Birkhoff's conjecture for the planar circular restricted three-body problem in the non-perturbative regime: dynamical convexity and disk-like global surfaces of section

In 1915 Birkhoff proved the existence of the retrograde periodic orbit in the planar circular restricted three-body problem (RTBP) and conjectured that, for energies below the first critical value, this orbit bounds a disk-like global surface of section - the object needed to reduce the flow to a Poincare return map and to prove the existence of the direct periodic orbit. Joung and van Koert (Nonlinearity 38, 025015, 2025) launched a program of computational symplectic topology toward this conjecture: validated-numerics proofs of a direct orbit for mass parameters $\mu\in[0.1,0.5]$ at Jacobi energies from the first critical value up to $c=2.1$, a certified classification of short-action symmetric periodic orbits for $\mu\in[0,0.5]$ in a $10^{-6}$-wide energy window, rigorous Conley-Zehnder index computations, and a computer-assisted proof that the regularized energy component is convex in that window, so that the retrograde and direct orbits do bind disk-like sections there. Liu and Salomao (2025-2026) subsequently proved the conjecture in Hill's lunar problem for all subcritical energies, at $\mu=1/2$ for all energies below the first Lagrange value, and for all mass ratios sufficiently close to $1/2$. The open problem is the conjecture in the remaining non-perturbative generality: whether, for every mass ratio and every subcritical Jacobi energy, the retrograde orbit in each regularized compact energy component bounds a disk-like global surface of section.

Background

The circular restricted three-body problem (RTBP) models a massless particle moving under the Newtonian gravity of two primaries of masses $1-\mu$ and $\mu$ that rotate about their common center of mass on circular orbits; in the planar version the particle moves in the primaries' orbital plane. In a frame rotating with the primaries the dynamics is governed by the Jacobi Hamiltonian $H=\frac{1}{2}|p|^{2}+q_{1}p_{2}-q_{2}p_{1}-\frac{1-\mu}{|q+\mu|}-\frac{\mu}{|q-1+\mu|}$ on $({\mathbb{R}}^{2}\setminus\{(-\mu,0),(1-\mu,0)\})\times{\mathbb{R}}^{2}$, and $H$ is conserved; one writes $\Sigma_{c}=H^{-1}(-c)$ and calls $c$ the Jacobi energy (in this convention larger $c$ means lower energy). The projection of an energy level to configuration space is the Hill region, whose topology changes at the critical values of $H$ attained at the Lagrange points (the equilibrium points of the rotating frame). Below the first critical value the bounded part of the Hill region splits into one component around each primary, and after Levi-Civita regularization - a complex squaring map that unfolds two-body collisions - each compact energy component lifts to a starshaped hypersurface in ${\mathbb{R}}^{4}$ (a hypersurface whose radial projection from the origin is a diffeomorphism, and which therefore carries a canonical contact form) diffeomorphic to $S^{3}$, a double cover of the Moser-regularized component $\mathbb{R}P^{3}$ (Moser regularization is an alternative regularization of both collision singularities that quotients out the two-fold symmetry introduced by the squaring map, so its energy component is a projective space rather than a sphere). The flow on a starshaped hypersurface is a Reeb flow, the natural dynamics associated with a contact form (here obtained from the Hamiltonian after regularization).

A planar periodic orbit encircling a primary is retrograde if it runs opposite to the primaries' rotation and direct otherwise. Birkhoff proved the existence of the retrograde orbit by a shooting method and asked whether it bounds a disk-like global surface of section: a properly embedded disk in the regularized energy component whose interior is transverse to the flow and which every orbit intersects in forward and backward time. Such a section reduces the continuous flow to an area-preserving return map on the disk, the framework Poincare introduced for organizing periodic orbits, and Birkhoff wanted it as a tool to prove the existence of the direct orbit. Conley proved the existence of direct and retrograde orbits for very large Jacobi constants ($c\gg 2$, corresponding to very low energies) by perturbative methods, and for small mass parameter the rotating Kepler problem (the $\mu=0$ limit of the RTBP, in which the lighter primary vanishes and the motion reduces to an integrable Kepler problem in the rotating frame) is integrable and perturbative arguments apply; in the non-perturbative regime even basic properties of the binding candidate, such as uniqueness and non-degeneracy of the retrograde orbit, are unproved.

The modern symplectic-topological route is dynamical convexity. The Conley-Zehnder index of a periodic Reeb orbit is an integer measuring the winding of the linearized flow; a starshaped hypersurface in ${\mathbb{R}}^{4}\cong\mathbb{C}^{2}$ is dynamically convex if every periodic Reeb orbit has Conley-Zehnder index at least $3$. Hofer, Wysocki and Zehnder proved that dynamically convex hypersurfaces in $\mathbb{R}^{4}$ admit disk-like global surfaces of section, and Hryniewicz sharpened this: on a dynamically convex starshaped hypersurface, an unknotted periodic orbit whose self-linking number (an integer invariant of a transverse knot in a contact 3-manifold; the binding of a disk-like section always has self-linking number $-1$) equals $-1$ bounds a disk-like section. Hence Birkhoff's conjecture follows from dynamical convexity of the regularized RTBP components together with certification of the retrograde orbit's topology. Convexity of the regularized embedding implies dynamical convexity, but convexity is delicate: Frauenfelder, van Koert and Zhao proved a convex symplectic embedding only near the rotating Kepler limit (below the first critical energy, outside small collision neighborhoods), and Liu and Salomao proved strict convexity at the equal-mass ratio $\mu=1/2$ for all energies up to the first Lagrange value.

Joung and van Koert combine these tools with validated numerics - interval arithmetic, in which every real quantity is enclosed in a machine-representable interval so that existence and non-degeneracy of orbits become finitely checkable certificates (implemented with the CAPD library, a C++ validated-numerics library, and the $C^{1}$-Lohner algorithm, an interval method that encloses a trajectory together with its derivative). They prove (Theorem 1.1) a non-degenerate symmetric direct periodic orbit for $\mu\in[0.1,0.5]$ and Jacobi energies between the first critical energy and $c=2.1$; (Theorem 1.2) for $\mu\in[0,0.5]$ and $c\in[2.1,2.1+10^{-6}]$, all symmetric periodic orbits of sufficiently small action are the retrograde or the direct orbit, which are non-degenerate, unknotted, of self-linking number $-1$, and of Conley-Zehnder index 3 and 5 in the Levi-Civita regularization (1 and 3 in the Moser regularization); and (Theorem 1.5, added in the published version) the compact Levi-Civita component is convex in the same window, so that both orbits bind disk-like global surfaces of section there. They also develop rigorous formulas for the Conley-Zehnder index that are robust under interval arithmetic. The paper proposes as next steps an exhaustive validated covering of the entire energy component up to a computable action bound, validated bounds on index growth - together sufficient to certify dynamical convexity - and, further out, a computational Floer homology (Floer homology being the Morse homology of the action functional on the loop space; a computational version would build its chain complexes and differentials by validated numerics) for the RTBP.

Subsequent work resolved several perturbative regimes but left the middle of the parameter space open. Liu and Salomao constructed finite energy foliations (pseudo-holomorphic-curve foliations of symplectizations that generalize surfaces of section) for the RTBP and proved Birkhoff's retrograde orbit conjecture for mass ratios sufficiently close to $1/2$ at all energies below the first Lagrange value, and for Hill's lunar problem - the scaled limit of a negligible mass ratio - at every energy below the critical value, in both cases via new convexity theorems. What remains is the non-perturbative intermediate regime: mass ratios away from $0$ and from $1/2$ (for example the Earth-Moon ratio $\mu\approx 0.012$) at general subcritical energies, where no convexity or dynamical convexity result is known and the conjectured surface of section remains unestablished.

Problem Statement

For the planar circular restricted three-body problem with mass parameter $\mu\in(0,1/2]$ (the full range, up to the symmetry $\mu\leftrightarrow 1-\mu$, which exchanges the two bounded Hill-region components) and Jacobi energy strictly below the first critical value - explicitly, $c>c_{1}(\mu)$, where $c_{1}(\mu)$ is the first critical value of the Jacobi Hamiltonian and, in the convention $\Sigma_{c}=H^{-1}(-c)$, larger $c$ means lower energy; the endpoint $c=c_{1}(\mu)$ is excluded because there $\Sigma_{c}$ contains the Lagrange equilibrium $L_{1}$, the two bounded Hill-region components meet, and the regularized component is no longer a smooth starshaped hypersurface carrying a Reeb flow - settle Birkhoff's conjecture in the generality that remains open: prove that, for every such $(\mu,c)$, every retrograde periodic orbit in each regularized compact energy component (existence of such an orbit is Birkhoff's theorem; uniqueness is not known in general, so the requirement applies to each retrograde orbit) - equivalently, its lift to the Levi-Civita double cover $S^{3}\subset{\mathbb{R}}^{4}$ of that component - bounds a disk-like global surface of section for the regularized Reeb flow; or refute the conjecture by a certified obstruction at an explicit parameter pair (for example, a certified knot-type or self-linking computation showing the retrograde orbit cannot bind a disk-like section, or a contact-topological obstruction showing no disk-like section exists for that flow). The two bounded components are the component $\Sigma_{\mu,c}$ around the heavier primary at $(-\mu,0)$, which is the one treated by the source paper's theorems, and the corresponding component around the lighter primary at $(1-\mu,0)$, which the $\mu\leftrightarrow 1-\mu$ symmetry identifies with the heavy-primary component at mass parameter $1-\mu$.

References

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