Smale’s sixth problem: finiteness of similarity classes of relative equilibria

Determine whether the number of similarity classes of relative equilibria in the Newtonian n-body problem with n positive point masses is finite for every integer n.

Background

The paper surveys central configurations in the Newtonian n-body problem and highlights longstanding unresolved issues. Relative equilibria are planar periodic solutions arising from central configurations and play a central role in explicit solutions of the n-body problem. Despite many partial results, only special cases have been fully classified, and the broader finiteness question is still unresolved.

Within this context, Smale’s sixth problem asks about the finiteness of similarity classes of relative equilibria for arbitrary n. This problem is connected to broader finiteness questions, including the Chazy–Wintner–Smale finiteness conjecture, and has been resolved only in limited cases (e.g., n=4).

References

General, basic problems on central configurations remain unsolved. An outstanding example is the celebrated Smale's sixth problem for the 21st century, which asks whether the number of similarity classes of relative equilibria of $n$ positive point masses is finite for all $n$.

— Moments and the Basic Equilibrium Equations  (2507.13865 - Leandro, 18 Jul 2025) in Section 1 (Introduction)

For five positive masses, finiteness is known; the corresponding question for six positive masses remains open.

— Finiteness of planar central configurations of six bodies for all positive masses: Part I, exceptional-mass first limits  (2610.23329 - Chen et al., 16 Oct 2026) in Section 1, Introduction

All-positive-mass finiteness, including $E_{40}$, is a distinct and unfinished Part II problem.

— Finiteness of planar central configurations of six bodies for all positive masses: Part I, exceptional-mass first limits  (2610.23329 - Chen et al., 16 Oct 2026) in Section 1, Introduction; Section 6, Section 2? Actually Section \ref{sec:partII} Part II boundary

This is a necessary condition: certain partial $K_4$ shapes still unfold across open positive mass sets but do not construct six-body germs. The positive zero set of the new mass equation, the shared-body higher jets and finiteness on exceptional masses remain open.

— Finiteness of planar central configurations of six bodies for all positive masses: Part I, exceptional-mass first limits  (2610.23329 - Chen et al., 16 Oct 2026) in Section 6, subsection No. 94; Section 7, Part II boundary