Canad. Math. Bull. Vol. 00 (0), 2020 pp. 1–13 http://dx.doi.org/10.4153/xxxx © Canadian Mathematical Society 2020 On Moser’s Regularization of the Kepler System. Positive and Negative Energies Sebastián Ferrer and Francisco Crespo Abstract. We present a generalization of Moser’s theorem on the regularization of Keplerian sys- tems that includes positive and negative energies. Our approach does not consider the hyperboloid’s geodesics embedded in a Lorentz space for the unbounded orbits, as it is previously done in the lit- erature. Instead, we connect the Keplerian positive and negative energy orbits with the harmonic oscillator with negative and positive frequencies. The connection is established through the canoni- cal extension of the stereographic projection, as it is done in Moser’s original paper. How we base our study reveals that KS and Moser regularizations are alternative ways of showing the spatial Kepler system as a sub-dynamics of the 4D harmonic oscillator. 1 Introduction and Main Result The spatial Kepler system describes the motion of a particle in a central potential and has the following energy function K μ = 1 2 | y| 2 - μ | x| , (1.1) where μ is the positive gravitational constant and x, y ∈ T * R 3 0 , with R n 0 = R n -{0}. In this paper, we consider the Kepler system’s regularization, which has been done in var- ious forms by several authors. After reparametrization of the independent variable and imposing several constraints, Moser [13, 15] and Kustaanheimo and Stiefel [9, 10] linked the Keplerian flow at each energy level with well-known linear systems. Each procedure has pros and cons; Moser’s technique is easily stated for arbitrary dimension, but it is not suitable for positive energy in its original formulation. The KS regularization is not restricted to bounded orbits; however, it works only for the spatial case. In this work, we focus on Moser’s procedure. The main result of this paper is the extension of the Moser theorem for the case of unbounded Keplerian orbits. This relies on a canonical stereographic-type transforma- tion ST , result of considering the stereographic projection in the simplectic contex, which will be detailed in Section 4. Furthermore, in our proof, the Keplerian orbits are embedded in the Hamiltonian flow given by H ω = 1 2  | p| 2 + ω | q| 2  , (1.2) 2020 Mathematics Subject Classification: 70F15,70F16,70H05. Keywords: Stereographic-type transformation, Kepler system, harmonic oscillator. 2020/12/13 09:51 This is a ``preproof'' accepted article for Canadian Mathematical Bulletin This version may be subject to change during the production process. DOI: 10.4153/S0008439520000983 Downloaded from https://www.cambridge.org/core. 28 Apr 2021 at 21:28:42, subject to the Cambridge Core terms of use.