We study the reduction and regularization processes of perturbed Keplerian systems from an astronomical point of view. Our approach connects axially symmetric perturbed 4-DOF oscillators with Keplerian systems, including the case of rectilinear solutions. This is done through a preliminary reduction recently studied by the authors. Then, the reduction program continues by removing the Keplerian energy. For each value of the semi-major axis, we explain the astronomical meaning of the sextuples defining the orbit space $$\mathbb {S}^2\times \mathbb {S}^2$$ and its connection with the orbital elements. More precisely, we present alternative sextuple coordinates for the set of bounded Keplerian orbits that ‘separate’ the node of the orbital plane from the argument of perigee giving the Laplace vector in that plane. Still, the reduction of the axial symmetry defined by the third component of the angular momentum is performed. For the thrice reduced space $$\varGamma _{0,L,H}$$ we propose the Cushman–Deprit coordinates, a variant to the set given by Cushman. The main feature of these variables is that they are all with the same dimensions, which is convenient for the normalization procedure. As an application of the proposed scheme, we study the spatial lunar problem.

中文翻译：

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开普勒系统中减少的 4D 振荡器和轨道元素：Cushman-Deprit 坐标

我们从天文学的角度研究扰动开普勒系统的归约和正则化过程。我们的方法将轴对称扰动 4 自由度振荡器与开普勒系统连接起来，包括直线解决方案的情况。这是通过作者最近研究的初步减少来完成的。然后，还原程序通过去除开普勒能量而继续。对于半长轴的每个值，我们解释了定义轨道空间 $$\mathbb {S}^2\times \mathbb {S}^2$$ 的六重星的天文意义及其与轨道元素的联系。更准确地说，我们提出了一组有界开普勒轨道的替代六重坐标，这些轨道将轨道平面的节点与给出该平面中拉普拉斯矢量的近地点参数“分离”。仍然，执行由角动量的第三个分量定义的轴对称的约简。对于三次缩减空间 $$\varGamma _{0,L,H}$$，我们提出了 Cushman-Deprit 坐标，这是 Cushman 给出的集合的变体。这些变量的主要特点是它们都具有相同的维度，便于归一化过程。作为所提出方案的应用，我们研究了空间月球问题。