The classical Euler problem on stationary configurations of elastic rod in the plane is studied in detail by geometric control techniques as a left-invariant optimal control problem on the group of motions of a two-dimensional plane E(2).
The attainable set is described, the existence and boundedness of optimal controls are proved. Extremals are parametrized by the Jacobi 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 the Euler 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 the Euler problem is obtained.