Yu and Zhu proved that the planar four-body problem has finitely many central configurations for any four nonzero real masses, settling a question of Hampton and Moeckel about negative masses; however, their proof normalizes the central-configuration multiplier to λ=±1\lambda=\pm 1 and explicitly excludes the case λ=0\lambda=0, in which the acceleration of every body — the net Newtonian force per unit mass — vanishes and the normalization is impossible. The excluded subcase remains open: for a 4-tuple of nonzero real masses, determine the set of planar collision-free configurations satisfying the zero-force equations, up to translations, rotations, and dilations — in particular, whether this set is finite for every such 4-tuple, or whether some mixed-sign 4-tuple admits a positive-dimensional continuum. The vortex analog of this question is well developed (finiteness of stationary configurations of four and five point vortices), which shows the subcase is a meaningful, separable question; no result on the Newtonian λ=0\lambda=0 four-body case is known.

Background

A central configuration of the Newtonian NN-body problem is a configuration of positions r1,…,rN∈R2\mathbf r_1,\dots,\mathbf r_N\in\mathbb R^2 with pairwise distinct positions and masses m1,…,mNm_1,\dots,m_N satisfying

λrk=∑1≤j≤N, j≠kmjrj−rk∣rj−rk∣3,k=1,…,N,\lambda\mathbf r_k=\sum_{1\le j\le N,\,j\ne k}m_j\frac{\mathbf r_j-\mathbf r_k}{|\mathbf r_j-\mathbf r_k|^3},\qquad k=1,\dots,N,

with the origin at the center of mass (whenever the total mass is nonzero; for mixed-sign masses it may vanish, and at λ=0\lambda=0 the equations below are translation-invariant, so this normalization is not needed). Central configurations are exactly the shapes that rotate rigidly and uniformly without deforming (relative equilibria) or collapse homothetically, that is, shrink rigidly along rays through the center of mass; they control basic qualitative features of the NN-body dynamics. A classical finiteness conjecture going back to Wintner (and popularized through Smale's problem list) asserts that for positive masses the number of central configurations, counted modulo translations, rotations, and dilations, is finite; it is known for N=3N=3 classically, for N=4N=4 by Hampton and Moeckel (2006), and for the planar problem with N=5N=5 by Albouy and Kaloshin (2012), the latter excepting an exceptional codimension-2 subvariety of the mass space (an algebraic subset cut out by two independent equations on the masses). When masses of mixed sign are allowed, new phenomena appear: Roberts (1999) constructed a continuum of relative equilibria of the planar five-body problem with one negative mass, and Hampton and Moeckel asked whether a continuum is possible for the four-body problem with negative masses.

Yu and Zhu (arXiv 2021; published version Zhu and Yu, Annali della Scuola Normale Superiore di Pisa, 2026) settled this question negatively: there are finitely many (real) central configurations of the four-body problem for any four nonzero real masses, and a complex continuum of relative equilibria exists only for two special families of signed masses (m1=m2=−m3=−m4m_1=m_2=-m_3=-m_4, and m1=m2m_1=m_2 with m3=−α2m1m_3=-\alpha^2 m_1 and m4=−(2−α2)m1m_4=-(2-\alpha^2)m_1, where α≈1.2407\alpha\approx 1.2407 is the unique positive root of an explicit algebraic equation in their paper). Their proof adapts the Albouy–Kaloshin diagrammatic analysis of singular sequences, extended to handle clusters whose total mass vanishes, which is unavoidable with mixed-sign masses. However, when masses may be negative, the multiplier λ\lambda in the defining equations can be positive, negative, or zero. For λ≠0\lambda\ne 0 the equations are invariant under rotation and dilation, so λ\lambda can be normalized to ±1\pm 1; the paper states verbatim that it will not study the case λ=0\lambda=0, and its Theorem 1.1 covers the normalized case only. The case λ=0\lambda=0 asks that the total Newtonian force on every body vanish simultaneously — a static equilibrium (zero-force configuration) of the system, which can only exist for mixed-sign masses: the classical virial identity ∑kmk rk⋅Fk=−∑i<jmimj/∣ri−rj∣\sum_k m_k\,\mathbf r_k\cdot\mathbf F_k=-\sum_{i<j}m_im_j/|\mathbf r_i-\mathbf r_j| forces the potential energy to vanish at any such configuration, which is impossible when all masses have the same sign. Because the λ=0\lambda=0 equations are unaffected by the normalization step and are invariant under translations, rotations, and dilations, the meaningful question is the structure of the solution set in the four-dimensional shape space of the four-body problem (the quotient of the eight position coordinates by the four similarity dimensions: two translations, one rotation, and one dilation). The analogous question for point vortices — a planar fluid model in which NN vortices of strengths (vorticities) Γ1,…,ΓN\Gamma_1,\dots,\Gamma_N interact through a logarithmic potential, and in which stationary configurations (equilibria, rigidly translating configurations, relative equilibria, and collapse configurations) replace central configurations — has an active literature: Hampton and Moeckel (2009) proved finiteness of stationary configurations of the four-vortex problem, Yu (2023) extended this to arbitrary nonzero vorticities with explicit bounds, and Yu and Zhu (2025) treated the five-vortex problem up to a codimension-2 exceptional subvariety. No corresponding Newtonian λ=0\lambda=0 result exists, leaving the four-body zero-multiplier subcase identified by the source paper untreated.

Problem Statement

For the planar Newtonian four-body problem, fix four nonzero real masses m1,m2,m3,m4m_1,m_2,m_3,m_4 (of arbitrary, mixed signs). Consider configurations of pairwise distinct positions r1,…,r4∈R2\mathbf r_1,\dots,\mathbf r_4\in\mathbb R^2 satisfying the zero-multiplier (λ=0\lambda=0) central configuration equations, i.e. the acceleration (net Newtonian force per unit mass) of every body vanishes:

0=∑1≤j≤4, j≠kmjrj−rk∣rj−rk∣3,k=1,2,3,4.0=\sum_{1\le j\le 4,\,j\ne k}m_j\frac{\mathbf r_j-\mathbf r_k}{|\mathbf r_j-\mathbf r_k|^3},\qquad k=1,2,3,4.

Determine the structure of the solution set modulo the similarity group of the plane (translations, rotations, and dilations), as the mass 4-tuple ranges over (R∖{0})4(\mathbb R\setminus\{0\})^4. In particular: (i) prove that for every 4-tuple of nonzero real masses the set of such zero-force configurations, up to similarity, is finite; or (ii) exhibit a 4-tuple of nonzero real masses together with a positive-dimensional family (a continuum) of collision-free zero-force configurations not related by similarity. This is the subcase explicitly excluded by Yu and Zhu's proof of finiteness of four-body central configurations for nonzero masses, which normalizes the multiplier to ±1\pm 1 and therefore covers only λ≠0\lambda\ne 0.

References

  1. Xiang Yu, Shuqiang Zhu, On the finiteness of four-body central configurations, arXiv:2103.08906 (2021), DOI 10.48550/arXiv.2103.08906, https://arxiv.org/abs/2103.08906
  2. Shuqiang Zhu, Xiang Yu, On the Finiteness of Four-Body Central Configurations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2026), DOI 10.2422/2036-2145.202501_008, https://doi.org/10.2422/2036-2145.202501_008
  3. Marshall Hampton, Richard Moeckel, Finiteness of relative equilibria of the four-body problem, Inventiones Mathematicae 163 (2006) 289–312, DOI 10.1007/s00222-005-0461-0, https://doi.org/10.1007/s00222-005-0461-0
  4. Alain Albouy, Vadim Kaloshin, Finiteness of central configurations of five bodies in the plane, Annals of Mathematics 176 (2012) 535–588, DOI 10.4007/annals.2012.176.1.10, https://doi.org/10.4007/annals.2012.176.1.10
  5. Gareth E. Roberts, A continuum of relative equilibria in the five-body problem, Physica D 127 (1999) 141–145, DOI 10.1016/S0167-2789(98)00315-7, https://doi.org/10.1016/S0167-2789(98)00315-7
  6. Marshall Hampton, Richard Moeckel, Finiteness of stationary configurations of the four-vortex problem, Transactions of the American Mathematical Society 361 (2009) 1317–1332, DOI 10.1090/S0002-9947-08-04685-0, https://doi.org/10.1090/S0002-9947-08-04685-0
  7. Xiang Yu, Finiteness of stationary configurations of the planar four-vortex problem, Advances in Mathematics 435 (2023), Article 109378, DOI 10.1016/j.aim.2023.109378, https://doi.org/10.1016/j.aim.2023.109378 (preprint arXiv:2103.06037)
  8. Xiang Yu, Finiteness of Stationary Configurations of the Planar Four-vortex Problem. II, arXiv:2111.07292 (2021), https://arxiv.org/abs/2111.07292
  9. Xiang Yu, Shuqiang Zhu, On finiteness of stationary configurations of the planar five-vortex problem, Mathematische Annalen 392 (2025) 4023–4069, DOI 10.1007/s00208-025-03183-w, https://doi.org/10.1007/s00208-025-03183-w