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REVIEW 2 major objections 8 minor 65 references

On the coordinate system-dependence of the accuracy of symplectic numerical methods

T0 review · 2 major / 8 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The coordinate system changes the distorted Hamiltonian that a symplectic integrator actually samples.

desk verdict A genuinely useful explicit computation for symplectic Euler (Eqs. 49 and 59) wrapped in a slightly too-broad claim about all symplectic methods; worth refereeing after the generalization is qualified or proven. read the letter →

arxiv 2506.21241 v1 pith:VPK4ZBSX submitted 2025-06-26 math.NA cs.NAphysics.class-phphysics.comp-ph

Donát M. Takács , Tamás Fülöp This is my paper · ORCID
classification math.NAcs.NAphysics.class-phphysics.comp-ph MSC 65P1037M1570H15
keywords symplecticintegratorsmodifiedHamiltoniandistortedcoordinatetransformationsfirstintegralscycliccoordinatesordercompensationbackwarderroranalysis
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

Symplectic integrators are trusted for long-time Hamiltonian simulations because they sample a nearby Hamiltonian system, not the original one — but the paper shows that this nearby system is not invariant under a change of coordinates. Rewriting the same physics in different generalized coordinates changes the distorted Hamiltonian that the method actually integrates, altering both the observed order of accuracy and which conserved quantities survive numerically. The paper derives explicit formulas for this coordinate dependence in the symplectic Euler method, including a necessary-and-sufficient condition for a cyclic-coordinate momentum to be preserved up to second order. It also shows that a carefully chosen coordinate transformation can eliminate the leading error term, turning a first-order method into a second-order one on the same time step.

What carries the argument

The load-bearing object is the distorted (modified) Hamiltonian $\tilde H$ from backward error analysis, whose first-order correction for symplectic Euler is the elementary Hamiltonian $H_p H_q$. The paper isolates the extra term $\Xi_{H_p H_q, Q}$ that appears when this elementary Hamiltonian is pulled back through a point transformation $Q$; this term measures the coordinate dependence. Poisson-bracket computations with this term convert the non-invariance into a concrete conservation test (condition 59), and the same bracket algebra shows that non-affine transformations generically fail to commute with the discretization.

What would settle it

Simulate the free planar point mass with symplectic Euler in polar coordinates for several step sizes $h$ and measure the drift of $p_x = p_r \cos\theta - (p_\theta/r)\sin\theta$: the paper predicts an $O(h^2)$ drift whenever $(p_y x - p_x y) \neq 0$, so observing exact preservation of $p_x$ (drift below machine precision across decreasing $h$) would refute condition (59).

Watch

Extended reading notes

Core claim

The central claim is that the distorted Hamiltonian $\tilde H$ produced by backward error analysis of a symplectic method is not invariant under canonical transformations induced by point transformations of the generalized coordinates. For the symplectic Euler method, whose distorted Hamiltonian is $\tilde H = H - \frac{h}{2} H_p H_q + O(h^2)$, the paper proves that after a change of coordinates $\bar q = Q(q)$ the leading elementary Hamiltonian becomes $\bar H_{\bar p}\bar H_{\bar q} = H_p H_q + \Xi_{H_p H_q, Q}$, so $\tilde H$ differs from the pullback of $\tilde{\bar H}$ unless the transformation is affine or the extra term happens to vanish. From this identity the paper derives condition (59): $\partial \Xi_{H_p H_q, Q}/\partial q^1 = 0$ is necessary and sufficient for preservation of the first integral $p_1$ up to second order, and it demonstrates the condition by showing that a free particle simulated in polar coordinates loses linear momentum while angular momentum survives a Cartesian simulation. A separate construction exhibits an order-compensating coordinate transformation for the harmonic oscillator that removes the first-order energy error, achieving second-order convergence with symplectic Euler.

Load-bearing premise

The derivation keeps only the first-order term $H_p H_q$ of the distorted Hamiltonian and discards the $O(h^2)$ remainder; if that remainder changed the sign or size of the leading coordinate-dependent drift, condition (59) and the predicted non-preservation could fail.

Editorial extensions

If this is right

  • Order compensation is real: a well-chosen coordinate transformation can cancel the leading term of the distorted Hamiltonian, raising symplectic Euler's observed order from one to two for the harmonic oscillator.
  • For any cyclic coordinate, the corresponding momentum is conserved up to second order by symplectic Euler exactly when condition (59) holds; otherwise the numerical solution drifts at $O(h^2)$.
  • Coordinate choice also affects stability, as polar-coordinate simulations of the elastic pendulum diverge for initial conditions that eventually pass through the origin.
  • Because higher-order elementary Hamiltonians are built from derivatives of the non-invariant pieces $H_q$, the non-invariance is expected to carry over to other symplectic methods, not just symplectic Euler.

Reading between the lines

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

  • The same mechanism implies that coordinates chosen to protect one first integral can destroy another, so coordinate selection is inherently a trade-off rather than a free improvement.
  • A practical extension would preprocess the initial condition with the compensating transformation and postprocess only output times, giving higher-order accuracy at overhead comparable to the processing methods reviewed in the paper.
  • Condition (59) could serve as a symbolic diagnostic: before long simulations, compute the extra term for candidate coordinate systems and reject any for which $\partial \Xi/\partial q^1 \neq 0$ for a desired conserved momentum.
  • Since the compensating maps are only $O(h^2)$-close to the identity, their benefits may be washed out by round-off or by higher-order terms at larger step sizes; testing across $h$ would reveal the practical envelope.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 8 minor

Summary. The manuscript studies how changes of generalized coordinates affect the accuracy and invariant-preservation properties of symplectic integrators. Sections 2–3 are tutorial in character: backward error analysis, order compensation by variable transformation (with explicit Euler examples), and processing methods (Rowlands time stepping). The new results are in Sections 4–5 and Appendix B. For the symplectic Euler method, Eq. (49) computes the pullback of the transformed first-order elementary Hamiltonian, H̄_{p̄}H̄_{q̄} = H_pH_q + Ξ_{H_pH_q,Q}, showing that the distorted Hamiltonian is not invariant under non-affine point-transformation-induced canonical transformations; Eq. (59) states that ∂Ξ_{H_pH_q,Q}/∂q^1 = 0 is necessary and sufficient for the O(h²)-preservation of a cyclic-coordinate momentum p1 by the transformed symplectic Euler method. These results are illustrated with the elastic pendulum (energy-error comparison in Cartesian versus polar coordinates) and a free point mass (linear momentum not preserved in polar coordinates, angular momentum preserved in Cartesian coordinates). Appendix B constructs an order-compensating coordinate transformation for the harmonic oscillator that turns the first-order symplectic Euler energy error into a second-order one. The explicit derivations (46)–(49) and (56)–(59) are parameter-free, direct calculations and appear correct.

Significance. If the results stand, the paper gives a useful systematic account, in the language of backward error analysis, of coordinate dependence of symplectic accuracy, a phenomenon usually treated ad hoc. The explicit identity (49), the clean condition (59), and the free-particle example are genuine contributions, and the authors are careful to restrict (59) to symplectic Euler. Strengths: explicit parameter-free calculations; falsifiable predictions (e.g., the O(h²)-per-step drift of linear momentum in polar coordinates) checked by direct calculation; and honest acknowledgement of limitations, including the polar-coordinate singularity at r = 0 and the difficulty of finding optimal transformations. Two stress-test concerns do not land on close reading: the O(h²) remainder discarded in Eq. (54) brackets with p1 only to O(h²) and so cannot cancel the leading-order condition (59), and the global-diffeomorphism limitation is flagged by the authors themselves. The main weakness is that the headline non-invariance claim for 'symplectic methods' is proven only for symplectic Euler; the extension to higher-order methods is a heuristic (Major Comment 1).

major comments (2)
  1. [Section 4 (Eqs. (44)–(49)); Abstract] The non-invariance of the distorted Hamiltonian is rigorously established only for the symplectic Euler method. The paragraph after Eq. (49) asserts that the reasoning 'extends naturally to higher-order elementary Hamiltonians' because H_q is not invariant under point transformations, and the Abstract promises 'a derivation for the non-invariance of the modified Hamiltonian of symplectic methods.' That extension is a heuristic, not a proof: the leading modified-Hamiltonian term of a second-order method (for example Störmer–Verlet or the implicit midpoint method) is a different combination of elementary Hamiltonians involving H_qq, H_pp, and mixed derivatives, and nothing in the manuscript rules out cancellation of the non-invariant parts for a particular non-affine Q; no example is given for any method beyond symplectic Euler. The Conclusion is more careful ('we have shown this rigorously for the symplectic Euler method'), so the gap is fixable: either prove or numerically verify non-invariance for at least one second-order method, or restrict the Abstract, the Introduction, and the Section 4 claims to symplectic Euler and label the general statement as a conjecture. Since this is the paper's headline claim, the mismatch between what is advertised and what is proven is load-bearing.
  2. [Appendix B (Eqs. (78)–(81), Fig. 8)] The order-compensation demonstration rests on two justifications that are not supplied. First, the transformation (81) is defined only for |q̂| ≤ 1 (it contains √(1−q̂²) and arcsin(q̂)); the paper asserts without proof that the choice k=2 guarantees |q̂| ≤ 1 for the numerical solution. Second, the paper does not explain why eliminating the first-order term along the exact solution, where p²+q²=1, yields an O(h²) energy error along the numerical trajectory, where p²+q²=1+O(h); the residual H̄_{p̄}H̄_{q̄} is O(h) on that trajectory, and an argument that this still removes the O(h) energy error is needed. A short proof is available for this example: the symplectic Euler map for the harmonic oscillator is linear and exactly conserves H̃ = (p²+q²)/2 − (h/2)pq, giving |q| ≤ (1−h²/4)^{−1/2} on the invariant ellipse, and (1−h²/4)^{−1/2} ≤ 1+2h² holds for all step sizes below h ≈ 1.99. Fig. 8 also does not state the range of step sizes for which second-order convergence is claimed. Please supply these justifications or soften the claim to what is actually demonstrated.
minor comments (8)
  1. [Section 2.3.1 (Eqs. (15)–(16))] The constant α is inconsistent: with Ψ(y) = (1/α) ln y, the transformed ODE is ˙ȳ = −1, not −α as stated after Eq. (15); choosing instead Ψ(y) = ln y gives ˙ȳ = −α. The conclusion (exact integration by explicit Euler) is unaffected, and Fig. 1 uses α = 1, but the displayed formulas should be reconciled.
  2. [Section 2.4 (Eq. (19))] The multidimensional compensation condition (19) is stated with 'it can be shown' but no derivation or reference; a short derivation would help, since this is the basis for the claim that the general problem is comparable in difficulty to solving a PDE.
  3. [Section 3.2 (Eq. (24))] The exponential-form definition of effective order mixes p, m, and k without stating which symbol denotes the effective order; please harmonize with the preceding sentence.
  4. [Section 4.1] The stability boundary condition 'r(0) ≥ 2l + 2mg/k cos(φ(0))' is ambiguous (which quantity multiplies the cosine?) and it is not stated whether it was derived or determined empirically; please clarify, and explain in the caption of Fig. 5 that the region beyond the white dashed line is excluded from the comparison.
  5. [Section 4 (text) and Conclusions] There are typos: 'a rich structure emerges even for for first-order methods' (Section 4) and 'While there exits a relatively developed choice' (Conclusions).
  6. [Section 5.2.1] The claim that 'it is straightforward to show that after a time step, p^{j+1}_x = p^j_x + O(h²)' should be backed by a one-line derivation (per-step change h{H̃̄,p_x} = −(h²/2)∂Ξ/∂x + O(h³) via Eq. (59)); as written it is an unproved assertion.
  7. [Section 5.1 (Eq. (53))] The paper defines 'preserved up to order r' through the Poisson-bracket condition (53) but never states how this translates to the numerical iterates: a per-step drift of O(h^{r+1}) accumulates to O(h^r) per unit time. One sentence making this precise would make the necessary-and-sufficient claim (59) unambiguous.
  8. [Figs. 3 and 8] The captions of Figs. 3 and 8 do not give the step sizes or the exact error definition for the convergence plots, so the claimed orders (four for Rowlands, two for the compensated symplectic Euler) cannot be reproduced from the text as it stands.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the non-invariance and first-integral derivations are self-contained, with only minor non-load-bearing self-citations.

full rationale

The paper's central derivation is not circular. The non-invariance claim rests on Eq. (44), the standard backward error analysis expansion for symplectic Euler, and Eq. (49), a direct computation of how the elementary Hamiltonian transforms under a point transformation. No parameter is fitted to data to obtain the non-invariance condition or the first-integral condition (59); both follow algebraically from the backward error expansion and the Poisson bracket calculation. The order-compensation example in Appendix B solves the ODE (79) for the compensating transformation rather than fitting it, and the parameter k in (81) is an illustrative choice that does not feed back into the theoretical claims. The only self-citations, e.g., [25] for the term 'distorted' and for the explicit Euler DVF in Eq. (10), are terminological or background references and are not load-bearing; the relevant DVF is also re-derived in Eqs. (10)-(11), and the elementary Hamiltonians are cited to external standard works [58, 59]. The generalization from symplectic Euler to all symplectic methods in Section 4 is an extrapolation and could be a correctness or scope risk, but it is not a circular reduction: the paper does not assume the conclusion it is proving. No step in the derivation equates its output to its input by construction, so there is no significant circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard backward error analysis and on smoothness/invertibility assumptions for the coordinate transformations. No new physical entities are introduced. The only hand-chosen parameter is k in the illustrative order-compensation example, which is not fitted to data and does not affect the non-invariance or first-integral conditions.

free parameters (1)
  • k in Eq. (81) = 2
    Chosen by hand in the harmonic oscillator example to guarantee |q̂| ≤ 1 for the numerical solution; purely illustrative, not fitted to data and not part of the central theoretical claims.
assumptions (5)
  • standard math A symplectic one-step method has an asymptotic modified Hamiltonian H̃ = H + h H_1 + h^2 H_2 + ... whose flow interpolates the numerical solution.
    Invoked in Section 3.1 and Section 4, Eqs. (44)-(45), from backward error analysis (Hairer, Lubich, Wanner [24]; Benettin and Giorgilli [23]). This is standard.
  • standard math For the symplectic Euler method, the first-order term of the modified Hamiltonian is -h/2 H_p H_q.
    Eq. (44), Section 4. This is a standard result for symplectic Euler.
  • domain assumption The coordinate transformation Q is a smooth, one-to-one point transformation and the induced canonical transformation is non-degenerate.
    Section 2.1 and Section 4, needed for the formulas (40)-(41). Fails at r=0 for polar coordinates, acknowledged in Section 4.1 and Section 5.
  • domain assumption Truncating the modified Hamiltonian at O(h^2) is sufficient to determine preservation of first integrals up to O(h^2).
    Section 5, around Eq. (54), where the O(h^2) remainder is discarded. This is standard in backward error analysis but not proved in the paper.
  • ad hoc to paper The numerical solution remains inside the domain of invertibility of the compensating transformation (|q̂| ≤ 1) for the harmonic oscillator example.
    Eq. (81), the parameter k=2 is chosen to enforce this condition; if the solution left the domain, the transformation would be invalid.

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Pith. "Pith review of On the coordinate system-dependence of the accuracy of symplectic numerical methods." pith.science (2026). https://pith.science/paper/VPK4ZBSX

@misc{pith2026250621241,
  author       = {Pith},
  title        = {Pith review of: On the coordinate system-dependence of the accuracy of symplectic numerical methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VPK4ZBSX}},
  note         = {Machine review of arXiv:2506.21241}
}
read the original abstract

Symplectic numerical methods have become a widely-used choice for the accurate simulation of Hamiltonian systems in various fields, including celestial mechanics, molecular dynamics and robotics. Even though their characteristics are well-understood mathematically, relatively little attention has been paid in general to the practical aspect of how the choice of coordinates affects the accuracy of the numerical results, even though the consequences can be computationally significant. The present article aims to fill this gap by giving a systematic overview of how coordinate transformations can influence the results of simulations performed using symplectic methods. We give a derivation for the non-invariance of the modified Hamiltonian of symplectic methods under coordinate transformations, as well as a sufficient condition for the non-preservation of a first integral corresponding to a cyclic coordinate for the symplectic Euler method. We also consider the possibility of finding order-compensating coordinate transformations that improve the order of accuracy of a numerical method. Various numerical examples are presented throughout.

Figures

Figures reproduced from arXiv: 2506.21241 by the authors.

Figure 1
Figure 1. Accuracy of the explicit Euler method for solving (14) in the original and in the optimal [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Accuracy of the explicit Euler method for solving (17) in the original and in the optimal [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Demonstration of the fourth-order convergence of the Rowlands method versus the second [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Relationship between the original Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Comparison of error in total energy between simulations of the elastic pendu [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Illustration of the mechanical model of the planar elastic pendulum. [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Trajectory of symplectic Euler simulation of the harmonic oscillator, with and without using [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Convergence of the error in the Hamiltonian for symplectic Euler simulations of the harmonic [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.