Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An intrinsic approach in the curved $ n$-body problem. The positive curvature case


Authors: Ernesto Pérez-Chavela and J. Guadalupe Reyes-Victoria
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
Published electronically: February 20, 2012
MathSciNet review: 2901235
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. A.V. Borisov, I.S. Mamaev, A.A. Kilin, Two-body problem on a sphere: Reduction, stochasticity, periodic orbits, Regul. Chaotic Dyn. 9, 3 (2004), 265-279. MR 2104172 (2005k:37199)
  • 2. J. Cariñena, M. Rañada, M. Santander, Central potentials on spaces of constant curvature: The Kepler problem on the two dimensional sphere $ \mathbb{S}^2$ and the hyperbolic plane $ \mathbb{H}^2$, J. Math. Phys. 46 (2005), 1-17. MR 2186791 (2007h:70020)
  • 3. F. Diacu, E. Pérez-Chavela, M. Santoprete, The n-body problem in spaces of constant curvature, arXiv:0807.1747, (2008).
  • 4. F. Diacu, On the singularities of the curved $ n$-body problem. Trans. Amer. Math. Soc. 363, 4 (2011), 2249-2264. MR 2746682
  • 5. F. Diacu, E. Pérez-Chavela, Homographic solutions of the curved $ 3$-body problem. Journal of Differential Equations 250, 1 (2011), 340-366. MR 2737846
  • 6. F. Diacu, Polygonal Homographic Orbits of the Curved $ n$-Body Problem. arXiv:1012.2490 (2010)
  • 7. M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, New Jersey, USA, 1976. MR 0394451 (52:15253)
  • 8. B. Dubrovin, A. Fomenko, P. Novikov, Modern Geometry, Methods and Applications, Vol. I, II and III, Springer-Verlag, New York, 1984, 1985, 1990. MR 0736837 (85a:53003); MR 0807945 (86m:53001); MR 1076994 (91j:55001)
  • 9. V. Guillemin, M. Golubitsky, Stable mappings and their singularities, Springer-Verlag, New York, USA, 1973. MR 0341518 (49:6269)
  • 10. V.V. Kozlov, A.O. Harin, Kepler's problem in constant curvature spaces, Celestial Mech. Dynam. Astronom 54 (1992), 393-399. MR 1188291 (93i:70010)
  • 11. A.V. Shchepetilov, Comment on: Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere $ {\bf S}^2$ and the hyperbolic plane $ {\bf H}^2$, J. Math. Phys. 46, 11 (2005), 052702. MR 2186790 (2007h:70019)
  • 12. A.V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. of Physics A: Mathematicas and general, 39, 5787-5806, 2006. MR 2238116 (2007c:70014)
  • 13. 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.
  • 14. T.G. Vozmischeva, Intergrable problems of celestial mechanics in spaces of constant curvature, Kluwer Acad. Publ., Dordrecht, 2003. MR 2027774 (2005a:37099)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 70F15, 34A26, 70F10, 70F07

Retrieve articles in all journals with MSC (2010): 70F15, 34A26, 70F10, 70F07


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

DOI: https://doi.org/10.1090/S0002-9947-2012-05563-2
Keywords: Differential geometry, Riemannian conformal metric
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.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society