Total Collision with Slow Convergence to a Degenerate Central Configuration

Abstract

For total collision solutions of the \(n\)-body problem, Chazy showed that the overall size of the configuration converges to zero with asymptotic rate proportional to \(|T-t|^{\frac{2}{3}}\) where \(T\) is the collision time. He also showed that the shape of the configuration converges to the set of central configurations. If the limiting central configuration is nondegenerate, the rate of convergence of the shape is of order \(O(|T-t|^{p})\) for some \(p>0\). Here we show by example that in the planar four-body problem there exist total collision solutions whose shape converges to a degenerate central configuration at a rate which is slower that any power of \(|T-t|\).