Research on the historical development of analytical mechanics that was applicable to the old quantum theory

本研究考察了世纪中期建立的哈密顿-雅可比理论的原型适用于旧量子理论的历史过程。在这个理论的各个方面，我们强调的是规范变换的概念。正则变换是将哈密顿-雅可比理论应用于旧量子理论的最重要的基本因素之一。我们注意到，在世纪，哈密顿-雅可比理论不仅在物理学中得到了有效的应用，而且在天体力学中得到了有效的应用。通过研究旧量子理论诞生之前发表的一些天体力学文章，我们发现以下事实：庞加莱在1892年出版的《梅卡尼克塞莱斯特》一书中首次提出了正则变换与哈密顿-雅可比理论完全解之间的关系。[99] Charlier在1902年和1907年出版的《机械论》（Mechanik des Himmel）一书中对这一关系给出了精确的证明。他事实上改变了典型的运动方程，并表明如何角变量，这往往是用于天体动力学，成为典型的元素。史瓦西参考了这个结果，并成功地构建了作用量和角动量，为解释斯塔克效应提供了合适的数学工具。史瓦西的新的正则变量有效地用于解释旧的量子理论基本上来自天体力学的结果。

This research examines the historical process that the so-called Hamilton-Jacobi theory, whose proto-type was established in the middle of nineteenth century, became applicable to the old quantum theory. Among the various aspects of this theory we highlight is the notion of the canonical transformation. The canonical transformation was one of the most important essential factors for the application of the theory to the old quantum theory.We note that the Hamilton-Jacobi theory was effectively used not in physics but in celestial mechanics in the nineteenth century. By examining some articles of the celestial mechanics published just before the birth of the old quantum theory, we find the following facts: It was Poincare that first mentioned the relation between the canonical transformation and the complete solution of the Hamilton-Jacobi theory in his large work of Mecanique Celeste published in 1892-99.Charlier gave an exact proof of this relation in his Mechanik des Himmel in 1902 and 1907. He in fact transformed the canonical equations of motion and showed how the angular variables, which often used in celestial dynamics, become the canonical elements. Schwarzschild referred to this result and succeeded in constructing the action and angular momentum that provided appropriate mathematical tools for explaining the Stark effects. Schwarzschild's new canonical variables effectively used in explaining the old quantum theory were essentially derived from the results of celestial mechanics.