Local abundance of periodic orbits near relative equilibria of the planar N-body problem: general central configurations, exceptional masses, and the critical Routh ratio

Xiang Yu (2020) proved that periodic orbits are measure-theoretically abundant near the classical relative equilibria of the planar Newtonian N-body problem: near the Lagrange equilateral relative equilibrium of the three-body problem for Routh-stable generic masses, and near every Euler–Moulton collinear relative equilibrium of the N-body problem for almost every mass choice, the relative measure of the closure of the set of periodic orbits in a polydisc of radius $\varepsilon$ on the center manifold is at least $1-O(\varepsilon^{1/4})$, improving to exponentially close to 1 for Diophantine frequencies. The theorem excludes by hypothesis exactly the regimes its method has not reached: relative equilibria from general (non-collinear, non-equilateral) central configurations, the resonant and degenerate mass sets removed by the genericity conditions, and the critical mass ratio $\beta=1/27$ (the Gascheau–Routh spectral-stability boundary), where even the existence of nontrivial nearby periodic orbits is unknown. The problem posed here is to determine the abundance of periodic orbits near every relative equilibrium of the planar N-body problem, for every choice of masses: prove the measure-theoretic abundance conclusion, or establish a rigorous alternative, in each of the three excluded regimes. The global Poincaré density conjecture that motivates this problem remains open.

Background

The Newtonian N-body problem asks for the motion of $N$ point masses moving in a plane under mutual gravitational attraction. A relative equilibrium is a solution in which the configuration of the bodies rotates rigidly at constant angular velocity; every relative equilibrium corresponds to a central configuration, a special shape for which the gravitational acceleration of each body is a common scalar multiple of its position relative to the center of mass. The two classical families are the Lagrange equilateral relative equilibria of the three-body problem and the Euler–Moulton collinear relative equilibria of the N-body problem. Near such a solution, the linearized flow has, besides the trivial Keplerian directions (the directions coming from the underlying two-body-type elliptic motion, which generate degenerate families of homographic periodic solutions), purely imaginary eigenvalues for an appropriate range of masses; the invariant manifold tangent to the imaginary eigenspaces, called the center manifold, carries all small recurrent dynamics and has dimension $2d$ ($d=3$ for the Lagrange case, $d=N-1$ for Euler–Moulton).

Two classical constructions give periodic orbits on this manifold: the Lyapunov center theorem and Weinstein's theorem each produce a one-parameter family of periodic orbits from every nonresonant imaginary eigenvalue pair (for the Lagrange relative equilibrium these families go back to Siegel). Periodic orbits of the N-body problem are otherwise famously hard to find — the variational approach, for instance, yielded its first emblematic result (the figure-eight orbit of Chenciner and Montgomery) only in 2000.

Yu's 2020 preprint adds a much stronger, measure-theoretic statement. Introducing a rotating coordinate system that removes the degeneracies coming from rotational and scaling symmetry, he computes the Birkhoff normal form of the reduced Hamiltonian on the center manifold — that is, he rewrites the Hamiltonian, up to a finite order, as a function of the action variables $\rho$ plus higher-order remainders, which is possible when the linear frequencies are nonresonant up to the required order — and then applies an observation going back to Conley and Zehnder: the invariant tori constructed by Kolmogorov–Arnold–Moser (KAM) theory, which carries quasi-periodic motion, lie in the closure of the set of periodic orbits supplied by the Birkhoff–Lewis fixed point theorem (a fixed-point counterpart of the Lyapunov center theorem, producing infinitely many periodic orbits near an elliptic equilibrium). If the normal form is nondegenerate (its Hessian determinant $\det(\omega_{jk})\neq 0$ in the actions), this yields abundance: the relative measure of the closure of the set of periodic orbits in a polydisc $\|\rho\|<\varepsilon$ on the center manifold is at least $1-O(\varepsilon^{1/4})$, and for Diophantine frequency vectors (a full-measure set of frequencies satisfying a quantitative irrationality condition) at least $1-O(\exp(-c\,\varepsilon^{-1/(\upsilon+1)}))$.

The theorem is proved for: (a) every Lagrange relative equilibrium with masses satisfying the Routh spectral-stability condition $\beta:=(m_1m_2+m_3m_2+m_1m_3)/(m_1+m_2+m_3)^2<1/27$, after removing the finite list of special resonant $\beta$-values $\{1/75,\,32/2187,\,16/675,\,1/36,\,64/1875\}$ and an algebraic degeneracy locus $V_{f_{deg}}$ (Theorem 5.1); (b) the Euler collinear relative equilibria of the three-body problem (Theorem 5.3); and (c) every Euler–Moulton relative equilibrium of the N-body problem for almost every choice of positive masses, with nondegeneracy proved outright only for sufficiently small $m_2,\dots,m_N$ (Theorem 6.1, Proposition 6.1). The value $\beta=1/27$ is the Gascheau–Routh boundary at which two imaginary eigenvalue pairs of the Lagrange relative equilibrium collide and the linearization loses ellipticity; there Yu writes that, apart from the trivial Keplerian family, "we don't know if there are other periodic orbits near the origin".

This local abundance is evidence for Poincaré's celebrated claim (1892) that periodic orbits are dense in the bounded phase space of the N-body problem — a famous problem that remains open in this generality; the strongest related statement, besides the weakened version of Gómez and Llibre (1981) for the restricted three-body problem, is Xue's recent closing-lemma result that for typical perturbations of Poincaré's original nearly-integrable setting periodic orbits are asymptotically dense as the perturbation size tends to zero, which does not address the gravitational N-body phase space itself. The question posed here is to remove the restrictions under which Yu's abundance theorem is proved.

Problem Statement

Complete the local abundance picture for periodic orbits near relative equilibria of the planar Newtonian N-body problem by removing the three restrictions under which Yu's abundance theorem is proved. The common objective is that the abundance conclusion — or a rigorous alternative when it fails — be established near every relative equilibrium of the planar N-body problem for every choice of masses. Here a rigorous alternative to the abundance conclusion is a proof that the corresponding measure bound fails, together with a rigorous description of the periodic orbits that do exist on the center manifold and how densely their closure fills the polydisc. Concretely, the problem has three linked parts:

1. General relative equilibria. For relative equilibria generated by central configurations other than the equilateral and collinear ones (in particular, non-collinear central configurations for $N\geq 4$ bodies), determine whether the abundance conclusion holds — i.e., whether on the center manifold of the linearized flow at the relative equilibrium the relative measure of the closure of the set of periodic orbits in a polydisc $\|\rho\|<\varepsilon$ is at least $1-O(\varepsilon^q)$ for some $q$ — or establish a rigorous alternative in the cases where the elliptic, definite structure of the reduced Hamiltonian on the center manifold fails.

2. Exceptional mass sets. Characterize the mass sets excluded from the proved theorems — the finite list of special resonant $\beta$-values and the degeneracy locus $V_{f_{deg}}$ for the Lagrange theorem, and the measure-zero exceptional set of the Euler–Moulton theorem — including their algebraic structure and dimensionality, whether they contain mass triples that are not in the closure of the degenerate limits $m_2,\dots,m_N\to 0$ used in the proofs, and the periodic-orbit abundance (or its failure) for masses lying on them.

3. The critical Routh ratio. For the Lagrange relative equilibrium at $\beta=1/27$, where two eigenvalue pairs of the linearization collide (a Hamiltonian–Hopf degeneracy at the Gascheau–Routh boundary), determine the periodic-orbit structure on the center manifold near the equilibrium — in particular whether any nontrivial (non-Keplerian) periodic orbits exist there, and whether any abundance or measure statement holds.

Each part removes one hypothesis from the same theorem, so the three regimes exhaust its hypotheses; determining abundance, or a rigorous alternative, in all three settles the local question of Poincaré's density claim near relative equilibria of the planar N-body problem.

References

1. Xiang Yu, "On Periodic orbits of the Planar N-body Problem", arXiv:2003.03208 (2020), https://arxiv.org/abs/2003.03208
2. Xiang Yu, "On the Stability of Lagrange Relative Equilibrium in The Planar Three-body Problem", arXiv:1911.12269 (2019, v4 2022), https://arxiv.org/abs/1911.12269
3. Gerard Gómez and Jaume Llibre, "A note on a conjecture of Poincaré", Celestial Mechanics 24, 335–343 (1981), doi:10.1007/BF01230393, https://doi.org/10.1007/BF01230393
4. C. Conley and E. Zehnder, "An index theory for periodic solutions of a Hamiltonian system", in Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Mathematics 987, Springer, 132–145 (1983), doi:10.1007/BFb0061415, https://doi.org/10.1007/BFb0061415
5. Alexander Mielke, "Hamiltonian and Lagrangian Flows on Center Manifolds", Lecture Notes in Mathematics 1489, Springer (1991), doi:10.1007/BFb0097544, https://doi.org/10.1007/BFb0097544
6. Alain Chenciner and Richard Montgomery, "A Remarkable Periodic Solution of the Three-Body Problem in the Case of Equal Masses", Annals of Mathematics 152, 881–902 (2000), doi:10.2307/2661357, https://doi.org/10.2307/2661357
7. Jinxin Xue, "Closing Lemma and KAM Normal Form", Acta Mathematica Sinica, English Series 42, 541–567 (2026), doi:10.1007/s10114-026-4598-7, arXiv:2207.06208, https://doi.org/10.1007/s10114-026-4598-7
8. Inmaculada Baldomá, Pau Martín, and Donato Scarcella, "Chaotic phenomena in generic unfoldings of the Hamilton Hopf bifurcation with emphasis on the restricted planar circular 3-body problem beyond the Gascheau-Routh mass ratio", arXiv:2511.05333 (2025), https://arxiv.org/abs/2511.05333