Birkhoff系统的广义正则变换及其基本形式

动力学方程的积分问题是分析力学研究的一个重要方面．由于求解一般的动力学方程往往会遇到很大困难，因此可利用变量变换，使方程变得容易求解．文章研究Birkhoff系统的广义正则变换．首先，建立Birkhoff系统的广义正则变换的充分必要条件；其次，基于该条件，给出Birkhoff系统的广义正则变换的六种基本形式，导出每一种情况下新旧变量之间的变换关系．作为特例，文中给出Hamilton方程的正则变换．文末，给出算例以说明结果的应用．

The Generalized Canonical Transformations of Birkhoffian Systems and Their Basic Formulations

The study of integration theory of dynamical equations is an important aspect of analytical mechanics. Since it is often very difficult to solve general dynamical equations, the method of variable transformation is used to solve the equations. In this paper, the generalized canonical transformations of a Birkhoffian system are studied. Firstly, the sufficient and necessary conditions for the generalized canonical transformations of the Birkhoffian system are established. Secondly, based on this condition, six basic forms of generalized canonical transformations of the Birkhoffian system are given, and the transformation relations between the old and the new variables in each case are derived. As a special case, the canonical transformations of Hamilton equations are given in this paper. Finally, two examples are given to illustrate the application of the results.