Exponential Mapping in Euler’s Elastic Problem
Exponential Mapping in Euler’s Elastic Problem
复制标题
欧拉弹性问题中的指数映射
DOI：
10.1007/s10883-014-9211-1
复制
发表时间：
2013
影响因子：
0.9
通讯作者：
E. Sachkova
中科院分区：
文献类型：
--
作者：
Y. Sachkov;E. Sachkova
研究了弹性杆在端点固定且端点处有切线的平面上的最优构型的欧拉问题。描述了参数化极值轨迹的指数映射的全局结构。证明了指数映射的原象和象中麦克斯韦地层切割出的开域是微分同构映射的。因此，具有给定边界条件的全局最优弹性的计算可以简化为求解在开放域中具有唯一解的代数方程组。对于某些特殊的边界条件，给出了最优弹性。
The Euler problem on optimal configurations of elastic rod in the plane with fixed endpoints and tangents at the endpoints is considered. The global structure of the exponential mapping that parameterises extremal trajectories is described. It is proved that open domains cut out by the Maxwell strata in the preimage and image of the exponential mapping are mapped diffeomorphically. As a consequence, computation of globally optimal elasticae with given boundary conditions is reduced to solving systems of algebraic equations having unique solutions in the open domains. For certain special boundary conditions, optimal elasticae are presented.