Yu.L. Sachkov, Closed Euler Elasticae

Title: Closed Euler Elasticae
Authors: Yu.L. Sachkov
Journal title: Proceedings of the Steklov Institute of Mathematics
Year: 2012
Volume: 278
Pages: 218–232

Yu. L. Sachkov, “Closed Euler elasticae”, Differential equations and dynamical systems, Collected papers, Tr. Mat. Inst. Steklova, 278, MAIK Nauka/Interperiodica, Moscow, 2012, 227–241


The classical Euler’s problem on stationary configurations of elastic rod in the 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 Pontryagin Maximum Principle. We prove that a closed elastica is either the circle or the figure 8 elastica, wrapped around itself several times. Finally, we study local and global optimality of closed elasticae: the figure 8 elastica is optimal only locally, while the circle is optimal globally.