V.I. Arnold's "pointwise" KAM Theorem

Comlan Edmond Koudjinan; Luigi Chierchia

REVIEW 1 major objections 5 minor 16 references

V.I. Arnold's "pointwise" KAM Theorem

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper shows that optimizing the original 1963 pointwise KAM scheme turns it into a quantitatively sharp machine: under mild Diophantine and non-degeneracy assumptions, a real-analytic invariant Lagrangian torus persists as soon as…

desk verdict A genuinely new quantitative KAM theorem for Arnold's pointwise scheme, with explicit constants; the proof structure is sound and the sharpness claim is credible, though the unverified algebraic chain of constants is the main risk. read the letter →

arxiv 1908.02523 v2 pith:R4PT4L76 submitted 2019-08-07 math.DS

Luigi Chierchia , Comlan Edmond Koudjinan This is my paper · ORCID

classification math.DS MSC 37J4070H0837J25

keywords KAMtheorypointwiseschemeinvariantLagrangiantorismalldivisorsDiophantinefrequenciesquantitativeestimatesHamiltonianperturbation

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

This paper takes a 1963 proof of the classical conservation theorem for conditionally periodic motions, the one that builds an invariant torus as a limit of symplectic transformations on action domains shrinking to a point, and asks how small the perturbation really has to be for the construction to work. The answer is a quantitative Theorem A: for $d\ge 2$, $\tau\ge d-1$, under the Diophantine and non-degeneracy conditions (12), the unperturbed torus persists as a real-analytic Lagrangian torus whenever $\alpha\le r/T$ and $\epsilon=KP\varepsilon/\alpha^2\le (s-s_*)^a/(C_*\theta^4)$. The smallness condition has the form $\epsilon/\alpha^2

What carries the argument

The engine is a quadratically convergent Newton scheme with a finite small-divisor cutoff. At step $j$ the Hamiltonian is $H_j=K_j+\varepsilon^{2^j}P_j$, with $K_j$ integrable, $K_j(y_j)=\omega$, and $K_j$ non-degenerate; a near-identity symplectic map $\varphi_j$ is built from a generating function $y'\cdot x+\varepsilon g(y',x)$, solving the homological equation only for Fourier modes $|k|_1\le\kappa_j$ with $\kappa_j\sim|\log(\varepsilon^{2^j}\|P_j\|)|$. The quadratic scheme makes the new perturbation of order $\varepsilon^{2^{j+1}}$, while the Fourier cutoff controls only finitely many small divisors per step. Two tunings carry the quantitative content: the action radii $r_j$ shrink to a point so that the Diophantine condition holds uniformly on the small domain, and the first step is isolated from the later steps so that a logarithmic factor does not contaminate the leading displacement estimate. The constants $K_j$, $T_j$ are kept within a factor $\sqrt2$ of their initial values, which is what fixes the constant $\theta$ in the final threshold.

What would settle it

Pick $d=2$, $\tau=1$, take $H=\frac12(y_1^2+y_2^2)+\epsilon(\cos x_1+\cos x_2)$ with $\omega=(1,\sqrt2)$, compute the Diophantine constant $\alpha$ and the constants in Appendix A, and check numerically whether the invariant torus exists and obeys (16) whenever (14) holds; a single counterexample below the stated threshold would refute the quantitative theorem.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the pointwise KAM scheme is quantitatively sharp. Theorem A states that for a real-analytic near-integrable Hamiltonian $H=K+\varepsilon P$ whose unperturbed frequency $\omega=K_y(y_0)$ is Diophantine and whose Hessian $K_{yy}(y_0)$ is invertible, there are constants $C,C_*$ depending only on $d$ and $\tau$ such that the smallness conditions $\alpha\le r/T$ and $\epsilon\le(s-s_*)^a/(C_*\theta^4)$ imply the existence of a real-analytic Lagrangian torus $\mathcal T_{\omega,\varepsilon}=\varphi_*(\mathbb T^d)$ invariant under the perturbed flow, with frequency $\omega$ and displacement bounds $\max\{\|u_*\|_{s_*},\frac{1}{2e}\|d_xu_*\|_{s_*},\frac{K}{\alpha}\|v_*\|_{s_*}\}\le C\theta^3(s-s_*)^{-a}\epsilon$. The displacement estimate is the advertised sharp form: the action oscillation is $O(\varepsilon/\alpha)$, exactly the size forced by the pendulum's separatrix. The torus is also Kolmogorov non-degenerate, and its dependence on parameters is real-analytic.

Load-bearing premise

The iteration must keep the Hessian of every intermediate integrable Hamiltonian $K_j$ invertible and keep its norm bounds within a factor of order $\sqrt2$ of the original values while the action domain shrinks to a point; if the choice of shrinking radii and Fourier cutoffs cannot enforce that, the construction collapses.

Editorial extensions

If this is right

If Theorem A is correct, the asymptotic threshold for persistence of a Diophantine torus is $\varepsilon\lesssim\alpha^2$, with displacement of order $\varepsilon/\alpha$; no hidden steep dependence on $\alpha$ appears.
The explicit constants make the smallness condition ready for quantitative applications, such as estimating the measure of the Kolmogorov set with complement of size $O(\sqrt\varepsilon)$ along the lines the paper discusses.
The constructed torus is Kolmogorov non-degenerate, so the perturbed Hamiltonian admits a normal form with non-degenerate quadratic part, allowing secondary KAM-type arguments to be applied nearby.
The torus depends real-analytically on external parameters whenever the Hamiltonian does, so the quantitative theorem extends to parametric families of perturbations.
The sharper estimates can be used to improve the exponentially long stability time estimates for nearly invariant tori, as the paper notes.

Reading between the lines

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

The same tuning of the scheme should transfer to the iso-energetically non-degenerate case, giving an equally explicit smallness condition on a fixed energy shell; the paper indicates the adaptation is routine.
The logarithmic corrections from the Fourier cutoff are the only non-sharp ingredient in the estimates; a sharper treatment of the Fourier tail might push the threshold closer to the pure $\varepsilon\lesssim\alpha^2$ form.
For concrete low-dimensional Hamiltonians the explicit constants could be evaluated numerically to produce breakdown thresholds for invariant tori, testable against computed invariant curves.
The device of isolating the first step to avoid logarithmic losses in the leading estimate is likely useful in other Newton-type KAM constructions.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

This paper does something real: it takes Arnold's 1963 pointwise scheme, separates the first step from the iteration, and proves sharp asymptotic smallness and oscillation bounds with all constants written out. The comparison with Villanueva's optimized Kolmogorov scheme is fair, and the authors are right that no one had shown Arnold's scheme achieves the same optimal epsilon/alpha-squared threshold and O(epsilon/alpha) displacement bound. The first-step treatment is a genuine technical device, not just bookkeeping; it removes a logarithmic loss that would otherwise spoil the displacement estimate. The pendulum example justifies the sharpness claim about the need for epsilon/alpha-squared smallness, and the paper correctly treats it as a benchmark rather than as a derivation. I believe the central theorem is new and significant for quantitative KAM theory.

The proof is detailed and follows standard machinery: Cauchy estimates, Fourier decay, an implicit function theorem, and a contraction argument for the inverse transformation. I did not verify every algebraic constant, and neither, apparently, did the stress-test note. That is the soft spot, and it is the only real one. The final constants C and C* are built from a long chain of nested definitions (C7, C9, C10, C14, C*), and a single transcription error or slipped exponent in Lemma 3's induction would compromise the precise form of (14). The authors say all constants are explicitly computed, but there is no machine-checked or computer-assisted verification. For a paper whose entire contribution is the quantitative statement, that is a legitimate referee concern, not a fatal one. Nothing in the logic looks circular or fitted; the smallness condition is derived from the scheme, and the constants are not post-hoc tuned to hit the pendulum bound.

I also think the paper is honest about its own limitations. It states that the threshold is not optimal (Remark 5), notes the iso-energetic extension is only discussed, and does not oversell the measure estimate for the Kolmogorov set, which it only mentions as a potential application. The citation pattern is reasonable; self-citations are used for background and not as support for the main theorem.

Who is this for? Researchers working on quantitative KAM, especially those who care about explicit smallness conditions or want to apply KAM to specific systems. The paper deserves a serious referee; the ideal referee would spend the time to check the constant chain in Lemma 3, especially the definitions of C10, C11, C12, and the final C*, and verify that (51) follows from (14). I would recommend sending it to peer review with that expectation. If the constants check out, it is a solid addition to the literature.

Referee Report

1 major / 5 minor

Summary. The paper revisits Arnold's 1963 proof of Kolmogorov's theorem and develops a quantitative, pointwise version of Arnold's scheme. The main result, Theorem A, asserts that under the Diophantine and non-degeneracy assumptions (12), if α ≤ r/T and the rescaled perturbation ǫ = K P ε/α² satisfies (14) with exponent a = 6τ+3d+8, then the unperturbed torus persists as a real-analytic Lagrangian torus and satisfies the displacement bounds (16). The proof is organized as an explicit one-step estimate (Lemma 1), a special first step (Lemma 2), an iterative construction with a super-exponentially convergent sequence (Lemma 3), and a final constant-collection argument in §3.3 and Appendix A. The paper also uses the pendulum to illustrate that the resulting ε/α² smallness condition and the ε/α oscillation bound are asymptotically sharp in their dependence on ε and α.

Significance. If correct, this is a valuable contribution: it turns Arnold's pointwise scheme into a fully quantitative theorem with explicit constants depending only on d and τ, it isolates the technical reason why the first step must be treated separately, and it demonstrates that the scheme reaches the same asymptotic scalings as the pendulum benchmark. The proof is self-contained, uses standard Cauchy and Fourier estimates plus an implicit function theorem, and the constants are collected in Appendix A. The paper also proves Kolmogorov non-degeneracy of the constructed torus in Appendix B. The main risk is the intricate chain of explicit constants, which is not machine-checked; I checked the closing inequalities of §3.3 and found them broadly consistent, with one important exception discussed below.

major comments (1)

[§3.2.2 and §3.3, Eq. (63) and footnote 14] The displayed inequality (63), namely (log t)^{4ν} ≤ (4ν e^{-1})^{4ν} √t for all t > 1, is false as stated; for ν = 1 and t = e^2 the left side is 16 and the right side is about 7.39. The related footnote claim in §3.2.2, "(log t)^{2s} ≤ t^{1/2} for all t ≥ e, s ≥ 1/4", is also false (take ν = 1, s = 1, t = e^2). These inequalities are used in the proof of the logarithmic bound (62) that controls κ_j in Lemma 3, and again in §3.3 to pass from ǫ^2 (log ǫ^{-1})^{2ν} to ǫ^{3/2}. The written proof therefore contains a gap at a load-bearing point. The needed estimates are weaker than the false statements and appear to follow from the explicit smallness conditions (41) and the size of C9, C14, and C*, so the gap is repairable, but the authors must replace the false inequalities with correct ones and justify them in the parameter range actually used.

minor comments (5)

[§3.2, definition of ¯s_j] In the block of definitions before Lemma 2, the formula for ¯s_j contains an undefined index i; it should read ¯s_j := s_j − 2σ_j/3.
[§3.1, Step 1 and Eq. (27)] The notation pκP for the Fourier truncation is easily confused with the product of the scalar κ and P; a clearer notation such as p_κ P or Π_κ P would improve readability.
[Introduction, paragraph b] The sentence 'deforms ... into a a Lagrangian torus' contains a duplicated article 'a'.
[Introduction, pendulum discussion] The sharpness comparison uses the pendulum, which is one-dimensional, while Theorem A is stated for d ≥ 2; a sentence clarifying that the sharpness refers to the scaling in ε and α rather than to a matching counterexample in the theorem's range would avoid a possible misunderstanding.
[Appendix A and §3.3] Given the length and nested structure of the constant list, a short computer-assisted verification of the algebraic inequalities involving C0, ..., C*, C, and C* would substantially increase confidence in the explicit-constant claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem A is derived from the stated Diophantine and non-degeneracy assumptions with explicitly computed constants; no fitted parameter is renamed as a prediction.

full rationale

The paper is a self-contained quantitative KAM theorem. The smallness condition (14) is obtained by iterating Lemma 1 and checking the inductive hypotheses in Lemma 3; the constants C0 through C15, C, and C* are explicit functions of d and tau collected in Appendix A. The proof shows directly that (14) implies the smallness conditions (41) and (51), and the displacement estimates (16) are bounded by the same constants that emerge from the Cauchy and Fourier estimates. No parameter entering the conclusion is fitted to the conclusion. The pendulum example in the introduction is used only as a benchmark to motivate optimality; it is not used to derive Theorem A, and the lower bound (6) is compared with the theorem's upper bound (8) only after the theorem is proved. Self-citations appear for standard background material such as Cauchy estimates, the implicit function theorem, and KAM reviews, not as the load-bearing justification of the main theorem. The possibility of an algebra slip in the explicit constants is a correctness risk, not a circularity, because none of the constants is defined in terms of the target torus or the target estimate.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof uses only standard analytic tools and the classical Arnold scheme; no new entities or fitted parameters are introduced. All constants are computed in Appendix A and depend only on dimension and Diophantine exponent.

assumptions (5)

standard math Cauchy estimates and Fourier coefficient decay for real-analytic functions (Lemma A.1).
Used throughout the basic step to bound derivatives and Fourier coefficients; quoted from the cited text.
standard math Implicit Function Theorem in the version of Lemma A.2.
Used to define the shifted action y1 and to invert the angle map; quoted from the cited text.
standard math Weierstrass theorem on uniform limits of analytic functions.
Used to conclude real-analyticity of the limiting torus embedding.
domain assumption The Hamiltonian H = K + epsilon P is real-analytic on the complex domain D_{r,s}(y0) and its frequency omega is Diophantine with non-degenerate Hessian.
This is the setup of Theorem A, Eq. (12).
domain assumption The constants K, T, P are finite bounds on the respective norms and the product theta = T K is at least 1.
These normalized bounds define the smallness parameter epsilon = K P epsilon / alpha^2 and are assumed in Lemma 1.

how reviews work

0 comments

Cite this review

Pith. "Pith review of V.I. Arnold's "pointwise" KAM Theorem." pith.science (2026). https://pith.science/paper/R4PT4L76

@misc{pith2026190802523,
  author       = {Pith},
  title        = {Pith review of: V.I. Arnold's "pointwise" KAM Theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R4PT4L76}},
  note         = {Machine review of arXiv:1908.02523}
}

abstract

We review V.I. Arnold's 1963 celebrated paper \cite{ARV63} {\sl Proof of A.N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian}, and prove that, optimizing Arnold's scheme, one can get "sharp" asymptotic quantitative conditions (as $\varepsilon\to 0$, $\varepsilon$ being the strength of the perturbation). All constants involved are explicitly computed.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[1]

V. I. Arnold, Proof of A. N. Kolmogorov’s theorem on the conservation of co nditionally periodic motions with a small variation in the Hamiltonian , Russian Math. Surv, 1963, vol. 18, no 5, pp. 9–36. 16As usual, Bl y :“ B|l|1 Byl1 1 ¨¨¨B y ld d , @y P Rd, lP Zd. 17Here, Dn r pz0q denotes the ball in Cn centered at z0 and with radius r. 32

work page 1963
[2]

V. I. Arnold, V.V. Kozlov, and A.I. Neishtadt (editor), Mathematical Aspects of Classical and Celestial Mechanics , Dynamical Systems III, Series: Encyclopaedia of Mathemat ical Sciences , Vol. 3, Springer-Verlag 3rd ed., 2006

work page 2006
[3]

Biasco and L

L. Biasco and L. Chierchia, Explicit estimates on the measure of primary KAM tori . Annali di Matematica Pura ed Applicata (1923-), 197(1):261–281, 2 018

work page 1923
[4]

Celletti and L

A. Celletti and L. Chierchia, A constructive theory of Lagrangian tori and computer-assi sted applications, In : Dynamics reported. Springer Berlin Heidelberg, 1995. p. 60-129

work page 1995
[5]

Chierchia, Kolmogorov’s 1954 paper on nearly integrable Hamiltonian s ystems, Regular and Chaotic Dynamics, 2008, vol

L. Chierchia, Kolmogorov’s 1954 paper on nearly integrable Hamiltonian s ystems, Regular and Chaotic Dynamics, 2008, vol. 13, no. 2, pp. 130–139

work page 1954
[6]

Chierchia, Kolmogorov–Arnold–Moser (KAM) Theory , In : Mathematics of Complexity and Dynamical Systems, Springer New York, 2012

L. Chierchia, Kolmogorov–Arnold–Moser (KAM) Theory , In : Mathematics of Complexity and Dynamical Systems, Springer New York, 2012. pp. 810–836

work page 2012
[7]

Chierchia and G

L. Chierchia and G. Gallavotti, Smooth prime integrals for quasi-integrable Hamiltonian systems, Il Nuovo Cimento, 67 B, 277-295 (1982)

work page 1982
[8]

Chierchia and E.C

L. Chierchia and E.C. Koudjinan, Structure of the Kolmogorov’s set

work page

Show all 16 references

[9]

Chierchia and M

L. Chierchia and M. Procesi, Kolmogorov-Arnold-Moser (KAM) Theory for Finite and Inﬁ- nite Dimensional Systems , Encyclopedia of Complexity and Systems Science Living Edi tion, Editors: Robert A. Meyers, Springer (New York), 2nd Edition (2018). doi: 10.1007/978-3- 642-27737-5 302-3

2018 doi
[10]

Delshams, P

A. Delshams, P. Guti´ errez, Eﬀective stability and KAM theory , J. Diﬀerential Equations, 128:415-490, 1996

1996
[11]

A. N. Kolmogorov, On the conservation of conditionally periodic motions unde r small per- turbation of the Hamiltonian , Dokl. Akad. Nauk. SSR 98 (1954), 527-530. English transla- tion by Helen Dahlby in: Stochastic behavior in classical an d quantum Hamiltonian systems, Volta...

1954
[12]

Koudjinan, Ph.D

C.E. Koudjinan, Ph.D. Thesis , Universit` a degli Studi Roma Tre, March 2019. arxiv.org/abs/1904.13062

2019 arXiv
[13]

P¨ oschel,Integrability of Hamiltonian systems on Cantor sets

J. P¨ oschel,Integrability of Hamiltonian systems on Cantor sets. Communic ations on Pure and Applied Mathematics , 35(5): 653–696, 1982

1982
[14]

A. I. Neishtadt, Estimates in the Kolmogorov theorem on conservation of cond itionally periodic motions, Journal of Applied Mathematics and Mechanics,1981, 45, no . 6, pp. 766– 772

1981
[15]

Salamon, The Kolmogorov-Arnold-Moser theorem

D.A. Salamon, The Kolmogorov-Arnold-Moser theorem. Math. Phys. Electron. J. 10 (2004), Paper 3, 37 pp. 33

2004
[16]

Villanueva

J. Villanueva. Kolmogorov theorem revisited, Journal of Diﬀerential Equations, 244(9):2251– 2276, 2008. 34

2008

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

V. I. Arnold, Proof of A. N. Kolmogorov’s theorem on the conservation of co nditionally periodic motions with a small variation in the Hamiltonian , Russian Math. Surv, 1963, vol. 18, no 5, pp. 9–36. 16As usual, Bl y :“ B|l|1 Byl1 1 ¨¨¨B y ld d , @y P Rd, lP Zd. 17Here, Dn r pz0q denotes the ball in Cn centered at z0 and with radius r. 32

work page 1963

[2] [2]

V. I. Arnold, V.V. Kozlov, and A.I. Neishtadt (editor), Mathematical Aspects of Classical and Celestial Mechanics , Dynamical Systems III, Series: Encyclopaedia of Mathemat ical Sciences , Vol. 3, Springer-Verlag 3rd ed., 2006

work page 2006

[3] [3]

Biasco and L

L. Biasco and L. Chierchia, Explicit estimates on the measure of primary KAM tori . Annali di Matematica Pura ed Applicata (1923-), 197(1):261–281, 2 018

work page 1923

[4] [4]

Celletti and L

A. Celletti and L. Chierchia, A constructive theory of Lagrangian tori and computer-assi sted applications, In : Dynamics reported. Springer Berlin Heidelberg, 1995. p. 60-129

work page 1995

[5] [5]

Chierchia, Kolmogorov’s 1954 paper on nearly integrable Hamiltonian s ystems, Regular and Chaotic Dynamics, 2008, vol

L. Chierchia, Kolmogorov’s 1954 paper on nearly integrable Hamiltonian s ystems, Regular and Chaotic Dynamics, 2008, vol. 13, no. 2, pp. 130–139

work page 1954

[6] [6]

Chierchia, Kolmogorov–Arnold–Moser (KAM) Theory , In : Mathematics of Complexity and Dynamical Systems, Springer New York, 2012

L. Chierchia, Kolmogorov–Arnold–Moser (KAM) Theory , In : Mathematics of Complexity and Dynamical Systems, Springer New York, 2012. pp. 810–836

work page 2012

[7] [7]

Chierchia and G

L. Chierchia and G. Gallavotti, Smooth prime integrals for quasi-integrable Hamiltonian systems, Il Nuovo Cimento, 67 B, 277-295 (1982)

work page 1982

[8] [8]

Chierchia and E.C

L. Chierchia and E.C. Koudjinan, Structure of the Kolmogorov’s set

work page