Noncollision Singularities in the Four-Body Problem

Noncollision Singularities: Do Four Bodies Suffice!

then we say that there is a collision between bodies i and j at t = t∗. Can there be a singularity without a collision? For example, Poincaré suggested, ri(t) might tend to infinity, or oscillate wildly (like sin 1 t ) as t→ t∗. Although Poincaré never wrote anything about noncollision singularities, Painlèvé gave him credit for being the first to ask this question. Painlèvé himself was able to...

Noncollision Singularities in the Newtonian N-body Problem: Results and Perspectives

Singularities in the Newtonian n-body Problem

In this note, we study the structure of singularities of the Newtonian n-body problem where we establish certain relations between collisions and noncollision singularities. Based on our results, we conjecture that the set of all accumulation points of collisions in the extended phase space is exactly the set of initial conditions that lead to a noncollision singularity.

ON THE SINGULARITIES OF THE CURVED n-BODY PROBLEM

We study singularities of the n-body problem in spaces of constant curvature and generalize certain results due to Painlevé, Weierstrass, and Sundman. For positive curvature, some of our proofs use the correspondence between total collision solutions of the original system and their orthogonal projection—a property that offers a new method of approaching the problem in this particular case.

Integral equations for the four-body problem