Abstract
The quasihomogeneous fields (featured by sums of homogeneous potentials) model a lot of concrete fields met in problems of nonlinear particle dynamics belonging mainly to physics and astronomy, but not only. The particularly important aspect of singularities of the n-body problem in such fields is being tackled. A Painlevé-type criterion for the existence of singularities and a necessary condition for collision singularities to occur are proved. An extension of the Lagrange–Jacobi relation to this case is proved, too. The main result of the paper proves that for the three-body problem all singularities are due to collisions. Some supplementary results are added: the impossibility of the simultaneous total collapse of the n bodies in infinite time, as well as the full understanding of the local behaviour of collision solutions within the framework of the two-body problem associated to quasihomogeneous fields.
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