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Total collison, completely parabolic motions and homothety reduction in the \(n\)-body problem. (Collisions totales, mouvements complètement paraboliques et réduction des homothéties dans le problème des \(n\) corps.) (French) Zbl 0973.70011

The author studies some interesting properties of the \(n\)-body problem. This problem simulates the motion of \(n\) particles under the influence of mutual force fields based, in general, on an inverse square law. As is well known, the two-body problem can be solved analytically, and the three-body problem is sufficiently complicated, so that only the planar restricted case can be simply treated. By considering \(2k\)-homogeneous potentials, the author gives a new look at some important results of Sundman, McGehee and Saari. In particular, using the homothety symmetry, a reduction procedure is given, and the singularities of the reduced vector field are analyzed.

MSC:

70F10 \(n\)-body problems
70F16 Collisions in celestial mechanics, regularization
70F15 Celestial mechanics
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