We present a generalization of Moser’s theorem on the regularization of Keplerian systems that include positive and negative energies. Our approach does not consider the geodesics of the hyperboloid embedded in a Lorentz space for the unbounded orbits, as it is previously done in the literature. 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 canonical extension of the stereographic projection, as it is done in Moser’s original paper. How we base our study reveals that Kustaanheimo–Stiefel map KS and Moser regularizations are alternative ways of showing the spatial Kepler system as a subdynamics of the 4D harmonic oscillator.

我们提出了关于包括正能量和负能量的开普勒系统正则化的 Moser 定理的推广。我们的方法不考虑嵌入在洛伦兹空间中的双曲面的测地线，用于无界轨道，正如之前在文献中所做的那样。相反，我们将开普勒正负能量轨道与具有负频率和正频率的谐振子连接起来。这种连接是通过立体投影的规范扩展建立的，正如 Moser 的原始论文中所做的那样。我们的研究基础表明，Kustaan​​heimo-Stiefel 映射 KS 和 Moser 正则化是将空间开普勒系统显示为 4D 谐振子的子动力学的替代方法。