On the singularities of the curved $n$-body problem

Abstract

Abstract. 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.

The n-Body Problem in Spaces of Constant Curvature. Part II: Singularities

“…The results of Painlevé don't remain intact in our problem, [23], [21], so whether pseudocollisions exist for κ = 0 is not clear. Nevertheless, we will now show that there are solutions ending in collision-antipodal singularities of the equations of motion, solutions these singularities repel, as well as solutions that are not singular at such configurations.…”

Homographic solutions of the curved 3-body problem

“…So the flow of system (12) consists in this case solely of homoclinic orbits to the fixed point (κ −1/2 , 0), orbits whose existence is claimed in Theorem 2(iv). Some of these trajectories may come very close to a total collapse, which they will never reach because only solutions with zero angular momentum (like the homothetic orbits) encounter total collisions, as proved in [4]. So the orbits cannot reach any singularity of the line r = 0, and neither can they begin or end in a singularity of the line r = κ −1/2 .…”

“…The discovery of new orbits of the curved 3-body problem, as defined here in the spirit of an old tradition, might help us extend our understanding of space to larger scales. So far, the only other existing paper on the curved n-body problem, treated in a unified context, deals with singularities [4], a subject we will not approach here. But relative equilibria can be put in a broader perspective.…”

On the Singularities of Generalized Solutions to n-Body-Type Problems

“…Such singularities play a fundamental role in the phase portrait (see, e.g. [20]) and strongly influence the global orbit structure, as they can be held responsible, among others, of the presence of chaotic motions (see, e.g. [16]) and of motions becoming unbounded in a finite time [35,54].…”