再谈拉格朗日函数非唯一性的物理意义:广义规范变换和正则变换的等价性证明

REMARKS ON THE NON-UNIQUENESS OF LAGRANGE FUNCTION:AN EQUIVALENCE PROOF BETWEEN THE GAUGE AND CANONICAL TRANSFORMATIONS

关于力学系统拉格朗日函数非唯一性问题所蕴含的物理意义,各种教学研究期刊都刊发了不少论文进行论述,但仍存在某些争议。本文从拉格朗日函数非唯一性等价于动力学系统广义规范选择的任意性这一基本性质出发,证明无论是拉格朗日方程还是哈密顿正则运动方程都具有这种规范变换的协变性。由此,澄清了近期一些文献和网络上关于文献中规范变换是否可以看作是一种正则变换的争议,供教学参考。

Although a series of papers have been published to discuss the non-uniqueness of Lagrange function in mechanics,certain arguments on this topic still exist.Here,based on the equivalence between the non-uniqueness of Lagrange function and the arbitrary of gauge choice,we show that both the Lagrangian equation and Hamiltonian canonical equation are gauge covariant,in detail.Specifically,we address the question under debate,i.e.,whether the transformation that leaves the Lagrange equation of motion invariant is also a canonical transformation,and show that it is Yes.Therefore,given a Lagrange function corresponds to choose a gauge,and dynamical equation is covariant under the gauge transformation.