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

2012; American Mathematical Society; Volume: 364; Issue: 7 Linguagem: Inglês

10.1090/s0002-9947-2012-05563-2

1088-6850

Ernesto Pérez-Chavela, J. Guadalupe Reyes-Victoria,

Black Holes and Theoretical Physics

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 $n=3$.