Maxwell strata in Euler's elastic problem
The classical Euler's problem on stationary configurations of elastic rod with fixed endpoints and tangents at the endpoints is considered as a left-invariant optimal control problem on the group of motions of a two-dimensional plane $\E(2)$. The attainable set is described, existence and boundedness of optimal controls are proved. Extremals are parametrized by Jacobi's elliptic functions of natural coordinates induced by the flow of the mathematical pendulum on fibers of the cotangent bundle of $\E(2)$. The group of discrete symmetries of Euler's problem generated by reflections in the phase space of the pendulum is studied. The corresponding Maxwell points are completely described via the study of fixed points of this group. As a consequence, an upper bound on cut points in Euler's problem is obtained.