No CrossRef data available.
Article contents
On Moser’s regularization of the Kepler system: Positive and negative energies
Part of:
Hamiltonian and Lagrangian mechanics
Published online by Cambridge University Press: 18 December 2020
Abstract
We present a generalization of Moser’s theorem on the regularization of Keplerian systems that include positive and negative energies. Our approach does not consider the geodesics of the hyperboloid embedded in a Lorentz space for the unbounded orbits, as it is previously done in the literature. Instead, we connect the Keplerian positive and negative energy orbits with the harmonic oscillator with negative and positive frequencies. The connection is established through the canonical extension of the stereographic projection, as it is done in Moser’s original paper. How we base our study reveals that Kustaanheimo–Stiefel map KS and Moser regularizations are alternative ways of showing the spatial Kepler system as a subdynamics of the 4D harmonic oscillator.
MSC classification
Primary:
70F15: Celestial mechanics
- Type
- Article
- Information
- Copyright
- © Canadian Mathematical Bulletin 2020
References
Belbruno, E. A., Two-body motion under the inverse square central force and equivalent geodesic flows. Celest. Mech. 15(1977), no. 4, 467–476.CrossRefGoogle Scholar
Crespo, F. and Ferrer, S., Alternative reduction by stages of Keplerian systems. Positive, negative, and zero energy. SIAM J. Appl. Dyn. Syst. 19(2020), no. 2, 1525–1539. http://dx.doi.org/10.1137/19M1264060.CrossRefGoogle Scholar
Cushman, R. H. and Bates, L. M., Global aspects of classical integrable systems. 2nd ed., Birkhäuser Verlag, Basel, 2015.CrossRefGoogle Scholar
Cushman, R. H. and Duistermaat, J. J., A characterization of the Ligon-Schaaf regularization map. Comm. Pure Appl. Math. 50(1997), no. 8, 773–787.3.0.CO;2-3>CrossRefGoogle Scholar
Ferrer, S. and Crespo, F., Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. J. Geom. Mech. 6(2014), no. 4, 479–502.CrossRefGoogle Scholar
Ferrer, S. and Crespo, F., Alternative angle-based approach to the
$\mathbf{\mathcal{KS}}$
-map. An interpretation through symmetry and reduction. J. Geom. Mech. 10(2018), no. 3, 359–372. http://dx.doi.org/10.3934/jgm.2018013.CrossRefGoogle Scholar
$\mathbf{\mathcal{KS}}$
-map. An interpretation through symmetry and reduction. J. Geom. Mech. 10(2018), no. 3, 359–372. http://dx.doi.org/10.3934/jgm.2018013.CrossRefGoogle ScholarGoldstein, H., Poole, C., and Safko, J., Classical mechanics. 3rd ed., Addison Wesley, New York, 2002.Google Scholar
Kummer, M., On the regularization of the Kepler problem. Commun. Math. Phys. 84(1982), no. 1, 133–152.CrossRefGoogle Scholar
Kustaanheimo, P., Spinor regularization of the Kepler motion. Annales Universitatis Turkuensis 73(1964), no. 3, 3–7.Google Scholar
Kustaanheimo, P. and Stiefel, E., Perturbation theory of Kepler motion based on spinor regularization. J. Reine Angew. Math. 218(1965), 204–219.Google Scholar
Ligon, T. and Schaaf, M., On the global symmetry of the classical Kepler problem. Rep. Math. Phys. 9(1976), no. 3, 281–300.CrossRefGoogle Scholar
Marsden, J. E. and Ratiu, T. S., Introduction to mechanics and symmetry. 2nd ed., Springer-Verlag, Inc., New York, 1999.CrossRefGoogle Scholar
Moser, J., Regularization of Kepler’s problem and the averaging method on a manifold. Commun. Pure Appl. Math. XXIII(1970), 609–636.10.1002/cpa.3160230406CrossRefGoogle Scholar
Moser, J. and Zehnder, E. J., Notes on dynamical systems. AMS and the Courant Institute of Mathematical Sciences at New York University, Providence, 2005.CrossRefGoogle Scholar
Osipov, Y. S., The Kepler problem and geodesic flows in spaces of constant curvature. Celest. Mech. 16(1977), no. 2, 191–208.CrossRefGoogle Scholar
van der Meer, J. C., Reduction of the harmonic oscillator and regularization of the Kepler problem. CASA-Rep. 20(2020), no. 1, 23.Google Scholar