On the secular evolution of the semi-major axis in canonical formalism

Barnab\'as Deme

REVIEW 4 major objections 4 minor 20 references

On the secular evolution of the semi-major axis in canonical formalism

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Second-order von Zeipel averaging can leave the semi-major axis slowly drifting, not merely oscillating, even without resonances.

desk verdict The Hamiltonian derivation is incomplete: Eq. (10) drops an O(epsilon^2) secular term that the toy model itself contains, so the claimed epsilon^3 scaling does not follow, though the underlying effect may be real. read the letter →

arxiv 2505.00207 v1 pith:3Q2LT3TD submitted 2025-04-30 astro-ph.EP

Barnabás Deme This is my paper · ORCID

classification astro-ph.EP MSC 70F1570H15

keywords secularevolutionsemi-majoraxisvonZeipeltransformationcanonicalperturbationtheorysecond-orderaveraginghierarchicalthree-bodyproblemDelaunayvariables

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Standard first-order secular perturbation theory says that, away from resonances, the semi-major axis only oscillates on the orbital timescale and does not drift. This paper tries to show that this conclusion can fail when the von Zeipel averaging is carried to second order: the relation between the true action $L=\sqrt{a}$ and the final constant action $L''$ can contain a term with no dependence on the fast angle. The formal condition is simple: in Eq. (10) the dangerous term carries $e^{i(k+m)l''}$, so it survives the averaging when $k=-m$. Working through a toy Hamiltonian, the paper obtains $L=L''+\cdots+\epsilon^3 {L''}^9/D^8\,\sin g''+\cdots$, and because $g''$ itself evolves slowly, it concludes that the semi-major axis evolves secularly. If the mechanism operates in real systems, it would overturn a standard piece of orbital lore and would match, at least in spirit, an independent derivation based on Lagrange's planetary equations.

What carries the argument

The engine of the argument is the second von Zeipel canonical transformation: a change of variables generated by $S'=\mathrm{id}+\epsilon^2\sum_m S_m(g',L'',G'')\,e^{iml'}$, chosen to remove the $O(\epsilon^2)$ oscillating pieces from the Hamiltonian. Its load-bearing identity is Eq. (10), where the substitution of both generating functions into $L=L'+\epsilon\sum_k ikS_k e^{ikl'}+\cdots$ leaves an $O(\epsilon^3)$ term with phase $e^{i(k+m)l''}$. Setting $k=-m$ freezes that phase, converting the constant action $L''$ and the secularly precessing angle $g''$ into a slow correction to $L=\sqrt{a}$. The toy model (12) supplies a concrete nonzero coefficient; the calculation isolates the term $\epsilon^3 {L''}^9/D^8\sin g''$ as the explicit secular channel.

What would settle it

Integrate the exact equations of motion of the toy Hamiltonian (12) numerically for many orbital periods and a set of initial $g$ values: if the mechanism is real, $L=\sqrt{a}$ should accumulate a drift of order $\epsilon^3 {L''}^9/D^8$ whose sign follows $\sin g''$, whereas a pure oscillation would average to zero. To test the physical relevance, compute the second-order von Zeipel generating function for a realistic hierarchical-three-body Hamiltonian and check whether any Fourier coefficient with $m=-k$ is nonzero; if none is, the claimed secular semi-major-axis evolution would not occur in that system.

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Extended reading notes

Core claim

The central discovery claimed is that a second-order von Zeipel canonical transformation can make the original action $L$ (hence the semi-major axis, since $L=\sqrt{a}$) depend on the slowly evolving angle $g''$ without any fast-angle factor. In the general formalism, after eliminating the $O(\epsilon^2)$ oscillations, the action relation (Eq. 10) contains mixed terms of order $\epsilon^3$ with phase $e^{i(k+m)l''}$; whenever $k=-m$, this phase becomes unity and the term survives as a purely secular correction. The paper demonstrates the mechanism on the toy Hamiltonian $H=-1/(2L^2)+(\epsilon/D^3)G\sin l+(\epsilon/D^2)\sin g$, obtaining $L=L''+\cdots+\epsilon^3 {L''}^9/D^8 \sin g''+\cdots$ alongside $\dot g''=-\frac{3}{2}\epsilon^2 {L''}^2 G''/D^6$. Since $g''$ evolves secularly and $L''$ is constant, the original semi-major axis acquires a long-term drift. The paper also notes that this drift is of order $\epsilon^3$ here, one power higher than the $\epsilon^2$ result of a Lagrange-planetary-equations treatment, attributing the difference to the use of doubly averaged generating functions.

Load-bearing premise

The load-bearing premise is that the simple toy Hamiltonian used for the demonstration is structurally similar enough to real second-order secular perturbations that its nonzero $k=-m$ coefficient actually occurs in nature; the paper itself notes the toy model has no direct physical meaning and does not reduce any real three-body Hamiltonian to this form.

Editorial extensions

If this is right

In a non-resonant Hamiltonian of the form (1), a second-order von Zeipel transformation can turn the constant action $L''$ and the slowly changing angle $g''$ into a secular correction to $L=\sqrt{a}$, so the semi-major axis is not necessarily secularly constant.
The effect is absent in a first-order (single-averaged) theory, which is why the standard result that $a$ is constant is correct only at that order.
The secular drift is controlled by the evolution of $g''$, coupling the semi-major axis to the precession of pericenter at $\epsilon^3$.
When the perturbation's Fourier coefficients make the prefactor of $e^{i(k+m)l''}$ in Eq. (10) vanish, the secular term disappears, so the phenomenon is conditional rather than universal.
The paper's toy-model drift scales as $\epsilon^3$, one order higher than the $\epsilon^2$ effect found with Lagrange's planetary equations; the author attributes this to double averaging, so the apparent order depends on the averaging scheme.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the toy Hamiltonian is explicitly not physical, the decisive next step is to evaluate the $k=-m$ Fourier coefficient in the second-order generating function for a real hierarchical-three-body Hamiltonian (for example, the quadrupole-squared terms); if that coefficient vanishes, the mechanism would not operate in those systems.
The paper's $\epsilon^3$ result and the literature's $\epsilon^2$ result may be two orderings of the same physical drift, since the paper attributes the difference to whether the first-order generating function is applied twice; reconciling the two conventions would pin down the actual size of the effect in nature.
If the effect is real, long-baseline observations of hierarchical triples (for example, an inner binary with a distant tertiary) could look for a cumulative change in the inner orbit's size over many secular cycles, a signature that standard first-order models would not predict.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper tries to give a Hamiltonian derivation of the secular semi-major axis drift that Conway and Will found by Lagrange equations. The general observation—that a second-order von Zeipel step can produce a term independent of the fast angle when k = -m—is a good one, and the toy model does exhibit such a term. But the central formal derivation has a hole that the stress-test note flags, and I think the note is right.

In Eq. (10), L is expressed through L'' and the generating functions, but the expression is taken from Eq. (5), which is only valid to first order in epsilon. When you invert the first transformation, you get O(epsilon^2) corrections, and those are missing from Eq. (10). The toy model shows it explicitly. In Eq. (19), L = L' - epsilon ... sin l' + 3 epsilon^2 ... cos^2 l' + ... . The cos^2 l' term contains a constant part, (3/2) epsilon^2 L'^5 G'^2/D^6. After the second transformation this becomes (3/2) epsilon^2 L''^5 G''^2/D^6, which is independent of l'' and depends on G''. Since G'' evolves under the averaged Hamiltonian, this is a secular contribution one order lower than the epsilon^3 term the paper highlights in Eq. (26). So the paper's stated scaling—epsilon^3—is not established, and the discrepancy with Conway & Will's epsilon^2 is more likely due to this missing term than to double averaging.

That said, the paper is not a waste of time. It is short, readable, and honest about the toy's limitations. The k = -m condition is a useful way to think about the effect, and the author correctly credits the earlier work. The main soft spots besides the missing term: the toy Hamiltonian is explicitly non-physical, and the full second generating function S' is never written out. So even after fixing Eq. (10), the astrophysical relevance would need a real hierarchical-three-body Hamiltonian, not a toy.

Who is this for? Someone working on second-order secular perturbation theory who wants a foothold in the Hamiltonian formalism. The current version does not quite get there. I would send it to peer review, because the question is timely and the flaw is technical and correctable. A referee should ask for a corrected Eq. (10), a full statement of the toy's leading secular term, and ideally an application to a physical Hamiltonian.

Referee Report

4 major / 4 minor

Summary. The paper studies second-order von Zeipel canonical perturbation theory for Hamiltonians of the form H = H0(L) + ε(H1 + H̃1), with Delaunay variables and a single fast angle l. It claims that after two canonical transformations the relation between the original action L and the final constant action L′′ can contain a term independent of the fast angle when k = −m, as written in Eq. (10). The author introduces a toy Hamiltonian (Eq. 12), carries out the two transformations explicitly, and obtains in Eq. (26) a non-oscillatory term ε³ L′′⁹/D⁸ sin g′′ in L. Since L is related to the semi-major axis, the paper concludes that the semi-major axis evolves secularly and discusses why this effect appears at ε³ rather than at the ε² found in Ref. [14].

Significance. The question addressed is interesting and timely: a Hamiltonian derivation of a secular drift of the semi-major axis in second-order perturbation theory would complement the Lagrange-based result of Ref. [14] and clarify the role of double averaging. The paper is self-contained, the toy-model calculation is explicit, and the author is candid that the toy Hamiltonian does not have direct physical meaning (footnote 2). The general idea that composition of two von Zeipel transformations can produce terms independent of the fast angle is worth articulating. However, the central quantitative claim is not currently supported: Eq. (10) omits O(ε²) inversion terms, and in the toy model the leading long-term variation of L is O(ε²), not O(ε³). The paper therefore needs substantial revision before its conclusions can be accepted.

major comments (4)

[Sec. 2, Eq. (10); Sec. 3, Eq. (19)] The derivation of Eq. (10) from Eq. (5) drops the O(ε²) remainder of the first von Zeipel inversion, but this remainder can contain terms independent of the fast angle after the second transformation. In the toy model, Eq. (19) contains 3ε² L′⁵ G′²/D⁶ cos² l′, whose constant part (3/2)ε² L′′⁵ G′′²/D⁶ survives the second transformation and is independent of l′′. Since dG′′/dt = −ε/D² cos g′′ + O(ε³) from Eq. (24), this term changes on the secular timescale and yields an O(ε²) long-term variation of L. Thus the leading secular effect in the toy model is one order lower than the ε³ term highlighted in Eq. (26) and Sec. 4. The author should redo the expansion including the O(ε²) inversion terms and reassess the comparison with Ref. [14].
[Sec. 2, Eqs. (4)–(6)] The inverse relations (4) and (6) have the wrong sign for the type-2 generating function (2). From the forward relations l′ = ∂S/∂L′ and g′ = ∂S/∂G′ one obtains l = l′ − ε Σ ∂S_k/∂L′ e^{ikl′} + O(ε²) and g = g′ − ε Σ ∂S_k/∂G′ e^{ikl′} + O(ε²), not the plus signs written in the paper. The toy model itself uses the minus signs in Eqs. (18) and (20), so the general equations are inconsistent with the application. These sign errors should be corrected, because Eq. (10) is obtained by composing such inverses.
[Sec. 3, Eq. (23)] Only one term of the second generating function S′ is displayed, so the reader cannot verify the claimed form of H′′ in Eq. (24). In particular, the O(ε²) oscillatory part of Eq. (22) also contains (3/4)ε² L′² G′²/D⁶ cos 2l′, which the displayed term ε² L′′⁶/D⁵ cos g′ sin l′ cannot cancel. The author should either display the complete S′ or list all terms needed to remove every O(ε²) oscillatory term in Eq. (22), and then confirm that H′′ has no residual O(ε²) dependence on l′′.
[Sec. 3, footnote 2; Sec. 4] The toy Hamiltonian is acknowledged to lack direct physical meaning, and the paper does not demonstrate that any realistic astrophysical Hamiltonian (for example, the hierarchical three-body Hamiltonian studied in Ref. [14]) has a non-zero prefactor for the k = −m term in Eq. (10). Without such a reduction, the astrophysical conclusion that the semi-major axis evolves secularly is not established. At minimum, the author should exhibit a concrete physical Hamiltonian for which the k = −m sum does not vanish, or state clearly that the result is only a formal possibility.

minor comments (4)

[Sec. 3, Eq. (26)] The term ε³ L′′⁹/D⁸ sin g′′ arises from sin² l′′, so its non-oscillatory part carries a factor 1/2; the coefficient in Eq. (26) should be checked.
[Sec. 1] In the Introduction, 'L ∝ √a' should read L ∝ √(μ a) in Delaunay variables.
[Sec. 2, Eq. (7)] Eq. (7) contains a typographical ambiguity: 'S_k(g′,L′,G′) ∂g′ e^{ikl′}' should be '∂S_k/∂g′ e^{ikl′}'.
[Abstract and Sec. 4] The abstract and Sec. 4 would benefit from a precise definition of 'secular evolution', in particular whether it means the constant part after averaging over the fast angle or the component independent of l′′ in the instantaneous expression for L.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a self-contained canonical perturbation calculation, and the toy-model artificiality is a physical-relevance caveat, not a circular step.

full rationale

The paper's central derivation is an explicit von Zeipel calculation on a stated toy Hamiltonian, with no fitted parameters, no data, and no load-bearing self-citations. The k = -m secular term in Eq. (10) is a general algebraic consequence of composing two generating functions, and the toy result in Eq. (26) follows from explicit substitution rather than from assuming the conclusion. The toy Hamiltonian is admittedly not directly physical, which limits the astrophysical extrapolation, but that is a scope limitation rather than circularity. The skeptical concern that Eq. (10) may omit an O(epsilon^2) contribution from inverting the first von Zeipel transformation is a possible correctness or ordering issue, not a circularity: even if a lower-order secular term exists, the derivation does not reduce to its input by construction. Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard von Zeipel machinery, a non-resonance assumption, and the representativeness of the toy Hamiltonian (12). The latter is the only genuinely ad hoc element, and it is explicitly non-physical. No data are fitted, and the paper introduces no new physical entities.

free parameters (2)

epsilon
Small perturbation parameter in the toy Hamiltonian (12); its smallness is assumed but no numerical value is given. It sets the order of the secular term (epsilon^3).
D
Dimensionful constant in Eq. (12) with magnitude similar to L and G, introduced by hand to give the perturbation terms the correct dimensions. The secular term in Eq. (26) scales as D^-8.

assumptions (4)

standard math von Zeipel canonical perturbation theory eliminates oscillating terms via generating functions (Eqs. 2 and 9).
The entire derivation assumes the validity and convergence of the von Zeipel procedure as described in Refs. [1,2,19,20].
domain assumption The perturbation is non-resonant, so the generating function (2) has no small denominators.
Stated in Sec. 2: 'if there are no resonances' (page 2). Resonant systems would require a different treatment.
ad hoc to paper The toy Hamiltonian (12) is structurally representative of physical second-order secular perturbation Hamiltonians.
The author writes that the toy model 'does not have a direct physical meaning' (Sec. 3, footnote 2), yet uses it to conclude that the semi-major axis evolves secularly in astrophysical configurations. This representativeness is assumed, not derived.
domain assumption The Keplerian relation L proportional to sqrt(a) connects the Delaunay action to the semi-major axis.
Used in Sec. 2 and Sec. 4 to translate L-variation into a-variation.

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Cite this review

Pith. "Pith review of On the secular evolution of the semi-major axis in canonical formalism." pith.science (2026). https://pith.science/paper/3Q2LT3TD

@misc{pith2026250500207,
  author       = {Pith},
  title        = {Pith review of: On the secular evolution of the semi-major axis in canonical formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3Q2LT3TD}},
  note         = {Machine review of arXiv:2505.00207}
}

read the original abstract

There are several astrophysical configurations where one is interested only in the long-term dynamical evolution. Although the first-order version of this approximation is usually sufficient in applications, second-order corrections may be relevant, too. Here we use the Hamiltonian formalism to show how such higher-order terms lead to the long-term evolution of the semi-major axis.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [14]

Landen Conway and Cliﬀord M. Will. Higher-order eﬀects in the dyn amics of hierarchical triple systems. II. Second-order and dotriacont apole-order eﬀects. Phys. Rev. D , 110(8):083022, October 2024

work page 2024

[2] [1]

von Zeipel

H. von Zeipel. Sur l’application des s´ eries de M. Lindstedt ` a l’´ etude du mou- vement des com` etes p´ eriodiques.Astronomische Nachrichten, 183(22):345, March 1910

work page 1910

[3] [2]

Mathematical methods of classical mechanics

Vladimir Igorevich Arnold. Mathematical methods of classical mechanics . New York: Springer, 1978

work page 1978

[4] [3]

Secular perturbations of asteroids with high inclin ation and eccentricity

Yoshihide Kozai. Secular perturbations of asteroids with high inclin ation and eccentricity. AJ, 67:591–598, November 1962

work page 1962

[5] [4]

M. L. Lidov. The evolution of orbits of artiﬁcial satellites of planet s under the action of gravitational perturbations of external bodies. Planetary and Space Science, 9(10):719–759, October 1962

work page 1962

[6] [5]

The Lidov-Kozai Oscillation and Hugo von Zeipel

Takashi Ito and Katsuhito Ohtsuka. The Lidov-Kozai Oscillation and Hugo von Zeipel. Monographs on Environment, Earth and Planets , 7(1):1–113, November 2019

work page 2019

[7] [6]

S. Naoz. The Eccentric Kozai-Lidov Eﬀect and Its Applications. Annual Review of Astronomy and Astrophysics , 54:441–489, September 2016

work page 2016

[8] [7]

Shevchenko

Ivan I. Shevchenko. The Lidov-Kozai Eﬀect - Applications in Exoplanet Research and Dynamical Astronomy , volume 441. Springer International Publishing Switzerland, 2017

work page 2017