Transformation is an important means to study problems in analytical mechanics. It is often difficult to solve dynamic equations, and the use of variable transformation can make the equations easier to solve. The theory of canonical transformations plays an important role in solving Hamilton’s canonical equations. Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. This paper attempts to extend the canonical transformation theory of Hamilton systems to Birkhoff systems and establish the generalized canonical transformation of Birkhoff systems. First, the definition and criterion of the generalized canonical transformation for the Birkhoff system are established. Secondly, based on the criterion equation and considering the generating functions of different forms, six generalized canonical transformation formulas are derived. As special cases, the canonical transformation formulas of classical Hamilton’s equations are given. At the end of the paper, two examples are given to illustrate the application of the results.

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Birkhoff系统的广义规范变换理论

转换是研究分析力学问题的重要手段。通常很难求解动态方程，使用变量变换可以使方程更容易求解。规范变换理论在求解汉密尔顿规范方程中起着重要作用。伯克霍夫力学是哈密顿力学的自然概括。本文试图将哈密顿系统的正则变换理论扩展到伯克霍夫系统，并建立伯克霍夫系统的广义正则变换。首先，建立了Birkhoff系统的广义规范变换的定义和判据。其次，基于准则方程，考虑不同形式的生成函数，推导了六个广义规范变换公式。作为特殊情况，给出了经典汉密尔顿方程的典范变换公式。在本文的最后，给出了两个例子来说明结果的应用。