OP-00520 Partially resolved

Finiteness of planar central configurations for fixed masses

Fix $N$ and masses $m_1, \dots, m_N > 0$. A normalised planar central configuration is a tuple of labelled points $q_1, \dots, q_N \in \mathbb{R}^2$, with $q_i \ne q_j$ for $i \ne j$, such that $$\sum_i m_i q_i = 0, \qquad \sum_i m_i |q_i|^2 = 1, \qquad \sum_{j \ne i} m_j \frac{q_j - q_i}{|q_j - q_i|^3} = -\lambda q_i \quad (i = 1, \dots, N)$$ for some $\lambda > 0$; the normalisations remove translation and scaling, and $SO(2)$ acts by rotating all points together. Central configurations give the relative equilibria of the Newtonian $N$-body problem. Hampton and Moeckel proved that there are finitely many orbits for $N = 4$ and all positive masses (2006). Albouy and Kaloshin proved it for $N = 5$ unless the masses lie in a certain subvariety of codimension $2$ of the mass space (2012), and Moczurad and Zgliczyński gave computer-assisted counts for several mass tuples in that exceptional set (2026).

Problem. For every integer $N \ge 5$ and every tuple of masses $m_1, \dots, m_N > 0$, including exceptional mass ratios, does the set of normalised planar central configurations consist of finitely many $SO(2)$-orbits?

Source: Chazy (1918) and Wintner (1941); Smale (1998), Problem 6, for every number of bodies. Stated for N ≥ 5 by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: DS — Dynamical Systems
Secondary: AG, MP
Keywords: AIM Open Applied Problems, central configurations, n-body problems, relative equilibria

References

@article{chazy1918,
  author  = {Chazy, Jean},
  title   = {Sur certaines trajectoires du probl{\`e}me des $n$ corps},
  journal = {Bull. Astron.},
  volume  = {35},
  year    = {1918},
  pages   = {321--389},
}
@book{wintner1941,
  author    = {Wintner, Aurel},
  title     = {The Analytical Foundations of Celestial Mechanics},
  series    = {Princeton Mathematical Series},
  volume    = {5},
  publisher = {Princeton University Press},
  address   = {Princeton, NJ},
  year      = {1941},
}
@article{smale1998,
  author  = {Smale, Steve},
  title   = {Mathematical problems for the next century},
  journal = {Math. Intelligencer},
  volume  = {20},
  number  = {2},
  year    = {1998},
  pages   = {7--15},
  doi     = {10.1007/BF03025291},
}
@article{hampton2006,
  author  = {Hampton, Marshall and Moeckel, Richard},
  title   = {Finiteness of relative equilibria of the four-body problem},
  journal = {Invent. Math.},
  volume  = {163},
  number  = {2},
  year    = {2006},
  pages   = {289--312},
  doi     = {10.1007/s00222-005-0461-0},
}
@article{albouy2012,
  author  = {Albouy, Alain and Kaloshin, Vadim},
  title   = {Finiteness of central configurations of five bodies in the plane},
  journal = {Ann. of Math. (2)},
  volume  = {176},
  number  = {1},
  year    = {2012},
  pages   = {535--588},
  doi     = {10.4007/annals.2012.176.1.10},
}
@misc{moczurad2026,
  author        = {Moczurad, Ma{\l}gorzata and Zgliczy{\'n}ski, Piotr},
  title         = {Central configurations with unequal masses: finiteness in several exceptional cases of five bodies},
  year          = {2026},
  eprint        = {2601.01165},
  archivePrefix = {arXiv},
}
@misc{aim-finite-planar-central-configurations,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Finiteness of planar central configurations for fixed masses}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/130-finite-planar-central-configurations.md},
  note         = {Entry 130 at commit a153f46, 2026-10-13},
}

Resolution claims

Smale's sixth problem for five bodies in the plane, with a proof formalized in Lean v1 arXiv ↗

Tejasvi Singh Tomar

We prove that for every choice of five positive masses the planar five-body problem has finitely many central configurations up to similarity. This settles Smale's sixth problem for five bodies in the plane. Albouy and Kaloshin proved it for masses outside a closed algebraic subset of codimension two. We follow their method. Infinitely many central configurations would give an algebraic …

Partial resolution · not verified · fidelity not established · posted on arXiv 15 Oct 2026

Status history

When	Change	By	Reason
2026-10-23	Open → Partially resolved	The registry's curators	Smale's sixth problem for five bodies in the plane, with a proof formalized in Lean v1 recorded from arXiv:2610.13100v2.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00520, "Finiteness of planar central configurations for fixed masses", clopen.solutions.