This article offers a review of the history of radar research and its application in the 20th century. After describing the wartime work of Sin-Itiro Tomonaga and his theory of the cavity magnetron, we formulate the equations of motion of an electron in a cavity magnetron using action-angle variables. This means following the electron’s path on its way from a cylindrical cathode moving toward a co-axial cylindrical anode in presence of a uniform magnetic field parallel to the common axis. After analyzing the situation without coupling to an external oscillatory electric field, we employ methods of canonical perturbation theory to find the resonance condition between the frequencies of the free theory ω _r , ω _ϕ and the applied perturbing oscillatory frequency ω. A long-time averaging process will then eliminate the periodic terms in the equation for the now time-dependent action-angle variables. The terms that are no longer periodic will cause secular changes so that the canonical action-angle variables (J, δ) change in a way that the path of the electron will deform gradually so that it can reach the anode. How the ensemble of the initially randomly distributed electrons forms spokes and how their energy is conveyed to the cavity-field oscillation is the main focus of this article. Some remarks concerning the importance of results in QED and the invention of radar theory and application conclude the article.

中文翻译：

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论Tomonaga的分裂阳极磁控管理论

本文回顾了雷达研究的历史及其在20世纪的应用。在描述Sin-Itiro Tomonaga的战时工作和他的腔磁控管理论之后，我们使用作用角变量来公式化电子在腔磁控管中的运动方程。这意味着在存在平行于公共轴的均匀磁场的情况下，沿着电子路径从圆柱形阴极向同轴圆柱形阳极移动。分析形势，而不耦合到外部振荡电场之后，我们采用经典的微扰理论的方法来寻找免费的理论频率之间的共振条件ω _[R ，ω _φ 和所施加的扰动振荡频率 ω。长时间的平均过程将消除方程中现在与时间相关的动作角度变量的周期项。不再周期性的项将引起长期变化，从而规范作用角变量（J，δ）发生变化，电子的路径将逐渐变形，从而可以到达阳极。最初随机分布的电子的整体如何形成辐条以及如何将其能量传递到腔场振荡是本文的主要重点。本文总结了有关结果在QED中的重要性以及雷达理论和应用发明的一些评论。