Chenciner, A. 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 Regul. Chaotic Dyn. 3, No. 3, 93-106 (1998). 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. Reviewer: David Martin de Diego (Madrid) Cited in 22 Documents MSC: 70F10 \(n\)-body problems 70F16 Collisions in celestial mechanics, regularization 70F15 Celestial mechanics Keywords:\(n\)-body problem; completely parabolic motions; homothety reduction; inverse square law; homogeneous potentials; homothety symmetry; singularities of reduced vector field PDFBibTeX XMLCite \textit{A. Chenciner}, Regul. Chaotic Dyn. 3, No. 3, 93--106 (1998; Zbl 0973.70011)