Almost-sure global existence in the Newtonian N-body problem

Newton's equations for point masses can break down in finite time: bodies may collide, or some may escape to infinity in finite time without colliding. The question is whether such breakdowns are exceptional, so that almost every choice of initial positions and velocities gives a motion that lasts forever.

Problem

Let $N \ge 2$ point masses $m_1, \dots, m_N > 0$ move in $\mathbb{R}^3$ under Newtonian gravity, with $G = 1$:

$$m_i \ddot r_i = \sum_{j \ne i} \frac{m_i m_j \,(r_j - r_i)}{|r_j - r_i|^3}, \qquad i = 1, \dots, N.$$

Initial data are distinct positions and any velocities. They form a point of $\mathbb{R}^{6N}$ and give a unique solution on a maximal interval $[0, T)$. If $T < \infty$, the singularity at $T$ is a collision when every $r_i(t)$ has a finite limit as $t \uparrow T$. Otherwise it is a non-collision singularity, and then $\sum_i |r_i(t)|^2 \to \infty$ (von Zeipel, Sperling).

Problem 1A. Prove that, for every $N$ and all masses, the initial conditions in $\mathbb{R}^{6N}$ whose solution is singular at a finite time form a set of Lebesgue measure zero.

Collisions are already known to be improbable, so the task is to show the same for non-collision singularities.

Background

This problem opens Barry Simon's 1984 list of fifteen problems in mathematical physics. Painlevé first studied these singularities, in lectures published in 1897. Simon called the measure of the non-collision set "the key question". He also recorded that some excellent mathematicians suspected an open set of initial conditions without global solutions. His reason to expect a positive answer comes from quantum mechanics: Kato proved that the Schrödinger equation with Coulomb potentials always has global solutions, and quantization should only smooth out null sets.

What is known

• Collisions. For every $N$, collisions occur only for a null set of initial conditions (Saari, 1972–73; see [1]). Fleischer and Knauf extended this to a wide class of pair interactions, including gravitational and Coulomb ones [5].
• $N \le 4$. Painlevé showed that three bodies have no non-collision singularities. Saari (1977) showed that four bodies have them only on a null set [2]. Simon noted that $N \ge 5$ is open.
• Existence. Non-collision singularities do occur. Xia found them for five bodies in space [3], and Xue for four bodies in the plane, the last open case of Painlevé's conjecture [4]. For $N \ge 5$ the open question is their measure.
• Recent progress. Quaschner (September 2026, unrefereed preprint) proves that singular orbits are improbable when every subsystem either collides totally at its centre of mass or has a non-collision singularity with exactly four particles, provided the asymptotic directions of these subsystems differ. He notes that the general problem remains open [6].

What would count as progress

• A proof for five bodies and all masses, the first open case.
• An extension of [6] to non-collision subsystems of five or more particles, or without its condition on directions.
• A refutation: for some $N$ and masses, a set of initial conditions of positive measure leading to non-collision singularities.

References

1. B. Simon, "Fifteen problems in mathematical physics", in Perspectives in Mathematics: Anniversary of Oberwolfach 1984, Birkhäuser, 1984, pp. 423–454. PDF
2. D. G. Saari, "A global existence theorem for the four-body problem of Newtonian mechanics", J. Differential Equations 26 (1977), 80–111. doi:10.1016/0022-0396(77)90100-0
3. Z. Xia, "The existence of noncollision singularities in Newtonian systems", Ann. of Math. 135 (1992), 411–468. doi:10.2307/2946572
4. J. Xue, "Noncollision singularities in a planar 4-body problem", Acta Math. 224 (2020), 253–388. doi:10.4310/ACTA.2020.v224.n2.a2
5. S. Fleischer and A. Knauf, "Improbability of collisions in n-body systems", Arch. Ration. Mech. Anal. 234 (2019), 1007–1039. doi:10.1007/s00205-019-01406-4
6. M. Quaschner, "A general improbability result for non-collision singularities as subsystems with at most four particles", preprint, 2026. arXiv:2609.00876

Source

Imported from Simon's problems in mathematical physics 1984-1A, curated by Barry Simon. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.