A computer-assisted proof of Saari's conjecture for the planar three-body problem

The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are t...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2005-08, Vol.357 (8), p.3105-3117
Main Author: Moeckel, Richard
Format: Article
Language:English
Subjects:
Algebra
Astronomy
Celestial mechanics (including n-body problems)
Coefficients
Coordinate systems
Earth, ocean, space
Exact sciences and technology
Fundamental areas of phenomenology (including applications)
Fundamental astronomy
Fundamental astronomy and astrophysics. Instrumentation, techniques, and astronomical observations
Global analysis, analysis on manifolds
Mathematical constants
Mathematics
Moments of inertia
Normal vectors
Physics
Polynomials
Polytopes
Sciences and techniques of general use
Solid dynamics (ballistics, collision, multibody system, stabilization...)
Solid mechanics
Three body problem
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Vertices
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Summary:The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-04-03527-5