An intrinsic approach in the curved $n$-body problem. The positive curvature case
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- by Ernesto Pérez-Chavela and J. Guadalupe Reyes-Victoria PDF
- Trans. Amer. Math. Soc. 364 (2012), 3805-3827 Request permission
Abstract:
We consider the gravitational motion of $n$ point particles with masses $m_1,m_2, \dots , m_n>0$ on surfaces of constant positive Gaussian curvature. Using stereographic projection, we express the equations of motion defined on the two-dimensional sphere of radius $R$ in terms of the intrinsic coordinates of the complex plane endowed with a conformal metric. This new approach allows us to derive the algebraic equations that characterize relative equilibria. The second part of the paper brings new results about necessary and sufficient conditions for the existence of relative equilibria in the cases $n=2$ and $n=3$.References
- A. V. Borisov, I. S. Mamaev, and A. A. Kilin, Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regul. Chaotic Dyn. 9 (2004), no. 3, 265–279. MR 2104172, DOI 10.1070/RD2004v009n03ABEH000280
- J. F. Cariñena, M. F. Ranada, and M. Santander, Response to: “Comment on: ‘Central potentials on spaces of constant curvature: the Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$’ ” [J. Math. Phys. 46 (2005), no. 11, 114101, 2 pp.; MR2186790] by A. V. Shchepetilov, J. Math. Phys. 46 (2005), no. 11, 114102, 3. MR 2186791, DOI 10.1063/1.2107287
- F. Diacu, E. Pérez-Chavela, M. Santoprete, The n-body problem in spaces of constant curvature, arXiv:0807.1747, (2008).
- Florin Diacu, On the singularities of the curved $n$-body problem, Trans. Amer. Math. Soc. 363 (2011), no. 4, 2249–2264. MR 2746682, DOI 10.1090/S0002-9947-2010-05251-1
- Florin Diacu and Ernesto Pérez-Chavela, Homographic solutions of the curved 3-body problem, J. Differential Equations 250 (2011), no. 1, 340–366. MR 2737846, DOI 10.1016/j.jde.2010.08.011
- F. Diacu, Polygonal Homographic Orbits of the Curved $n$–Body Problem. arXiv:1012.2490 (2010)
- Manfredo P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. Translated from the Portuguese. MR 0394451
- B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern geometry—methods and applications. Part I, Graduate Texts in Mathematics, vol. 93, Springer-Verlag, New York, 1984. The geometry of surfaces, transformation groups, and fields; Translated from the Russian by Robert G. Burns. MR 736837, DOI 10.1007/978-1-4684-9946-9
- M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518
- Valeriĭ V. Kozlov and Alexander O. Harin, Kepler’s problem in constant curvature spaces, Celestial Mech. Dynam. Astronom. 54 (1992), no. 4, 393–399. MR 1188291, DOI 10.1007/BF00049149
- A. V. Shchepetilov, Comment on: “Central potentials on spaces of constant curvature: the Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$” [J. Math. Phys. 46 (2005), no. 5, 052702, 25 pp.; MR2143000] by J. F. Cariñena, M. F. Rañada and M. Santander, J. Math. Phys. 46 (2005), no. 11, 114101, 2. MR 2186790, DOI 10.1063/1.2107267
- Alexey V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A 39 (2006), no. 20, 5787–5806. MR 2238116, DOI 10.1088/0305-4470/39/20/011
- A.F. Stevenson, Note on the Kepler problem in a spherical space, and the factorization method for solving eigenvalue problems, Phys. Rev. 59 (1941), 842-843.
- Tatiana G. Vozmischeva, Integrable problems of celestial mechanics in spaces of constant curvature, Astrophysics and Space Science Library, vol. 295, Kluwer Academic Publishers, Dordrecht, 2003. MR 2027774, DOI 10.1007/978-94-017-0303-1
Additional Information
- Ernesto Pérez-Chavela
- Affiliation: Departamento de Matemáticas, UAM-Iztapalapa, Av. San Rafael Atlixco 186, México, D.F. 09340, Mexico
- Email: epc@xanum.uam.mx
- J. Guadalupe Reyes-Victoria
- Affiliation: Departamento de Matemáticas, UAM-Iztapalapa, Av. San Rafael Atlixco 186, México, D.F. 09340, Mexico
- Email: revg@xanum.uam.mx
- Received by editor(s): November 26, 2010
- Received by editor(s) in revised form: January 25, 2011
- Published electronically: February 20, 2012
- Additional Notes: Both authors thank the anonymous referees for their deep review of the original version and for their valuable comments and suggestions that helped us to improve this work. This work has been partially supported by CONACYT, México, Grant 128790.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 3805-3827
- MSC (2010): Primary 70F15, 34A26; Secondary 70F10, 70F07
- DOI: https://doi.org/10.1090/S0002-9947-2012-05563-2
- MathSciNet review: 2901235