Geodesic
Closed Euler elasticae
Informatics and Automation, Differential equations and dynamical systems, Tome 278 (2012), pp. 227-241

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{TRSPY_2012_278_a20,
     author = {Yu. L. Sachkov},
     title = {Closed {Euler} elasticae},
     journal = {Informatics and Automation},
     pages = {227--241},
     publisher = {mathdoc},
     volume = {278},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2012_278_a20/}
}
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Yu. L. Sachkov. Closed Euler elasticae. Informatics and Automation, Differential equations and dynamical systems, Tome 278 (2012), pp. 227-241. http://geodesic.mathdoc.fr/item/TRSPY_2012_278_a20/