Lagrange系统的Lie对称性与动力学逆问题

Lie symmetry and inverse problem of dynamics for Lagrange system

研究Lagrange系统的Lie对称性以及与Lie对称性相关的动力学逆问题.一个动力学逆问题总会涉及一个动力学正问题.因此,先给出系统动力学正问题的提法和解法.为研究逆问题,首先,将已知积分当作由Lie对称性导致的Noether守恒量.其次,无限小变换的生成元可由Noether逆定理得到.第三,利用Killing方程建立系统Lagrange函数,无限小生成元和规范函数之间的关系.最后,验证所得生成元是否Lie的.

A Lie symmetry and an inverse problem of dynamics related to the Lie symmetry for the Lagrange system are studied. An inverse problem of dynamics relates often to a direct problem of dynamics. Therefore, at first, the direct problem of dynamics of the system is proposed and is solved. To study the inverse problem, at first, a given integral is considered as a Noether conserved quantity obtained by Lie symmetry. Second, the generators of infintesimal transformations can be obtained by the inverse Noether theorem. Third, the Killing equations are used to establish the relation between the Lagrangian, the infinitesimal generators and the gauge function. Finally, to verify the generators obtained are or not Lie symmetrical.