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Mechanical presymplectic structures and Marsden-Weinstein reduction of time-dependent Hamiltonian systems

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.11997 v2 pith:XOZRF5PH submitted 2024-11-18 math.DG math-phmath.MP

I. Gutierrez-Sagredo , D. Iglesias Ponte , J. C. Marrero , E. Padrón This is my paper · ORCID
classification math.DGmath-phmath.MP MSC 53D2070G4570G6570H0570H33
keywords cosymplecticstructurestime-dependentHamiltoniansystemsMarsden-WeinsteinreductionReebdynamicspresymplecticmechanicalevolutionvectorfieldmomentummap
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 / 4 minor

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.

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 (2)
  1. [§6.2, after Eq. (46)] 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.
  2. [§5, Theorem 5.2 and §4, Definition 4.3(ii)] 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.
minor comments (4)
  1. [Abstract and title of §6.2] There are several typos: 'Motived' should be 'Motivated', 'monocromatic' should be 'monochromatic', and 'dinamics' in the diagram before Remark 5.3 should be 'dynamics'.
  2. [Theorem 4.10(i)] 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.
  3. [Remark 4.11] 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.
  4. [Appendix A] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the reduction theorem is proved from definitions and linear algebra, with only a non-load-bearing self-citation.

full rationale

The central derivation is self-contained. Theorem 4.10 is proved directly from Definition 4.1, Lemma 4.8, and Proposition 4.9: the non-tangency condition (Definition 4.3(ii)) is a hypothesis, not an output, and the conclusion that the projected Reeb field R_mu spans ker(omega_mu) is obtained by linear algebra from that hypothesis. No fitted parameter is renamed as a prediction; the reduced 2-form is characterized by pi_mu^* omega_mu = iota_mu^* omega and the momentum maps satisfy i_{xi_M} omega_H = d(J_H)_xi. The relation to [14] is explicit in Remark 4.11: the authors concede that part (i) is a consequence of [14] for corank-1 structures and include a proof for self-containedness, while part (ii) differs because [14] assumes the opposite kernel condition. The only self-citation, [18], is background on the extended symplectic formalism and is not used in the proofs. A correctness caveat exists in Section 6.2: the statement that no points of C belong to J_H^{-1}(mu) is vacuous because C was removed from the domain, and the subsequent unrestricted equality for J_H^{-1}(mu) fails for mu = mc^2/2 and mu = mc^2/2 + e^2 A_0^2/(2m), where C intersects the would-be level set. This is a singularity/hypothesis-checking problem for those exceptional values, not a circularity in the derivation chain.

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

The central theorem is built on linear algebra (Lemma 4.8) and manifold arguments; the main assumptions are the standard topological hypotheses of Marsden-Weinstein reduction plus the non-tangency of the Reeb field to group orbits. No free parameters are fitted to data, and the only new entity is the named mathematical structure itself.

assumptions (5)
  • domain assumption The quotient J^{-1}(μ)/G_μ is a smooth manifold and the canonical projection is a submersion
    Assumed in Theorems 4.10 and 5.2 rather than derived; standard in Marsden-Weinstein reduction but must hold for the examples.
  • domain assumption The action is infinitesimally free and the Reeb vector field is nowhere tangent to the group orbits (Definition 4.3(ii))
    Used in Proposition 4.7 and Theorem 4.10 to guarantee J is a submersion and the reduced Reeb field spans the kernel; in the examples this forces removal of the set C.
  • domain assumption Completeness of the Reeb flow and exactness of the left-invariant 1-form associated to the cocycle c_η
    Assumed in Proposition 3.7 to construct the modified cosymplectic action; the paper notes this limits the practical usefulness of that approach.
  • standard math Standard background in differential geometry, Lie group actions, de Rham cohomology, and stable Hamiltonian structures
    Invoked throughout, including the Betti-number argument showing S^{2n+1} admits no cosymplectic structure.
  • domain assumption The plane-wave perturbation is small (A0 << 1) so the quadratic term H2 is neglected
    Section 6.2, around equation (43); the reduction is applied to the approximate Hamiltonian, not to the full nonlinear system.
invented entities (1)
  • mechanical presymplectic structure
    purpose: A corank-1 closed 2-form with a chosen kernel-generating Reeb vector field, used to reduce time-dependent Hamiltonian dynamics without fixing a 1-form η
    This is a definitional term for a known type of mathematical object rather than an empirically testable entity; it has no falsifiable handle outside the paper.

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Pith. "Pith review of Mechanical presymplectic structures and Marsden-Weinstein reduction of time-dependent Hamiltonian systems." pith.science (2026). https://pith.science/paper/XOZRF5PH

@misc{pith2026241111997,
  author       = {Pith},
  title        = {Pith review of: Mechanical presymplectic structures and Marsden-Weinstein reduction of time-dependent Hamiltonian systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XOZRF5PH}},
  note         = {Machine review of arXiv:2411.11997}
}
read the original abstract

In 1986, Albert proposed a Marsden-Weinstein reduction process for cosymplectic structures. In this paper, we present the limitations of this theory in the application of the reduction of symmetric time-dependent Hamiltonian systems. As a consequence, we conclude that cosymplectic geometry is not appropriate for this reduction. Motived for this fact, we replace cosymplectic structures by more general structures: mechanical presymplectic structures. Then, we develop Marsden-Weinstein reduction for this kind of structures and we apply this theory to interesting examples of time-dependent Hamiltonian systems for which Albert's reduction method doesn't work.

Figures

Figures reproduced from arXiv: 2411.11997 by the authors.

Figure 1
Figure 1. Cut with p3 = 0, q2 = 0, q3 = 0 of the level sets J −1 H (0) for eA0 = 1 (left) and eA0 = 0.1 (right). We have used unit such that m = c = 1. Now, we consider the 5-dimensional closed submanifold (47) N = J −1 H (µ) ∩ {q 1 = 0} ⊂ R 7 given by (48) N = ( (q i , pi , t) ∈ R 7 | 2mcp1 − X 3 i=1 p 2 i + 2eA0p2 cos(ct) = 2mµ) . Deriving the previous constraint, we deduce that the points of this submanifold satisfy (49) i… view at source ↗

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Cited by 1 Pith paper

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