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

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

2011-09-13

arxiv.org/abs/1109.2652v1

J. Differential Equations 252 (2012), 4529-4562

Mathematics

Dynamical Systems

44 pages

Scientific paper

10.1016/j.jde.2012.01.002

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 Poincar\'e disk, where we study the elliptic relative equilibria. Then we obtain the equations of motion in the Poincar\'e 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.

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