Abstract
We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari’s conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically.
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Communicated by P. Newton.
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Diacu, F., Pérez-Chavela, E. & Santoprete, M. The n-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria. J Nonlinear Sci 22, 247–266 (2012). https://doi.org/10.1007/s00332-011-9116-z
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DOI: https://doi.org/10.1007/s00332-011-9116-z