Conformal Regularization

Abstract

In Part I we have discussed in detail the relevant points of the Kepler Problem in a pure computational way, without trying to understand its intrinsic structure. In this way one can work out explicitly the whole argument with relatively elementary mathematical tools. In Part II we change somewhat the presentation, which will require more sophisticated mathematical tools, including differential geometry and Lie group theory⁽’⁾but will provide (hopefully) a clearer understanding.

The crucial point lies in Souriau (1974) and Souriau (1983), which we have already partially reviewed: the phase space of the regularized n-dimensional Kepler Problem is symplectomorphic to a coadjoint orbit of the Lie group S00(2, n + 1), the (double covering of the) conformal group of (n + 1)-dimensional Minkowski space. Moreover, all relevant quantities of the problem (the Hamiltonian, the angular momentum, the Runge-Lenz-Laplace vector and the Fock or Bacry-Györgyi parameters) fit well into this construction. They are, in fact, components of the associated moment map. This will be explained in this chapter, which is in fact intended to be the most important of the entire book. However, as proposed in Cordani (1986) and Cordani (1988), we reverse the order of the exposition with respect to Souriau: we start from the conformal group and onlya posterioriwill we recover, by a judicious choice of the coordinates, the Kepler Problem, in such a way as to focus on what is