Alternative Reduction by Stages of Keplerian Systems. Positive, Negative, and Zero Energy

Abstract

This work deals with the full reduction of the spatial Kepler system for bounded and unbounded motions. Precisely, we consider the four-dimensional oscillator associated to the Kepler system and carry out our program in three stages: axial-axial-energy rather than energy-axial-axial as is customary. This approach reveals the true role of the $\mathcal{KS}$ map and the bilinear relation. Our development allows for a global analysis at each reduction stage providing a complete description of each reduced space. The first reduced space is described by means of only six invariants leading to a nonsingular six-dimensional Poisson manifold. Then, the classical bilinear relation is replaced by the momentum map of a geometric reduction. Furthermore, the second reduced space is given as the product of two hyperboloids and has cones as singular strata. For the last stage, we distinguish among the possible sign of the energy. The positive and zero cases bring new noncompact reduced spaces which are described in detail.