非完整映射理论与刚体定点转动的几何描述  

Nonholonomic mapping theory and geometric formulation for rotation of a rigid body with one fixed point

作　　者：王勇[1] 郭永新[2] 吕群松[3] 刘畅[4] 

机构地区：[1]广东医学院基础学院,东莞523808 [2]辽宁大学物理学院,沈阳110036 [3]广东医学院实验管理处,东莞523808 [4]北京理工大学应用力学系,北京100081

出　　处：《物理学报》2009年第8期5142-5149,共8页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:10872084和10472040);辽宁省优秀青年科研人才培养基金(批准号:3040005);辽宁省高校科研基金(批准号:2008S098);辽宁省高等学校优秀人才支持计划(批准号:2008RC20);辽宁省重点实验室建设计划(批准号:2008403009)资助的课题~~

摘　　要：利用非完整映射方法,从一个已知Riemann空间构造一个嵌入其中的Riemann-Cartan空间.作为特例,研究从Euclidean空间构造Weitzenbock空间的方法.基于dAlembert-Lagrange原理和非完整映射,将一个Riemann空间的测地线对应于另一个Riemann-Cartan空间的自平行线.把这种非完整映射理论应用到刚体定点转动问题上,得到了刚体运动的欧拉方程是欧拉角描述的Riemann位形空间的测地线方程,而在刚体角速度对应的准坐标空间上是常挠率Riemann-Cartan空间的自平行线方程的结论.The method of nonholonomic mapping is adopted to construct a Riemann-Cartan space embedded in a known Riemann space. As a special case, Weitzenbock space is embedded in an Euclidean space. By means of the nonholonomic mapping and d＇ Alembert-Lagrange principle a geodesic in a Riemann space is mapped to an autoparallel in a Riemann-Cartan space. The mapping theory is applied to the problem of rotation of a rigid body with a fixed point. It is proved that Euler equations for the rigid body are equations of geodesic in the Riemann configuration space described by Euler angles, whereas the equations in the pseudo-coordinate space corresponding to angular velocities of the rigid body are equations of autoparallel in the Riemann-Cartan space with constant torsion.

关 键 词：欧拉角 非完整映射 Riemann-Cartan空间 自平行线 

分 类 号：O186.1[理学—数学] O411[理学—基础数学]