже текст является ознакомительной версией и может не соответствовать печатной

Readers community rating: votes 0

1. Ardentov A.A., Sachkov Yu.L. Solution to Euler’s Elastic Problem // Autom. Remote Control. 2009. Vol. 70. No. 4. P. 633–643.

2. Lyav A. Matematicheskaya teoriya uprugosti. M.: ONTI, 1935.

3. Bernulli 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. P. 262–276.

4. Bernoulli D. The 26th Letter to Euler (October 1742) // Correspondence Math´ematique et Physique de Quelques C´el`ebres G´eom`etres du XVIII`eme Si`ecle (Fuss P.H., ed.). V. 2. St. Petersburg: Academia Imperiale des Sciences, 1843.

5. Euler L. Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, sive Solutio Problematis Isoperimitrici Latissimo Sensu Accepti. Lausanne: Bousquet, 1744.

6. Saalschu¨tz L. Der Belastete Stab unter Einwirkung einer Seitlichen Kraft. Leipzig: Teubner, 1880.

7. Born M. Stabilita¨t der Elastischen Linie in Ebene und Raum. G¨ottingen: Dieterich, 1906.

8. Levien R. The Elastica: a Mathematical History // Technical Report No. UCB/ EECS-2008-103. 2008. P. 1–25.

9. Sachkov Yu.L. Maxwell Strata in the Euler Elastic Problem // J. Dynam. Control Syst. 2008. V. 14. No. 2. P. 169–234.

10. Sachkov Yu.L. Closed Euler Elasticae // Different. Equat. Dynam. Syst. Collected Papers, Tr. Mat. Inst. Steklova. 2012. V. 278. P. 227–241.

11. Sachkov Yu.L., Sachkova E.F. Exponential Mapping in Euler’s Elastic Problem // J. Dynam. Control Syst. 2014. V. 20. No. 4. P. 443–464.

12. Kushner A.G., Lychagin V.V., Rubtsov V.N. Contact Geometry and Nonlinear Differential Equations. Cambridge: Cambridge Univer. Press, 2007.

13. Agrachev A.A., Sachkov Yu.L. Geometricheskaya teoriya upravleniya. M.: Fizmatlit, 2005.

14. Akhiezer N.I. Ehlementy teorii ehllipticheskikh funktsij. M.: Nauka, 1970.

15. Krantz S.G., Parks H.R. The Implicit Function Theorem: History, Theory, and Appl. Basel: Birkauser, 2003.

16. Arnol'd V.I. Matematicheskie metody klassicheskoj mekhaniki. M.: Nauka, 1989