Conformal Geometry of the Kepler Orbit Space
Abstract
We present here a group theoretical analysis of the structure of the space Ω of orbits in the classical (plane) Kepler problem, and relate it to the description of the Kepler orbits as curves in configuration and in velocity spaces. A Minkowskian parametrization in Ω is introduced which allows us a clear description of many aspects of this problem. In particular, this parametrization suggests us the introduction in Ω of a Lorentzian metric, whose conformal group SO(3, 2) contains a sevendimensional subgroup which is induced by point transformations in the configuration space X. A SO(2, 1) subgroup of this group still acts transitively on X, which is thus identified as a homogeneous space for SO(2,1); each regular Kepler orbit is the trace of a onedimensional subgroup whose canonical parameter automatically equals to the classical anomalies. These results are somehow a configuration space analogous of the geometrical structure of the Kepler problem in the velocity space previously known.
 Publication:

Celestial Mechanics and Dynamical Astronomy
 Pub Date:
 December 1991
 DOI:
 10.1007/BF00048449
 Bibcode:
 1991CeMDA..52..307C
 Keywords:

 Conformal Mapping;
 Kepler Laws;
 Orbital Elements;
 Two Body Problem;
 Circular Orbits;
 Computational Astrophysics;
 Riemann Manifold;
 Astrophysics;
 Two body problem;
 conformal geometry