Richard Moeckel

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 $|T − t|$.

Solutions of the planar Kepler problem with fixed energy $h$ determine geodesics of the corresponding Jacobi–Maupertuis metric. This is a Riemannian metric on $\mathbb{R}^2$ if $h\geqslant 0$ or on a disk $\mathcal{D}\subset \mathbb{R}^2$ if $h<0$. The metric is singular at the origin (the collision singularity) and also on the boundary of the disk when $h<0$. The Kepler problem and the corresponding metric are invariant under rotations of the plane and it is natural to wonder whether the metric can be realized as a surface of revolution in $\mathbb{R}^3$ or some other simple space. In this note, we use elementary methods to study the geometry of the Kepler metric and the embedding problem. Embeddings of the metrics with $h\geqslant0$ as surfaces of revolution in $\mathbb{R}^3$ are constructed explicitly but no such embedding exists for $h<0$ due to a problem near the boundary of the disk. We prove a theorem showing that the same problem occurs for every analytic central force potential. Returning to the Kepler metric, we rule out embeddings in the three-sphere or hyperbolic space, but succeed in constructing an embedding in four-dimensional Minkowski spacetime. Indeed, there are many such embeddings.

A chaotic invariant set is constructed for the planar three-body problem. The orbits in the invariant set exhibit many close approaches to triple collision and also excursions near infinity. The existence proof is based on finding appropriate "windows" in the phase space which are stretched across one another by flow-defined Poincaré maps.