Mechanics of particles and systems Particle systems and celestial mechanics Theorem

Painlevé conjecture on noncollision singularities

Builds on n-body problem Three-body problem

Statement

Consider the n-body problem with positive masses in $\R^3$ . Let a solution $q(t) = (q_1(t), \ldots, q_n(t))$ be defined on a maximal interval $[0, t^*)$ with $t^* < \infty$ , so that $t^*$ is a singularity. The singularity is a collision singularity if $q(t)$ has a limit as $t \to t^*$ , and a noncollision singularity otherwise.

(Painlevé) For $n = 3$ , every singularity is a collision singularity.
(Xia) For $n = 5$ , there are positive masses and initial conditions whose solution has a noncollision singularity.
(Xue) For $n = 4$ , there are positive masses and initial conditions in the plane whose solution has a noncollision singularity.

Remarks

Painlevé showed that $t^*$ is a singularity exactly when the minimum of the mutual distances tends to $0$ as $t \to t^*$ . In his 1895 Stockholm lectures he conjectured that noncollision singularities exist for $n \ge 4$ . By a theorem of von Zeipel, at a noncollision singularity the configuration is unbounded: some body escapes to infinity in finite time. Xia's example uses two binary pairs and a fifth body that bounces between them and gains energy at each close passage, so the system expands without bound in finite time. Xia published his example in 1992, and Xue's planar four-body result was published in 2020. Gerver gave earlier constructions in the plane with many bodies. By item 1, three bodies are not enough for this behaviour, which is why the conjecture is about $n \ge 4$ .

Going deeper

For a student

The result shows that the $n$ -body flow can be incomplete in a way that no regularisation of collisions repairs. It contrasts with Sundman's analysis of three bodies, where all singularities are collisions and binary ones can be regularised. The constructions use repeated near-collisions to transfer unbounded energy to one body, a mechanism that needs at least four bodies.

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First integral concept
Hamiltonian system concept
Ordinary differential equation concept
n-body problem problem
Three-body problem problem
Painlevé conjecture on noncollision singularities statement

Tag	`PBS0PYZ7BE`
URI	`m2://PBS0PYZ7BE`
Kind	statement, theorem
Writing	W0 stub
MSC	70F16, 70F10
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About	n-body problem
Uses	n-body problem, Three-body problem
Same as	Wikidata Q18386040 Wikipedia Painlevé conjecture
Changed	3 November 2026