Following the work of Chazy on hyperbolic motions for the Newtonian \(n\)-body problem [J. Chazy, Ann. Sci. Éc. Norm. Supér. (3) 39, 29–130 (1922; JFM 48.1074.04)], there are many interesting researches in hyperbolic motions. Hyperbolic motions are solutions of \(n\)-body problem such that “all its interbody distances tend to infinity with time, and do so asymptotically linearly”. The main aim of the paper under review is to investigate the hyperbolic scattering problem for the Newtonian \(n\)-body problem: “for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related?”
By introducing a kind of McGehee (blown-up) coordinates, “which partially compactifies \(n\)-body phase space by adding a boundary at infinity”, the problem is in a well setting:

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first, for the blown-up flow, hyperbolic motions form the stable or unstable manifolds of normally hyperbolic equilibrium points in the added boundary manifold at infinity, and the solutions which are hyperbolic in both forward and backward time are exactly the heteroclinic orbits;

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second, the flow near the added boundary manifold can be analytically linearized locally, and this yields a new proof of Chazy’s classical asymptotic formulas;

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at last, near the added boundary manifold, that is, “when the bodies stay far apart and interact only weakly”, after obtaining a kind of continuity property of the scattering relation, the authors arrive at some elegant results on the scattering relation; in particular, they proved that for almost every initial asymptotic velocities \(A\) without collisions, there are at least a nonempty open subset of the energetically possible asymptotic velocities such that every element \(A'\) of it can be connected by a hyperbolic orbit with \(A\).

The results are important contributions to a deeper understanding of hyperbolic motions. Needless to say, the paper will be one of the fundamental literature on hyperbolic motions for the Newtonian \(n\)-body problem.