{"id":"fe71cf38-e506-412c-85a4-d8ad124cbbca","arxiv_id":"2411.11997","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Marsden-Weinstein reduction for corank-1 presymplectic structures with a chosen Reeb field is developed and applied to reduce the evolution dynamics of time-dependent Hamiltonian systems.","lead":"Time-dependent Hamiltonian systems with natural moving-observer symmetries often fail the condition required by Albert's 1986 cosymplectic reduction. The authors introduce mechanical presymplectic structures and prove a Marsden-Weinstein reduction theorem that covers these cases.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The non-tangency hypothesis in Theorem 5.2 is not verified on the wave-example level sets: in §6.2, J_H is constant on C1 and C2 with values mc²/2 and mc²/2+e²A0²/(2m), so for those μ the reduced space is singular and the assertion that C is absent from J_H^{-1}(μ) is false.","rationale":"The reader's weakest assumption is the right one. The core Theorem 4.10 is internally consistent: under R∉T_x(G·x), Proposition 4.9 gives ker(ι_μ*ω)=T_x(G_μ·x)⊕⟨R(x)⟩, and the projection argument correctly yields ker ω_μ=⟨R_μ⟩. I found no gap in the theorem itself. The load-bearing issue is whether the non-tangency condition can actually be met in the paper's advertised examples. Section 6.2 explicitly exhibits E_H tangent to the orbits on C, removes C, and then claims C never intersects J_H^{-1}(μ). The computation above shows two critical values where that claim fails and the reduced section N is singular. This does not disprove Theorem 5.2, but it invalidates the example as stated for μ=μ1,μ2 and shows the global non-tangency assumption is not a harmless technicality: it selects regular momentum values in a way the paper does not track. The harmonic-oscillator example is less problematic, since there μ=mv²/2 gives an empty level set after removing C; the wave example carries the concern. The verdict remains CONDITIONAL: the central reduction theorem appears correct, but the application needs qualification or a singular-reduction treatment.","tokens_in":24620,"tokens_out":11872,"duration_ms":115358,"concrete_test":"Check μ=μ1=mc²/2 in §6.2: write F=(p1−mc)²+(p2−eA0 cos(q1−ct))²+p3²−e²A0² cos²(q1−ct), so J_H^{-1}(μ1)={F=0}⊂R^7. Evaluate dF at a point of C1, for example q1=ct+π/2, p1=mc, p2=0, p3=t=0. If dF=0, the level set is not a smooth submanifold and Theorem 5.2 does not apply at that μ; if dF≠0, the objection is refuted. Repeat the same gradient check at a point of C2 for μ2=mc²/2+e²A0²/(2m).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reduction theorem is correct conditional on Definition 4.3(ii), that R (or E_H in Theorem 5.2) is nowhere tangent to the G-orbits; without it, the projected Reeb field can vanish and ker ω_μ need not be the line spanned by R_μ. The vulnerable point is the application in Section 6.2. After defining C = C1 ∪ C2 where E_H(q,p,t) ∈ ⟨1_M(q,p,t)⟩, the authors remove C and then state that 'none of the points in C belong to J_H^{-1}(μ)', so the removal can be ignored. This is false. For the wave Hamiltonian, J_H = c p1 − (1/(2m))(p1²+p2²+p3²) + (eA0/m) p2 cos(q1−ct), with 1_M = c ∂_{q1} + ∂_t. On C1 = {q1=ct+(2n−1)π/2, p1=mc, p2=0, p3=0}, cos(q1−ct)=0, hence J_H = mc²/2 =: μ1. On C2 = {q1=ct+nπ, p1=mc, p2=(−1)^n eA0, p3=0}, cos(q1−ct)=(−1)^n, and J_H = mc²/2 + e²A0²/(2m) =: μ2. Thus C1 ⊂ J_H^{-1}(μ1) and C2 ⊂ J_H^{-1}(μ2). At these points E_H is tangent to the orbit, so Theorem 5.2's hypothesis fails on the level set. Moreover the gradient of the defining equation for J_H^{-1}(μ) vanishes there, so the level set is not a smooth submanifold; the section N = J_H^{-1}(μ) ∩ {q1=0} is likewise singular. The recipe must exclude μ1 and μ2, or invoke singular reduction, before claiming a mechanical presymplectic reduced manifold for all μ.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper argues that Albert's Marsden-Weinstein reduction for cosymplectic structures is ill-suited to symmetric time-dependent Hamiltonian systems, because the condition η(ξ_M)=0 often fails and Noether-type first integrals in the cosymplectic framework are necessarily time-independent. To overcome this, the authors introduce mechanical presymplectic structures, i.e. odd-dimensional manifolds with a closed corank-one 2-form ω and a Reeb vector field R spanning ker ω. They prove a reduction theorem for Hamiltonian presymplectic actions under a non-tangency hypothesis on R (Theorem 4.10), then apply it to evolution dynamics by passing to the modified form ω_H=ω+dH∧η and modified momentum map J_H=J-Hc_η (Theorem 5.2). Two examples are treated: the N-dimensional harmonic oscillator seen by a moving observer, and an electron perturbed by a linearly polarized plane wave. The central reduction theorem is proved cleanly and the algebraic Lemma 4.8 is deferred to an appendix, but the plane-wave example contains a load-bearing gap concerning the non-tangency condition on the level sets of J_H.","tokens_in":25097,"tokens_out":3513,"duration_ms":36275,"significance":"If the main theorem and its applications are correct, the paper makes a useful contribution: it identifies a genuine limitation of Albert's cosymplectic reduction, proposes a workable generalization that covers systems outside the cosymplectic framework, and includes a transparent comparison with the presymplectic reduction of Echeverría-Enríquez et al. in Remark 4.11. The proofs are largely self-contained, the algebraic lemma is proved in the appendix, and the worked examples are computed in detail. The main reduction theorem is internally consistent and the comparison with [14] is explicit rather than circular. The significance is somewhat tempered by the fact that the non-tangency hypothesis in Definition 4.3(ii) is essential and is not verified in one of the two main examples for all momentum values.","major_comments":[{"comment":"The assertion that 'none of the points in C belong to J_H^{-1}(μ)' is false for two momentum values, and this invalidates the application of Theorem 5.2 on the corresponding level sets. For the wave Hamiltonian, J_H = c p1 − (1/(2m))(p1²+p2²+p3²) + (eA0/m) p2 cos(q1−ct). On C1 = {q1=ct+(2n−1)π/2, p1=mc, p2=0, p3=0}, cos(q1−ct)=0, so J_H = mc²/2 =: μ1. On C2 = {q1=ct+nπ, p1=mc, p2=(−1)^n eA0, p3=0}, cos(q1−ct)=(−1)^n, so J_H = mc²/2 + e²A0²/(2m) =: μ2. Thus C1⊂J_H^{-1}(μ1) and C2⊂J_H^{-1}(μ2). At these points E_H is tangent to the G-orbits, so the hypothesis E_H(x)∉T_x(G·x) of Theorem 5.2 fails on the level set; moreover the gradient of the defining equation for J_H^{-1}(μ) vanishes there, so the level set is not a smooth submanifold and the slice N=J_H^{-1}(μ)∩{q1=0} is singular. The authors must either exclude the exceptional values μ1 and μ2, or invoke singular reduction, before claiming that (N,(ωH)_μ,(E_H)_μ) is a reduced mechanical presymplectic manifold for all μ.","section":"§6.2, after Eq. (46)"},{"comment":"The non-tangency condition R(x)∉T_x(G·x) is load-bearing for the reduction theorem: without it, the projected Reeb field can vanish and ker ω_μ need not be the line spanned by R_μ, as the proof of Theorem 4.10 itself shows. In the application of Theorem 5.2 to the plane-wave example, the authors remove the set C where E_H is tangent to the orbits, but they do not check that J_H^{-1}(μ) avoids C. As shown above, for μ=μ1,μ2 this fails. The paper should state explicitly which values of μ are admissible, or should prove that the non-tangency condition holds on the entire level set before rewriting J_H^{-1}(μ) as a subset of R^7 and defining the reduced manifold N. This is not merely a regularity technicality; it is exactly the hypothesis that makes the reduced Reeb field well defined and of rank one.","section":"§5, Theorem 5.2 and §4, Definition 4.3(ii)"}],"minor_comments":[{"comment":"There are several typos: 'Motived' should be 'Motivated', 'monocromatic' should be 'monochromatic', and 'dinamics' in the diagram before Remark 5.3 should be 'dynamics'.","section":"Abstract and title of §6.2"},{"comment":"The inclusion map is written ι_μ : J^{-1}(μ) → T^*M; it should be ι_μ : J^{-1}(μ) → M. This is a typo that could confuse readers comparing (30) with the surrounding text.","section":"Theorem 4.10(i)"},{"comment":"The text refers to 'the second part of Theorem 4.1' but the intended reference is Theorem 4.10, since that is the theorem whose second part concerns the reduced Reeb vector field and ker ω_μ.","section":"Remark 4.11"},{"comment":"The expression 'dim A⊥ = 1 − dim(Ao + ♭(V )) + dim Ao + dim ♭(V )' is correct only after taking dim(ker ♭|_A⊥)=1; the line is easy to misread because '1' is the dimension of the kernel and not a typo. A short parenthetical clarification would improve readability.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The flaw in §6.2 is localized and fixable by restricting the allowed momentum values or by using singular reduction; I do not regard it as grounds for rejection. The authors should, however, be asked to state explicitly that the plane-wave example is valid only for μ not equal to mc²/2 or mc²/2+e²A0²/(2m), or to handle those cases separately. The main reduction theorem itself appears sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main reduction theorem (Theorem 4.10) is solid, and the application to evolution dynamics via the modified momentum map J_H = J - c_eta H is genuinely new. But the paper's showcase example, Section 6.2, contains a concrete error: the claim that no point of C lies in J_H^{-1}(mu) is false for mu = mc^2/2 and mu = mc^2/2 + e^2 A_0^2/(2m). Those exceptional values make the level set singular and the reduced space ill-defined. This does not sink the main theorem, but the example as stated overclaims.\n\nWhat the paper does well: it identifies real limitations of Albert's cosymplectic reduction for time-dependent systems -- the condition eta(xi_M)=0 is often violated, and Noether first integrals end up time-independent, which is physically unsatisfying. The replacement of cosymplectic structures by mechanical presymplectic structures (closed corank-1 2-form plus a vector field spanning its kernel) is a modest but useful move. The reduction theorem is proved cleanly, with the algebraic lemma in the appendix; I spot-checked Lemma 4.8 and Proposition 4.9 and the logic holds. The authors are also honest about the relationship to the earlier presymplectic reduction of Echeverria-Enriquez et al.: the form-reduction part of Theorem 4.10 is a special case of their Theorem 2, and this is said explicitly. The genuinely new content is the reduction of the evolution dynamics when the Reeb field is not tangent to the group orbits.\n\nSoft spots, in proportion:\n\n1. The Section 6.2 error is not a typo. On C1 the Hamiltonian satisfies J_H = mc^2/2, and on C2 it satisfies J_H = mc^2/2 + e^2A_0^2/(2m). The gradient of the defining equation vanishes at those points, so J_H^{-1}(mu) is not a smooth submanifold for those mu, and the slice N = J_H^{-1}(mu) cap {q1=0} is singular as well. The recipe must exclude those mu values or invoke singular reduction. This is fixable, but the current text says the opposite of what is true.\n\n2. The non-tangency condition, Definition 4.3(ii) and its dynamical analogue in Theorem 5.2, is load-bearing. When E_H becomes tangent to the orbit, the projected Reeb field can vanish and ker omega_mu is no longer a line spanned by R_mu. The paper states the condition but does not discuss what happens when it fails; the examples show that failure is not pathological. A clearer treatment of the exceptional set would strengthen the paper.\n\n3. Minor: the name \"mechanical presymplectic structure\" rebrands a corank-1 presymplectic form with a chosen Reeb field, but the authors are upfront about the comparison with [14], so this is not misleading.\n\nThe main theorem holds up; the example needs revision. The paper deserves a serious referee, not a desk reject.\n\nRecommendation: send to peer review, with the expectation that the authors fix the level-set issue in Section 6.2 and clarify the consequences of the non-tangency hypothesis.","headline":"Sound reduction theorem for corank-1 presymplectic structures with a genuinely new dynamic-reduction application, but the plane-wave example contains a real level-set error that needs fixing.","tokens_in":25634,"tokens_out":2124,"would_cite":true,"duration_ms":21547,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D20","70G45","70G65","70H05","70H33"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a Marsden-Weinstein reduction for a new class of odd-dimensional structures, mechanical presymplectic manifolds, and uses it to reduce time-dependent Hamiltonian dynamics when cosymplectic reduction fails.","keywords":["cosymplectic structures","time-dependent Hamiltonian systems","Marsden-Weinstein reduction","Reeb dynamics","presymplectic structures","mechanical presymplectic structures","evolution vector field","momentum map"],"falsifier":"Compute $\\ker(\\omega_H)_\\mu$ on the reduced manifold $N$ for a momentum value $\\mu$ whose level set $J_H^{-1}(\\mu)$ intersects the tangency set $C=\\{x \\mid E_H(x)\\in T_x(G\\cdot x)\\}$: if the kernel is not the line spanned by the projected evolution field, or if the quotient is not a smooth manifold, the theorem fails exactly at those exceptional values, which the worked examples omit.","tokens_in":24432,"feed_emoji":"⚙️","tokens_out":8208,"duration_ms":72446,"temperature":0.7,"pith_summary":"The paper establishes a Marsden-Weinstein-type reduction for a class of odd-dimensional geometries it calls mechanical presymplectic structures: a closed two-form of corank one together with the vector field spanning its kernel. On this footing the authors prove that, under a Hamiltonian group action with equivariant momentum map, an infinitesimally free action, and a smooth quotient, the level set of the momentum map carries a unique reduced closed two-form whose kernel is spanned by the projected Reeb vector field. They then apply this to time-dependent Hamiltonian systems by twisting the cosymplectic form, $\\omega_H = \\omega + dH \\wedge \\eta$, so that the evolution vector field becomes the Reeb field, with momentum map $J_H = J - c_\\eta H$. This yields a reduction of evolution dynamics in examples where an older cosymplectic reduction theorem cannot be applied, such as a harmonic oscillator described from a uniformly moving observer and an electron perturbed by a monochromatic plane wave. The point of the construction is that conserved quantities used in the reduction need only be first integrals of the evolution vector field, not of the separate Reeb or Hamiltonian fields.","feed_headline":"New reduction theorem covers time-dependent Hamiltonian symmetries","feed_subtitle":"Mechanical presymplectic structures reduce evolution dynamics where older cosymplectic methods fail.","key_machinery":"The central object is the mechanical presymplectic structure $(M,\\omega,R)$: a closed two-form of corank one with $\\ker\\omega = \\langle R\\rangle$. The carrying identity is the deformation $\\omega_H = \\omega + dH\\wedge\\eta$, which turns the evolution vector field $E_H = X_H + R$ into the Reeb vector field of $(\\omega_H,\\eta)$, together with the modified momentum map $J_H = J - c_\\eta H$, which makes the same group action Hamiltonian for $\\omega_H$. The algebraic engine is Lemma 4.8: for a tangent subspace $A$, the double orthogonal $(A^\\perp)^\\perp$ equals $A\\oplus\\langle R\\rangle$ when $R\\notin A$, which is what forces the kernel of the reduced form to be exactly the line spanned by the projected Reeb field.","core_discovery":"On a $(2n+1)$-dimensional manifold $M$, a mechanical presymplectic structure is a closed 2-form $\\omega$ whose kernel is the one-dimensional distribution generated by a vector field $R$. The paper's Theorem 4.10 says: if a Lie group $G$ acts preserving $\\omega$ and $R$, the action is Hamiltonian with a $G$-equivariant momentum map $J$, the action is infinitesimally free, and $J^{-1}(\\mu)/G_\\mu$ is a manifold with submersion projection, then there is a unique closed 2-form $\\omega_\\mu$ on the quotient satisfying $\\pi_\\mu^*\\omega_\\mu = \\iota_\\mu^*\\omega$, and $R$ projects to a vector field $R_\\mu$ with $\\ker\\omega_\\mu = \\langle R_\\mu\\rangle$. Thus the quotient is again a mechanical presymplectic manifold. Theorem 5.2 transfers this to evolution dynamics: for a $G$-invariant Hamiltonian $H$ on a cosymplectic manifold, replacing $\\omega$ by $\\omega_H = \\omega + dH\\wedge\\eta$ and $J$ by $J_H = J - c_\\eta H$ makes the evolution vector field $E_H$ a Reeb field, and the same reduction produces a mechanical presymplectic structure on $J_H^{-1}(\\mu)/G_\\mu$ whose Reeb field is the projected evolution dynamics.","pith_inferences":["A natural testable extension is to include the exceptional momentum values $\\mu$ for which $J_H^{-1}(\\mu)$ meets the tangency set $C$; a singular or stratified reduction would be needed there, and the examples suggest the reduced space loses smoothness exactly at those values.","The same deformation $\\omega_H = \\omega + dH\\wedge\\eta$ could be used to reduce any evolution vector field that becomes a Reeb field after twisting, so the method likely applies beyond cosymplectic systems to other corank-one presymplectic geometries with a parallelizable characteristic foliation.","Because the modified momentum map $J_H$ encodes the observer, the framework offers a geometric way to pass between inertial observers: changing the observer changes $c_\\eta$ and $J_H$, and the reduced manifold changes accordingly, which could provide a systematic reduction for frame changes in classical mechanics."],"forward_implications":["For a $G$-invariant time-dependent Hamiltonian, the reduced dynamics on $J_H^{-1}(\\mu)/G_\\mu$ is governed by the projected evolution vector field, which is the Reeb field of the reduced mechanical presymplectic structure.","Conserved quantities used in the reduction may depend explicitly on time: they are first integrals of $E_H$ without being first integrals of the Reeb field or the Hamiltonian field separately.","Systems where the older cosymplectic reduction fails, because the action does not satisfy $\\eta(\\xi_M)=0$ or because the natural conserved quantities depend on time, become reducible by this method.","In the two worked examples the reduction produces an explicit reduced manifold $N$ with a reduced two-form and reduced evolution field: an ellipsoid in the moving-oscillator case and a level-set manifold in the plane-wave case.","The price is the non-tangency condition: in the examples the set $C$ where the evolution field is tangent to the orbits must be removed before reduction."],"supporting_citations":[{"why":"Supplies the cosymplectic Marsden-Weinstein reduction theorem whose hypotheses and limitations motivate the paper's new construction.","marker":"[3]"},{"why":"Gives the earlier reduction theorem for presymplectic manifolds with symmetry that Theorem 4.10 is compared against, especially its opposite kernel assumption.","marker":"[14]"},{"why":"Records the fact that every cosymplectic evolution dynamics can be re-expressed as Reeb dynamics, which the paper uses before its reduction step.","marker":"[17]"},{"why":"Develops the extended-formalism reduction of symplectic principal $\\mathbb{R}$-bundles, the alternative route to time-dependent reduction that the paper contrasts with its direct presymplectic method.","marker":"[18]"},{"why":"Provides the original symplectic Marsden-Weinstein reduction that the mechanical presymplectic theorem generalizes.","marker":"[22]"}],"fun_headline_variants":["Mechanical presymplectic reduction covers time-dependent Hamiltonians","New reduction theorem for mechanical presymplectic structures","Presymplectic reduction fixes time-dependent symmetry cases","Beyond cosymplectic: reduction for time-dependent dynamics","Generalized reduction for time-dependent Hamiltonian systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction relies on the Reeb vector field (or the evolution vector field in the dynamical reduction) never being tangent to the symmetry group orbits; when that fails, the projected field can vanish and the kernel of the reduced form is no longer one-dimensional, so the reduction argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Mechanical presymplectic reduction covers time-dependent Hamiltonians","New reduction theorem for mechanical presymplectic structures","Presymplectic reduction fixes time-dependent symmetry cases","Beyond cosymplectic: reduction for time-dependent dynamics","Generalized reduction for time-dependent Hamiltonian systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000264,"raw_usage":{"total_tokens":1602,"prompt_tokens":944,"completion_tokens":658,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":560,"completion_tokens_details":{"reasoning_tokens":591}},"tokens_in":560,"tokens_out":658,"duration_ms":6368,"temperature":1.0,"reasoning_tokens":591,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:06:09.738991+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\ker(\\omega_H)_\\mu$ on the reduced manifold $N$ for a momentum value $\\mu$ whose level set $J_H^{-1}(\\mu)$ intersects the tangency set $C=\\{x \\mid E_H(x)\\in T_x(G\\cdot x)\\}$: if the kernel is not the line spanned by the projected evolution field, or if the quotient is not a smooth manifold, the theorem fails exactly at those exceptional values, which the worked examples omit.","supporting_citations":[{"cited_title":"Albert, Le th´ eor` eme de r´ eduction de Marsden-Weinstein en g´ eom´ etrie cosymplectique et de contact,Journal of Geometry and Physics , 6 (1989), no","cited_arxiv_id":null,"evidence_quote":"Supplies the cosymplectic Marsden-Weinstein reduction theorem whose hypotheses and limitations motivate the paper's new construction."},{"cited_title":"Echeverr ´ ıa Enr ´ ıquez, M.C","cited_arxiv_id":null,"evidence_quote":"Gives the earlier reduction theorem for presymplectic manifolds with symmetry that Theorem 4.10 is compared against, especially its opposite kernel assumption."},{"cited_title":"Jovanovi´ c, and K","cited_arxiv_id":null,"evidence_quote":"Records the fact that every cosymplectic evolution dynamics can be re-expressed as Reeb dynamics, which the paper uses before its reduction step."},{"cited_title":"Lacirasella, J","cited_arxiv_id":null,"evidence_quote":"Develops the extended-formalism reduction of symplectic principal $\\mathbb{R}$-bundles, the alternative route to time-dependent reduction that the paper contrasts with its direct presymplectic method."},{"cited_title":"Marsden, A","cited_arxiv_id":null,"evidence_quote":"Provides the original symplectic Marsden-Weinstein reduction that the mechanical presymplectic theorem generalizes."}],"review_version":1}