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Transactions of the American Mathematical Society
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
Published electronically: February 20, 2012
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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$.


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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: http://dx.doi.org/10.1090/S0002-9947-2012-05563-2
PII: S 0002-9947(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.