Clustering motions in N-body systems – free-fall motions in the Gaussian three-body problem

Clustering motions are typical and universal phenomena in N-body systems. Basic mechanisms leading to escaping and/or to trapping of particles are pursued in the analysis of a global structure for the three-body problem. The global structure of the three-body problem is numerically studied under the short range Gaussian interaction potential. As the Gaussian potential does not have any singularities at zero distance, we can avoid the computational errors in the long time simulations. Main concerns are the analysis of the collinear three-body problem, and the result compared with the case of gravitational potential. The distributions of periodic orbits are precisely searched and their stability is determined by the linear stability analysis. The collapsing of quasi-periodic motions is correlated to the destabilization of the three-body cluster in the case of the free-fall motions, and that the boundary for the collapsing tori displays fractal curves. Finally the escape diagram for two-dimensional three-body problems are discussed in comparison with the case of gravitational potential, where the remarkable difference near the triple collision is pointed out.