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Conformal geometry of the Kepler orbit space. (English) Zbl 0751.70006

Summary: We present a group theoretical analysis of the structure of the space \(\Omega\) 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 \(\Omega\) is introduced, which allows us a clear description of many aspects of this problem. In particular, this parametrization suggests us the introduction of a Lorentzian metric in \(\Omega\), whose conformal group \(\text{SO}(3,2)\) contains a seven-dimensional subgroup, which is induced by point transformations in the configuration space \({\mathcal X}\). A \(\text{SO}(2,1)\) subgroup of this group still acts transitively on \({\mathcal X}\), which is thus identified as a homogeneous space for \(\text{SO}(2,1)\); each regular Kepler orbit is a trace of a one-dimensional subgroup, whose canonical parameter automatically equals to the classical anomalies. These results are somehow a configuration space analogue of the geometrical structure of the Kepler problem in the velocity space previously known.

MSC:

70F05 Two-body problems
58J70 Invariance and symmetry properties for PDEs on manifolds
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