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.