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

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
252
Issue
8
Identifiers
DOI: 10.1016/j.jde.2012.01.002
Disciplines
  • Mathematics

Abstract

Abstract 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.

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