Asymptotic velocity for four celestial bodies

Abstract

Asymptotic velocity is defined as the Ces\`aro limit of velocity. As such, its existence has been proven for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here we show for a class of pair potentials including the homogeneous ones of degree -a for 0<a<2, that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface.