Asymptotic velocity for four celestial bodies

Asymptotic velocity is defined as the Cesaro limit of velocity. As such, its existence has been proved for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more b o d Here, we show for n class of pair potentials including the homogeneous ones of degree -alpha for alpha is an element of (0,2), that asymptotic velocities exist for up to tour bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface. This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.