The KS regularization connects the dynamics of the harmonic oscillator to the dynamics of bounded Kepler orbits. Using orbit space reduction, it can be shown that reduced harmonic oscillator orbits can be identified with re-parametrized Kepler orbits by factorizing the KS map as reduction mapping followed by a chart on the reduced phase space. In this note, we will show that also other regularization maps can be obtained this way. In particular, we will show how Moser’s regularization and Ligon–Schaaf regularization are related to KS-regularization. All regularizations are a result of choosing the right invariants to represent the reduced phase space, which is isomorphic to \(T^+S^3\), and a chart on this reduced phase space. We show how this opens the way to directly reduce the KS transformed Kepler system and find other regularization maps that are valid for all values of the Keplerian energy similar to Ligon–Schaaf regularization.

KS 正则化将谐振子的动力学连接到有界开普勒轨道的动力学。使用轨道空间缩减，可以证明缩减的谐振子轨道可以通过重新参数化的开普勒轨道识别，方法是将 KS 映射分解为缩减映射，然后是缩减相空间上的图表。在本笔记中，我们将展示其他正则化图也可以通过这种方式获得。特别是，我们将展示 Moser 正则化和 Ligon-Schaaf 正则化如何与 KS 正则化相关。所有正则化都是选择正确的不变量来表示缩减的相空间的结果，它同构于\(T^+S^3\)，以及关于这个减少的相空间的图表。我们展示了这如何打开直接减少 KS 转换开普勒系统的方法，并找到其他正则化映射，这些映射对类似于 Ligon-Schaaf 正则化的开普勒能量的所有值都有效。