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 seven-dimensional 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 one-dimensional 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.
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References
Hamilton, W. R.: 1846, Proc. Roy. Irish Acad. 3, 344.
Györgi, G.: 1968, Nuovo Cim. 53A, 717.
Moser, J.: 1970, Commun. Pure Appl. Math. 23, 609.
Osipov, Yu. S.: 1972, Uspehi Math. Nauk 27 (2), 161.
Belbruno, E. A.: 1977, Cel. Mech. 15, 467.
Belbruno, E. A.: 1977, Cel. Mech. 16, 191.
Milnor, J.: 1983, Amer. Math. Monthly 90, 353.
Crampin, M. and Pirani, F. A. E.: 1986, Applicable Differential Geometry, Cambridge Univ. Press.
Laplace, P. S.: 1799, Traité de Mechanique celeste, Paris.
Runge, C.: 1919, Vektoranalysis, Dutton, New York.
Herrick, S.: 1971, Astrodynamics, Vol. 1, Van Nostrand Reinhold.
Evans, N. W.: 1991, Phys. Rev., in press.
Devaney, R. L.: 1982, Am. Math. Monthly 89, 535.
Pedoe, D. 1977, A course of Geometry.
Yaglom, I. M.: 1979, A Simple Non-Euclidean Geometry and its Physical Basis, Berlin: Springer.
Cariñena, J. F., del Olmo, M. A., and Santander, M.: 1985, J. Phys. A: Math Gen. 18, 1855.
Arnold, V.: 1986, Dynamical Systems III, Springer, Berlin, p. 13.
Penrose, R. and Rindler, W.: 1986, Spinors and Space-Time, Vol. 2, Cambridge University Press, Chap. 9.
Boya, J., Cariñena, J. F., and Santander, M.: 1975, J. Math. Phys. 16, 1813.
Beckers, J. et al.: 1978, J. Math. Phys. 19, 2126.
Golubitsky, M.: J. Differential Geometry 7, 175.
Klein, J.: 1982, New Developments in Analytical Mechanics, Proc. of the lUTAM, Torino.
Juárez, M. and Santander, M.: 1982, J. Phys A: Math. Gen. 15, 3411.
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On leave from Depto. de Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain
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Cariñena, J.F., López, C., del Olmo, M.A. et al. Conformal geometry of the Kepler orbit space. Celestial Mech Dyn Astr 52, 307–343 (1991). https://doi.org/10.1007/BF00048449
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DOI: https://doi.org/10.1007/BF00048449