Almost-everywhere global existence for Newtonian gravitation
Fix $N \ge 5$ and masses $m_1, \dots, m_N > 0$, and let $$\mathcal{P} = \{ (q_1, \dots, q_N, v_1, \dots, v_N) \in \mathbb{R}^{6N} : q_i \ne q_j \ (i \ne j) \}.$$ For each initial state in $\mathcal{P}$ solve $$\dot q_i = v_i, \qquad \dot v_i = \sum_{j \ne i} m_j \frac{q_j - q_i}{|q_j - q_i|^3}$$ on its maximal forward interval $[0, T_{\max})$ of classical existence. A solution with $T_{\max} < \infty$ ends in a collision or in a noncollision singularity, and both count. Painlevé proved that for three bodies every singularity is a collision. Saari proved that the initial states leading to a collision form a set of Lebesgue measure zero (1971, 1973), and that for four bodies the initial states with $T_{\max} < \infty$ do (1977). Xia constructed noncollision singularities for five bodies in space (1992), and Xue for four bodies in the plane (2020).
Problem. Is the set of initial states in $\mathcal{P}$ with $T_{\max} < \infty$ of $6N$-dimensional Lebesgue measure zero?
Source: Simon (1984), the first of his fifteen problems in mathematical physics; see Xue, On the Painlevé conjecture (2020), Conjecture 1.1. Stated for N ≥ 5 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- CA, MP
- Keywords
- AIM Open Applied Problems, celestial mechanics, n-body problems, noncollision singularities
References
@incollection{simon1984,
author = {Simon, Barry},
title = {Fifteen problems in mathematical physics},
booktitle = {Perspectives in Mathematics: Anniversary of {O}berwolfach 1984},
editor = {J{\"a}ger, Willi and Moser, J{\"u}rgen and Remmert, Reinhold},
publisher = {Birkh{\"a}user},
address = {Basel},
year = {1984},
pages = {423--454},
}
@incollection{xue2020cdm,
author = {Xue, Jinxin},
title = {On the {P}ainlev{\'e} conjecture},
booktitle = {Current Developments in Mathematics 2020},
publisher = {International Press},
address = {Somerville, MA},
year = {2020},
pages = {157--206},
doi = {10.4310/CDM.2020.v2020.n1.a4},
}
@book{painleve1897,
author = {Painlev{\'e}, Paul},
title = {Le{\c{c}}ons sur la th{\'e}orie analytique des {\'e}quations diff{\'e}rentielles},
publisher = {Hermann},
address = {Paris},
year = {1897},
}
@article{saari1971,
author = {Saari, Donald G.},
title = {Improbability of collisions in {N}ewtonian gravitational systems},
journal = {Trans. Amer. Math. Soc.},
volume = {162},
year = {1971},
pages = {267--271},
doi = {10.1090/S0002-9947-1971-0295648-8},
}
@article{saari1973,
author = {Saari, Donald G.},
title = {Improbability of collisions in {N}ewtonian gravitational systems. {II}},
journal = {Trans. Amer. Math. Soc.},
volume = {181},
year = {1973},
pages = {351--368},
doi = {10.1090/S0002-9947-1973-0321386-0},
}
@article{saari1977,
author = {Saari, Donald G.},
title = {A global existence theorem for the four-body problem of {N}ewtonian mechanics},
journal = {J. Differential Equations},
volume = {26},
number = {1},
year = {1977},
pages = {80--111},
doi = {10.1016/0022-0396(77)90100-0},
}
@article{xia1992,
author = {Xia, Zhihong},
title = {The existence of noncollision singularities in {N}ewtonian systems},
journal = {Ann. of Math. (2)},
volume = {135},
number = {3},
year = {1992},
pages = {411--468},
doi = {10.2307/2946572},
}
@article{xue2020acta,
author = {Xue, Jinxin},
title = {Noncollision singularities in a planar 4-body problem},
journal = {Acta Math.},
volume = {224},
number = {2},
year = {2020},
pages = {253--388},
doi = {10.4310/ACTA.2020.v224.n2.a2},
}
@misc{aim-almost-sure-global-newtonian-motion,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Almost-everywhere global existence for Newtonian gravitation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/131-almost-sure-global-newtonian-motion.md},
note = {Entry 131 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00498, "Almost-everywhere global existence for Newtonian gravitation", clopen.solutions.