准坐标下约束Hamilton系统的Noether对称性与守恒量研究

Noether Symmetries and Conserved Quantities of Constrained Hamilton Systems with Quasi Coordinates

本文研究在相空间中的准坐标下非保守奇异系统的Noether对称性和守恒量。首先,将奇异性导致的内在约束按外在非完整约束等效处理,利用Euler-Lagrange方程变换得到准坐标下的约束Hamilton系统的正则方程;其次引进时间、准坐标和广义动量的无限小变换,得到系统Hamilton作用量在此变换下的Noether广义准对称性的定义、判据和定理,并研究了该系统的Noehter对称性逆问题。研究结果表明,准坐标下的约束力学系统比广义坐标下的约束力学系统更具有普遍性,准坐标可使奇异系统表达更简洁。

We study the Noether symmetries and conserved quantities of non-conservative singular systems with quasi coordinates in phase space.Firstly,the internal constraints induced by singularity are equivalent considered as extrinsic non-holonomic constraints,the canonical equations of constrained Hamilton systems with quasi coordinates are obtained by using transform to the Euler-Lagrange equations.Secondly,the infinitesimal transformations of time,quasi coordinates and generalized momentum are introduced.The definition,criterion and Noether theorem are obtained according to the regular action quantity keep generalized quasi invariance under the transformation,meanwhile,the inverse problem of Noehter symmetry is also studied.Finally,an example is given to illustrate the application.The results show that the constrained mechanical system with quasi-coordinates is more universal than with generalized coordinates,and it can make the structure description of singular systems easier.