Nonlinear stability of the triangular points in the circular spatial restricted three-body problem at exceptional frequency arithmetic: Liouville frequency vectors, Pythagorean mass ratios, and rigorous steepness verification

For the triangular equilibria $L_4/L_5$ of the circular spatial restricted three-body problem, Cárcamo-Díaz, Palacián, Vidal and Yanguas (2020) proved Lie stability for almost all mass ratios below Routh's critical value and exponential-time adiabatic-invariant estimates under a Diophantine hypothesis on the vector $\sigma$ of formal-integral coefficients, and proved instability at two resonant mass ratios — but their analysis leaves precisely classified gaps: Liouville (non-Diophantine) frequency vectors, where their Theorem 2 estimates do not apply; Pythagorean mass ratios, where the quartic normal form is sign-indefinite and the true Lyapunov stability status (Lyapunov stable, or unstable — possibly only on an extremely slow diffusion time scale) is unknown; and the lack of fully rigorous high-order steepness verifications. This problem asks to complete the rigorous nonlinear-stability classification of $L_4/L_5$ across the whole mass range and all arithmetic types of the frequency data, closing exactly those gaps.

Background

In the circular spatial restricted three-body problem (CRTBP) an infinitesimal particle moves under the Newtonian attraction of two primaries, of masses $1-\mu$ and $\mu$, that revolve around their center of mass on circular orbits; the mass ratio $\mu$ is the problem's only parameter. Besides three collinear equilibria, the problem has the two triangular equilibrium points $L_4$ and $L_5$ (found by Lagrange), which in the spatial, three-degree-of-freedom formulation are equilibrium points of a natural Hamiltonian. They are elliptic equilibria — the eigenvalues of the linearized flow are purely imaginary, $\pm i\omega_1$, $\pm i\omega_2$, $\pm i$ — precisely when $\mu$ is below Routh's critical value $\mu_R=\frac{1}{2}(1-\frac{\sqrt{69}}{9})\approx 0.03852$. The frequency vector is $\omega=(\omega_1,\omega_2,1)$ (Definition 1 of the source paper), with $\omega_1^2+\omega_2^2=1$, a function of $\mu$.

Linear stability does not by itself imply stability of the nonlinear dynamics, and the modern picture of $L_4/L_5$ combines several perturbative mechanisms. The Birkhoff normal form is the averaged polynomial Hamiltonian obtained by iterated near-identity canonical changes of variables that remove the angle dependence order by order around the equilibrium. Moser's twist argument gives Lie stability — stability of the truncated normal form, which confines orbits for all time when the leading averaged term is sign-definite on suitable invariant subspaces — while Kolmogorov–Arnold–Moser (KAM) theory constructs quasiperiodic invariant tori of dimensions 2 and 3 filling most (but not all) of a neighborhood. Nekhoroshev theory gives effective stability: every orbit starting within distance $\varepsilon$ of the equilibrium stays close for a time growing faster than any power of $1/\varepsilon$, provided the averaged Hamiltonian is steep, a transversality condition introduced by Nekhoroshev ensuring that the averaged actions cannot drift fast in any direction; convexity implies steepness, and steepness is the sharp hypothesis. After Birkhoff normalization the quadratic part of the Hamiltonian is written in terms of independent formal first integrals $F_1,\dots,F_d$ (constructed from a basis of the resonance $\mathbb{Z}$-module of $\omega$, with $d\in\{1,2,3\}$) as $\mathcal{H}^2(I)=\sum_{k=1}^{d}\sigma_kF_k(I)$; the coefficient vector $\sigma=(\sigma_1,\dots,\sigma_d)$, whose entries are linear combinations of $\omega_1,-\omega_2,1$, is the object to which the source paper's arithmetic hypotheses apply (its Statement 2 and Definition 6). $\sigma$ is Diophantine if $|k\cdot\sigma|\ge c\,|k|^{-\nu-1}$ for all nonzero $k\in\mathbb{Z}^d$ with some $c>0$ and $\nu\ge d-1$ (condition (2.2) of the source paper); Liouville-type vectors are those failing every such inequality, i.e. admitting arbitrarily good rational approximations. When $d=1$ — the Pythagorean case — no Diophantine condition is needed. Around $L_4$, two of the three actions behave as adiabatic invariants (quantities that change very slowly), and controlling their drift over long times is the route to rigorous confinement.

Benettin, Fassò and Guzzo (1998) proved Nekhoroshev stability of $L_4/L_5$ in the spatial problem for all $0<\mu<\mu_R$, with the possible exception of five mass values, by extending the Nekhoroshev theorem to directionally quasi-convex and weakly steep Hamiltonians. Cárcamo-Díaz, Palacián, Vidal and Yanguas (2020) sharpened the picture: they proved Lie stability for all $\mu\in(0,\mu_R)$ except three types of mass ratios — the two resonant values at which $\omega_1=2/\sqrt{5}$ and $\omega_1=3/\sqrt{10}$ (rationally resonant frequencies), where the resonant normal form is indefinite and the equilibrium is shown to be unstable, and the Pythagorean mass values $\mu\in(\mu_1,\mu_2)$: here $\mu_1$ and $\mu_2\approx0.016376755355816$ are (Notations 2–3 of the source paper, following Benettin–Fassò–Guzzo) the mass values corresponding respectively to $\omega_1^{*}\approx0.959622914$ and $\omega_1^{]}\approx0.935871439$, the unique roots in $(1/\sqrt{2},1)$ of the coefficients $\beta_1$ and $\beta_3$ of the restricted quartic normal form $\mathcal{H}^4(I)=\beta_1I_1^2+\beta_2I_1I_3+\beta_3I_3^2$, and the exceptional values are the countable set of $\mu$ inside this interval for which $\omega=(\omega_1,\omega_2,1)$ is associated with a Pythagorean triple — i.e. $\omega_1=m/n$ in lowest terms with $n^2-m^2$ a perfect square, equivalently $\omega_2\in\mathbb{Q}$, so that $(\omega_1,\omega_2)$ is proportional to an integer triple $(a,b,c)$ with $a^2+b^2=c^2$, such as 3–4–5 (Definition 2 of the source paper). At those values the quartic normal form $\mathcal{H}^4$ changes sign on the invariant set $S=\{I:F_1(I)=\cdots=F_d(I)=0\}$ (Definition 5 of the source paper) on which their Theorem 1 tests sign-definiteness, and their Theorem 1(B) shows higher-order truncations cannot restore sign-definiteness, so Lie stability fails, although the sixth-order truncation $\mathcal{H}^6$ is known (from Benettin–Fassò–Guzzo and the authors' own analysis) to be steep at all Pythagorean triples, which still gives Nekhoroshev-type bounds. Under the Diophantine condition (2.2) on $\sigma$, their Theorem 2 gives exponential-time estimates for the adiabatic invariants; for Liouville-type $\sigma$ these estimates do not apply, since Liouville vectors admit rational approximations too good for the standard small-divisor/averaging scheme, and the authors fall back on the Benettin–Fassò–Guzzo bounds. The general theory nearby includes Lochak's simultaneous-approximation (periodic-averaging) method, Pöschel's Nekhoroshev estimates for elliptic lower-dimensional tori, and the recent results of Bounemoura–Fayad–Niederman that steep real-analytic elliptic equilibria are exponentially stable (confirming a conjecture of Nekhoroshev) and that super-exponential (doubly exponential in $1/\varepsilon$) stability is generic among real-analytic elliptic equilibria — but these are generic-in-the-system results whose hypotheses have not been verified for the fixed CRTBP Hamiltonian, and they do not yield pointwise bounds at a named equilibrium with a Liouville frequency vector.

Problem Statement

Complete the rigorous nonlinear-stability classification of the triangular equilibria $L_4/L_5$ of the circular spatial restricted three-body problem for mass ratios $\mu\in(0,\mu_R)$ across all arithmetic types of the frequency data — the frequency vector $\omega=(\omega_1,\omega_2,1)$ and the coefficient vector $\sigma$ of $\mathcal{H}^2$ on which the source paper's Diophantine condition (2.2) is imposed — by resolving the three gaps left open by Cárcamo-Díaz–Palacián–Vidal–Yanguas (2020), unified by this single classification objective:

(i) Liouville frequencies. Establish rigorous long-time confinement (Nekhoroshev-type) estimates for the adiabatic invariants around $L_4/L_5$ when $\sigma$ is a Liouville-type vector that strictly strengthen the Benettin–Fassò–Guzzo bounds the source currently falls back on — i.e. extend or replace the exponential-time estimates of Theorem 2 of the source paper so that they no longer require the Diophantine condition (2.2) on $\sigma$ while matching the quantitative bounds those estimates give.

(ii) Pythagorean mass ratios. For the countable set of mass values $\mu\in(\mu_1,\mu_2)$ leading to a Pythagorean triple — the interval $(\mu_1,\mu_2)$ and the invariant set $S$ being as defined above and in the source paper — at which the quartic normal form $\mathcal{H}^4$ is sign-indefinite on $S$ and Lie stability therefore fails, determine the true Lyapunov stability status of the equilibrium, the source's own "pending issue": is $L_4$ Lyapunov stable (does every orbit starting sufficiently close remain in a fixed neighbourhood for all time), or is it Lyapunov unstable, with the drift of the actions then at worst on an extremely slow (Arnold-diffusion-type) time scale? Restating the already-known Nekhoroshev-type effective-stability bound that follows from steepness of $\mathcal{H}^6$ does not by itself resolve this question. The source notes that $\mathcal{H}^6$ is steep at all Pythagorean triples, so that if instability holds the diffusion mechanism would be very slow.

(iii) Rigorous verification. Support (i)–(ii) with fully rigorous high-order normal-form and steepness verifications at the exceptional mass ratios — symbolic, or computer-assisted with validated error bounds — in place of the floating-point steepness and normal-form computations on which the current estimates rest and whose related asymptotic estimates, as the source notes, have not been determined so far; sharper bounds in the setting of Nekhoroshev theory via higher-order normal forms are the stated motivation.

References

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