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An intrinsic approach in the curved n-body problem: The negative curvature case

2012; Elsevier BV; Volume: 252; Issue: 8 Linguagem: Inglês

10.1016/j.jde.2012.01.002

1090-2732

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

Stellar, planetary, and galactic studies

We consider the motion of n point particles of positive masses that interact gravitationally on the 2-dimensional hyperbolic sphere, which has negative constant Gaussian curvature. Using the stereographic projection, we derive the equations of motion of this curved n-body problem in the Poincaré disk, where we study the elliptic relative equilibria. Then we obtain the equations of motion in the Poincaré upper half plane in order to analyze the hyperbolic and parabolic relative equilibria. Using techniques of Riemannian geometry, we characterize each of the above classes of periodic orbits. For n=2 and n=3 we recover some previously known results and find new qualitative results about relative equilibria that were not apparent in an extrinsic setting.