An intrinsic approach in the curved body problem. The positive curvature case
Authors:
Ernesto PérezChavela and J. Guadalupe ReyesVictoria
Journal:
Trans. Amer. Math. Soc. 364 (2012), 38053827
MSC (2010):
Primary 70F15, 34A26; Secondary 70F10, 70F07
Published electronically:
February 20, 2012
MathSciNet review:
2901235
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Abstract: We consider the gravitational motion of point particles with masses on surfaces of constant positive Gaussian curvature. Using stereographic projection, we express the equations of motion defined on the twodimensional sphere of radius 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 and .
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 2.
 J. Cariñena, M. Rañada, M. Santander, Central potentials on spaces of constant curvature: The Kepler problem on the two dimensional sphere and the hyperbolic plane , J. Math. Phys. 46 (2005), 117. MR 2186791 (2007h:70020)
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 4.
 F. Diacu, On the singularities of the curved body problem. Trans. Amer. Math. Soc. 363, 4 (2011), 22492264. MR 2746682
 5.
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Additional Information
Ernesto PérezChavela
Affiliation:
Departamento de Matemáticas, UAMIztapalapa, Av. San Rafael Atlixco 186, México, D.F. 09340, Mexico
Email:
epc@xanum.uam.mx
J. Guadalupe ReyesVictoria
Affiliation:
Departamento de Matemáticas, UAMIztapalapa, Av. San Rafael Atlixco 186, México, D.F. 09340, Mexico
Email:
revg@xanum.uam.mx
DOI:
http://dx.doi.org/10.1090/S000299472012055632
PII:
S 00029947(2012)055632
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.
