In our unitary description (Vivarelli in Meccanica 50:915–925, 2015) of the Kepler problem (obtained via the introduction of a simple structure, the primigenial sphere \(S_{p^{-1}}\)), we have shown that this sphere encompasses, in a sort of inbred order of its elements, several fundamental elements of the Kepler problem. In this paper, we show that also the mechanical energy of an elliptic Kepler orbit is an element embedded in the sphere through a peculiar point, the energy point \(P^*\). We show that this point in its circular motion on the sphere has a velocity which is strictly linked to so-called Sundman–Levi-Civita regularizing time transformation (Levi-Civita in Opere matematiche, 1973). Moreover in this spherical scenario, we reconsider both the two regularizations of the Kepler problem, namely the Bohlin–Burdet (Burdet in Z Angew Math Phys 18:434–438, 1967) and the Kustaanheimo and Stiefel (KS) regularizations (J Reine Angew Math 218:204–219, 1965): we present a geometrical interpretation of the first one, and we show an explicit link between their regularizing fundamental equations.

中文翻译：

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开普勒问题：能量点、Levi-Civita、Burdet 和通过原始球面的 KS 正则化

在我们对开普勒问题的幺正描述（Vivarelli in Meccanica 50:915–925, 2015）中（通过引入一个简单的结构，原始球体 \(S_{p^{-1}}\)），我们有表明这个球体以其元素的一种近交顺序包含开普勒问题的几个基本元素。在本文中，我们还表明椭圆开普勒轨道的机械能是通过一个特殊点嵌入球体中的元素，能量点 \(P^*\)。我们表明，该点在球体上的圆周运动中具有与所谓的 Sundman-Levi-Civita 正则化时间变换（Levi-Civita in Opere matematiche，1973）严格相关的速度。此外，在这个球形场景中，我们重新考虑开普勒问题的两个正则化，