Maxwell strata in the Euler elastic problem
Maxwell strata in the Euler elastic problem
复制标题
欧拉弹性问题中的麦克斯韦地层
DOI：
10.1007/s10883-008-9039-7
复制
发表时间：
2008
影响因子：
0.9
通讯作者：
Y. Sachkov
中科院分区：
文献类型：
--
作者：
Y. Sachkov
利用几何控制技术，详细研究了经典的弹性杆平面静止构型的欧拉问题，作为二维平面E(2)运动群上的左不变最优控制问题，描述了可达集，证明了最优控制的存在性和有界性。极值由数学摆在E(2)余切丛纤维上的流动引起的自然坐标的雅可比椭圆函数参数化。研究了由摆的相空间中的反射产生的欧拉问题的离散对称群。通过对这组不动点的研究，完整地描述了相应的麦克斯韦点。结果，获得了欧拉问题中割点的上限。
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.