Saari's constant-inertia conjecture for planar gravitation
Fix $N \ge 4$ and masses $m_1, \dots, m_N > 0$, and let $q_i : J \to \mathbb{R}^2$ be a collision-free solution of the planar Newtonian $N$-body problem on an open time interval $J$, with centre of mass at the origin: $$\ddot q_i = \sum_{j \ne i} m_j \frac{q_j - q_i}{|q_j - q_i|^3}, \qquad \sum_i m_i q_i = 0.$$ Its moment of inertia is $I(t) = \sum_i m_i |q_i(t)|^2$. A relative equilibrium, in which the configuration rotates rigidly about the origin at constant angular velocity, has constant $I$, and Saari conjectured that no other motion does. Moeckel proved this for three bodies, in the plane (2005) and in every dimension (2008), and Diacu, Pérez-Chavela and Santoprete for collinear motions of any number of bodies (2005).
Problem. If $I$ is constant, must the solution be a relative equilibrium? That is, must there be a constant $\omega$ such that $q_i(t) = R_{\omega (t - t_0)}\, q_i(t_0)$ for all $i$ and all $t, t_0 \in J$, where $R_\theta$ is the rotation of the plane by $\theta$?
Source: Saari (1970), for the Newtonian n-body problem; stated for planar motions of N ≥ 4 bodies 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, relative equilibria
References
@incollection{saari1970,
author = {Saari, Donald G.},
title = {On bounded solutions of the $n$-body problem},
booktitle = {Periodic Orbits, Stability and Resonances},
editor = {Giacaglia, G. E. O.},
publisher = {Reidel},
address = {Dordrecht},
year = {1970},
pages = {76--81},
doi = {10.1007/978-94-010-3323-7_7},
}
@article{moeckel2005,
author = {Moeckel, Richard},
title = {A computer-assisted proof of {S}aari's conjecture for the planar three-body problem},
journal = {Trans. Amer. Math. Soc.},
volume = {357},
number = {8},
year = {2005},
pages = {3105--3117},
doi = {10.1090/S0002-9947-04-03527-5},
}
@article{moeckel2008,
author = {Moeckel, Richard},
title = {A proof of {S}aari's conjecture for the three-body problem in {$\mathbb{R}^d$}},
journal = {Discrete Contin. Dyn. Syst. Ser. S},
volume = {1},
number = {4},
year = {2008},
pages = {631--646},
doi = {10.3934/dcdss.2008.1.631},
}
@article{diacu2005,
author = {Diacu, Florin and P{\'e}rez-Chavela, Ernesto and Santoprete, Manuele},
title = {Saari's conjecture for the collinear $n$-body problem},
journal = {Trans. Amer. Math. Soc.},
volume = {357},
number = {10},
year = {2005},
pages = {4215--4223},
doi = {10.1090/S0002-9947-04-03606-2},
eprint = {0909.4909},
archivePrefix = {arXiv},
}
@misc{aim-saari-constant-inertia,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Saari's constant-inertia conjecture for planar gravitation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/129-saari-constant-inertia.md},
note = {Entry 129 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-00550, "Saari's constant-inertia conjecture for planar gravitation", clopen.solutions.