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Differential Integral Equations, 5(1):103–136, 1992. 43 [18] F. Diacu. Painlev´e’s conjecture. Math. Intelligencer, 15(2):6–12, 1993. [19] F. Diacu. Near-collision dynamics for particle systems with quasihomogeneous potentials. J. Differential Equations, 128(1):58–77, 1996. [20] F. Diacu. Singularities of the N -body problem. In Classical and celestial mechanics (Recife, 1993/1999), pages 35–62. Princeton Univ. Press, Princeton, NJ, 2002. [21] F. Diacu, E. P´erez-Chavela, and M. Santoprete. Central configurations and total collisions for quasihomogeneous n-body problems.

As a consequence, the usual analysis of collision and near collision motions can not be extended to this case. Example 6 (N –body potential reduced by a symmetry group satisfying the rotating circle property). The paper [51] deals with minimal trajectories to the spatial 2N –body problem under the hip–hop symmetry, where the configuration is constrained at all time to form a regular antiprism. This problem has three degrees of freedom and the reduced potential of a configuration generated by the point of coordinates (u, ζ) ∈ C × R R3 decomposes as K(N ) U (u, ζ) = + U0 (u, ζ), |u|α where K(N ) = N −1 1 , sinα ( kπ N ) k=1 N U0 (u, ζ) 1 = k=1 sin2 (2k−1)π 2N |u|2 + ζ 2 α 2 , The first term comes from the interaction among points of the same N –agon and is singular at simultaneous partial collisions on the ζ–axis.

Systems, 25(3):921–947, 2005. [42] P. Painlev´e. Le¸cons sur la th´eorie analytique des ´equations diff´erentielles. Hermann, Paris, 1897. [43] H. G. Saari. Singularities of the n-body problem. I. Arch. Rational Mech. , 30:263–269, 1968. [44] H. G. Saari. Singularities of the n-body problem. II. In Inequalities, II (Proc. S. , 1967), pages 255–259. Academic Press, New York, 1970. [45] H. Riahi. Study of the generalized solutions of n-body type problems with weak forces. , 28(1):49–59, 1997. [46] H.