The Kepler problem: the energy point, the Levi-Civita, the Burdet and the KS regularizations via the primigenial sphere

Abstract

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.