{"id":"c0e37956-e50e-49dc-b5de-9578d4fec71d","arxiv_id":"2506.21241","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Symplectic Euler's modified Hamiltonian is not invariant under non-affine coordinate changes, and a new condition predicts when a cyclic-coordinate first integral is lost.","lead":"Symplectic numerical methods are popular for long-term simulation of Hamiltonian systems because they keep energy stable, but their accuracy turns out to depend on the coordinates the equations are written in. The paper proves this for symplectic Euler, derives a test for when a conserved momentum survives a coordinate change, and shows coordinate choices can raise a method's effective order.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"General non-invariance claim for all symplectic methods is extrapolated from symplectic Euler; higher-order leading corrections may behave differently.","rationale":"The reader's weakest_assumption concerned the O(h^2) truncation in the backward error analysis and the global diffeomorphism assumption. Those are real but secondary: the leading-order non-invariance result for symplectic Euler does not depend on the discarded remainder in any way that would overturn it, and the local nature of the computation is acknowledged in the polar-coordinate example. The more load-bearing issue is the gap between the rigorously proven symplectic Euler statement and the general claim about symplectic methods in the abstract and conclusions. The paper's own text in Section 4 concedes that a deeper investigation of higher-order methods is beyond its scope, yet the abstract promises a derivation for 'symplectic methods' without qualification. My concrete check would test the generalization on Störmer–Verlet, a canonical second-order symplectic method, by comparing its modified Hamiltonian across a non-affine point transformation. A nonzero difference would support the generalization; a zero difference would force a restriction of the claim. This is consistent with the reader's CONDITIONAL verdict, though for a different reason than the stated weakest assumption, hence 'partial' agreement.","tokens_in":18905,"tokens_out":17669,"duration_ms":202615,"concrete_test":"Derive the O(h^2) term of the modified Hamiltonian for the Störmer–Verlet method for a Hamiltonian under a non-affine point transformation, e.g. the planar elastic pendulum in Cartesian vs polar coordinates, using the standard SV modified-Hamiltonian formula (Hairer–Lubich–Wanner, Geometric Numerical Integration, Ch. IX). Symbolically compute \\tilde H_SV(q,p) and compare it with the pullback of \\tilde{\\bar H}_SV(\\bar q,\\bar p) via Q. If the difference is nonzero for a generic non-affine Q, the generalization is supported; if it vanishes in all tested cases, the abstract claim must be restricted to symplectic Euler.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central advertised result, that the distorted Hamiltonian of a symplectic method is coordinate-dependent, is rigorously established only for the symplectic Euler method (Eqs. (42)-(44), (49)). The extension to general symplectic methods in Section 4 is a plausibility argument: it observes that H_q is not invariant under point transformations and infers that all higher-order elementary Hamiltonians, and hence all symplectic methods, inherit the non-invariance. This is not a proof. For a second-order method such as Störmer–Verlet, the leading correction to the modified Hamiltonian is a different combination of elementary Hamiltonians (involving H_{qq}, H_{pp}, etc.), and nothing rules out that this combination is invariant for some non-affine Q even though H_q alone is not. The abstract's phrase 'symplectic methods' thus overreaches the theorems actually proven. The first-integral condition (59) is also derived only for symplectic Euler, and the paper correctly restricts it, so the overclaim is limited to the modified-Hamiltonian non-invariance. If the general claim is intended, it needs either a proof via the BCH composition of the method's elementary Hamiltonians or an explicit restriction to symplectic Euler.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19069,"tokens_out":51859,"duration_ms":525842,"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":[{"comment":"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.","section":"Section 4 (Eqs. (44)–(49)); Abstract"},{"comment":"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.","section":"Appendix B (Eqs. (78)–(81), Fig. 8)"}],"minor_comments":[{"comment":"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.","section":"Section 2.3.1 (Eqs. (15)–(16))"},{"comment":"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.","section":"Section 2.4 (Eq. (19))"},{"comment":"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.","section":"Section 3.2 (Eq. (24))"},{"comment":"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.","section":"Section 4.1"},{"comment":"There are typos: 'a rich structure emerges even for for first-order methods' (Section 4) and 'While there exits a relatively developed choice' (Conclusions).","section":"Section 4 (text) and Conclusions"},{"comment":"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.","section":"Section 5.2.1"},{"comment":"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.","section":"Section 5.1 (Eq. (53))"},{"comment":"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.","section":"Figs. 3 and 8"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The core calculations appear sound, and I believe the paper is publishable in a numerical-analysis journal after revision. The main points are: (i) the title and Abstract promise non-invariance for 'symplectic methods' while the theorems are for symplectic Euler only; this can be fixed by restriction or by adding a higher-order example; (ii) the tutorial material of Sections 2–3 is competent but long relative to the new results, and the editor may ask the authors to tighten it; (iii) the Appendix B example needs the missing justification described in Major Comment 2, including the step-size range for Fig. 8. I found no citation or novelty concerns beyond the overstatement noted; the processing-methods literature is credited appropriately."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real contribution here is the explicit algebra in Section 4: equation (49) shows in closed form how the first-order elementary Hamiltonian H_p H_q changes under a point transformation for symplectic Euler, and equation (59) turns that into a necessary and sufficient condition for the cyclic-coordinate momentum to survive to second order. I checked the derivation and it is direct, correct, and parameter-free. That is genuinely new, as far as the cited literature goes, and it is useful for practitioners who want to know whether their favorite coordinates are costing them a first integral.\n\nThe paper does other things well too. The order-compensation example for the harmonic oscillator (Appendix B) is self-consistent and honestly flagged as relying on knowledge of the solution domain. The review of processing methods is competent. The numerical demonstrations, including the elastic pendulum with Cartesian versus polar coordinates, are clear and support the main point.\n\nThe soft spot is the gap between what is proven and what the abstract promises. The abstract says \"non-invariance of the modified Hamiltonian of symplectic methods,\" but the rigorous derivation only covers symplectic Euler. The extension to general symplectic methods in Section 4 is a plausibility argument: H_q is not invariant, so higher-order elementary Hamiltonians, which involve derivatives of H_q, are probably not invariant either. That is reasonable but not a proof. For Störmer–Verlet, for example, the leading correction is a different combination of elementary Hamiltonians, and nothing rules out an accidental cancellation for some non-affine coordinate change. The condition (59) is also derived only for symplectic Euler, but the paper is careful to restrict it there.\n\nTwo smaller issues: the paper discards O(h^2) remainders in the first-integral argument, which is fine for a leading-order statement but should be stated more precisely; and there are enough typos in long formulas that a careful reader will have to reconstruct some equations by hand. No code is provided, but the computations are simple enough to reproduce.\n\nOverall, this is a solid paper with one clearly identified, load-bearing but fixable overclaim. My recommendation is to send it to peer review. A good referee should ask the authors to either prove the general non-invariance for all symplectic methods or explicitly restrict the abstract and conclusions to symplectic Euler and closely related first-order methods. The core result deserves to be in the literature.","headline":"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.","tokens_in":19664,"tokens_out":2062,"would_cite":true,"duration_ms":27108,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65P10","37M15","70H15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The coordinate system changes the distorted Hamiltonian that a symplectic integrator actually samples.","keywords":["symplectic integrators","modified Hamiltonian","distorted Hamiltonian","coordinate transformations","first integrals","cyclic coordinates","order compensation","backward error analysis"],"falsifier":"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).","tokens_in":18608,"feed_emoji":"📐","tokens_out":6577,"duration_ms":63003,"temperature":0.7,"pith_summary":"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.","feed_headline":"A symplectic integrator's accuracy depends on your coordinates","feed_subtitle":"The distorted Hamiltonian a symplectic method samples changes with coordinates; a careful choice can raise the observed order.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes that near-to-identity symplectic maps have a Hamiltonian interpolation, the distorted Hamiltonian whose non-invariance is the paper's subject.","marker":"[23]"},{"why":"Supplies the standard backward error analysis of symplectic methods, the symplectic Euler expansion (44), and the P-series result that quadratic first integrals are preserved.","marker":"[24]"},{"why":"Defines point transformations, induced canonical phase-space transformations, cyclic coordinates, and first integrals used throughout the derivation.","marker":"[39]"},{"why":"Shows the distorted Hamiltonian is an $h$-dependent linear combination of elementary Hamiltonians built from derivatives of $H$, which the non-invariance argument extends to higher orders.","marker":"[58]"},{"why":"Provides the Runge-Kutta theorem that symplectic methods preserve quadratic first integrals, used to explain exact angular-momentum preservation in Cartesian coordinates.","marker":"[60]"},{"why":"The converse half of the quadratic-invariant theorem for Runge-Kutta schemes, completing the classification the paper invokes for symplectic P-series.","marker":"[61]"}],"fun_headline_variants":["Symplectic accuracy shifts with coordinate choice","Coordinates can make or break a symplectic method","The distorted Hamiltonian is coordinate-dependent","A coordinate change upgrades symplectic Euler's order","Non-invariance: symplectic methods bend to coordinates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Symplectic accuracy shifts with coordinate choice","Coordinates can make or break a symplectic method","The distorted Hamiltonian is coordinate-dependent","A coordinate change upgrades symplectic Euler's order","Non-invariance: symplectic methods bend to coordinates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000839,"raw_usage":{"total_tokens":3667,"prompt_tokens":962,"completion_tokens":2705,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":2633}},"tokens_in":578,"tokens_out":2705,"duration_ms":21491,"temperature":1.0,"reasoning_tokens":2633,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:30:17.275249+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":"OntheHamiltonianinterpolationofnear-to-theidentitysymplectic mappingswithapplicationtosymplecticintegrationalgorithms","cited_arxiv_id":null,"evidence_quote":"Establishes that near-to-identity symplectic maps have a Hamiltonian interpolation, the distorted Hamiltonian whose non-invariance is the paper's subject."},{"cited_title":"Hairer, C","cited_arxiv_id":null,"evidence_quote":"Supplies the standard backward error analysis of symplectic methods, the symplectic Euler expansion (44), and the P-series result that quadratic first integrals are preserved."},{"cited_title":"Goldstein, C","cited_arxiv_id":null,"evidence_quote":"Defines point transformations, induced canonical phase-space transformations, cyclic coordinates, and first integrals used throughout the derivation."},{"cited_title":"Order Conditions for Canonical Runge–Kutta Schemes","cited_arxiv_id":null,"evidence_quote":"Shows the distorted Hamiltonian is an $h$-dependent linear combination of elementary Hamiltonians built from derivatives of $H$, which the non-invariance argument extends to higher orders."},{"cited_title":"Hairer, S","cited_arxiv_id":null,"evidence_quote":"Provides the Runge-Kutta theorem that symplectic methods preserve quadratic first integrals, used to explain exact angular-momentum preservation in Cartesian coordinates."}],"review_version":1}