Regularization and Symmetry

Abstract

In Chapter 2 the physical time t, i.e., the independent variable in the equation of motion, has been replaced with the eccentric anomaly s. This substitution has the nice property of transforming the equation of motion into a regular one. In fact, the Hamiltonian vector field of the Kepler Problem is not complete in its natural phase space, as collision orbits reach the singularity of the potential in a finite time and with a divergent velocity. On the contrary, all solutions of the transformed equation remain finite for every value of the eccentric anomaly.

This is an instance of a regularization process, which in general amounts to embedding the incomplete system into a complete one as a dense open subset, or, loosely speaking, to “adding” some orbits to those of the original system. At first sight this seems not well founded, since one may object that the topology of the phase space is changed. In Souriau (1974) the situation is clarified as follows. Define the evolution space as the product of the phase space by the time. Every solution of the equations of motion is described by a curve on it. Identify all points in the evolution space that belong to the same solution curve and define the space of motions as the quotient under this equivalence relation: each point of the space of motions is thus a trajectory and the information on the global structure of the mechanical system is encoded into its topology. A fixed t = to section of the evolution space is the phase space at time to. It may happen however that the mapping: “section at to — space of motions” is not onto. An example is just the Kepler