  We solve the N-body problems in which the total potential energy is any
function of the mass-weighted root-mean-square radius of the system of N point
masses. The fundamental breathing mode of such systems vibrates non-linearly
for ever. If the potential is supplemented by any function that scales as the
inverse square of the radius there is still no damping of the fundamental
breathing mode. For such systems a remarkable new statistical equilibrium is
found for the other coordinates and momenta, which persists even as the radius
changes continually.
