标题： 欧拉弹性问题中的麦克斯韦流形 显示英文标题

标题： Maxwell strata in Euler's elastic problem

摘要： 经典欧拉问题关于固定端点和端点切线的弹性杆的平衡配置被考虑为二维平面运动群上的左不变最优控制问题$\E(2)$。 可达集被描述，证明了最优控制的存在性和有界性。 极值通过由余切丛上数学摆流动引起的自然坐标下的雅可比椭圆函数参数化$\E(2)$。 研究了由摆相空间中的反射生成的欧拉问题的离散对称群。 通过研究该群的不动点，完全描述了相应的马克斯韦点。 作为结果，得到了欧拉问题中截断点的上界。 显示英文摘要

摘要： 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.