We provide a detailed classical description of the oscillatory-precessional motion of an electron in the field of an electric dipole. Specifically, we demonstrate that in the general case of the oscillatory-precessional motion of the electron (the oscillations being in the meridional direction (θ-direction) and the precession being along parallels of latitude (φ-direction)), both the θ-oscillations and the φ-precessions can actually occur on the same time scale—contrary to the statement from the work by another author. We obtain the dependence of φ on θ, the time evolution of the dynamical variable θ, the period Tθ of the θ-oscillations, and the change of the angular variable φ during one half-period of the θ-motion—all in the forms of one-fold integrals in the general case and illustrated it pictorially. We also produce the corresponding explicit analytical expressions for relatively small values of the projection pφ of the angular momentum on the axis of the electric dipole. We also derive a general condition for this conditionally-periodic motion to become periodic (the trajectory of the electron would become a closed curve) and then provide examples of the values of pφ for this to happen. Besides, for the particular case of pφ = 0 we produce an explicit analytical result for the dependence of the time t on θ. For the opposite particular case, where pφ is equal to its maximum possible value (consistent with the bound motion), we derive an explicit analytical result for the period of the revolution of the electron along the parallel of latitude.

中文翻译：

---

里德堡电子绕极分子的振荡-进动运动

我们提供了电子在电偶极子场中的振荡进动运动的详细经典描述。具体来说，我们证明了在电子的振荡进动运动的一般情况下（振荡在子午方向（θ 方向）和进动沿纬度的平行线（φ 方向）），θ-振荡和 φ 进动实际上可以在同一时间尺度上发生——与另一位作者的作品中的陈述相反。我们获得了 φ 对 θ 的依赖性、动力变量 θ 的时间演化、θ 振荡的周期 Tθ 以及角变量 φ 在 θ 运动的一个半周期内的变化——所有形式一般情况下的一重积分并用图解说明。我们还为角动量在电偶极子轴上的投影 pφ 的相对较小的值生成了相应的显式解析表达式。我们还推导出这种有条件的周期性运动变为周期性的一般条件（电子的轨迹将变为闭合曲线），然后提供 pφ 值的示例以使其发生。此外，对于 pφ = 0 的特殊情况，我们对时间 t 对 θ 的依赖性产生了明确的分析结果。对于相反的特殊情况，其中 pφ 等于其最大可能值（与束缚运动一致），我们推导出了电子沿平行纬线旋转周期的明确分析结果。