On the final evolution of the n-body problem

The Newtonian n-body problem is studied. The main result gives the asymptotic properties of the distances between particles as time (t) approaches infinity. It is shown how particles can separate into subsystems between which the distances are asymptotic to constant multiples of time, and within which the mutual distances are at most of the order $\text{t}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Each subsystem behaves asymptotically like a pure n-body problem in the sense that an energy and an angular momentum relationship are asymptotically satisfied. These techniques allow some three-body results found by Birkhoff and by Sundman to be extended to the n-body problem. Finally, some n-body results are derived for a motion which expands faster than time.