A relation between k-symplectic and k-contact Hamiltonian systems

S. Vilari\~no

REVIEW 2 major objections 5 minor 5 references

A relation between k-symplectic and k-contact Hamiltonian systems

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Contactification lifts k-symplectic Hamiltonian systems to k-contact systems, and projectable solutions of the latter project back to solutions of the former.

desk verdict The projection result is new and correct under the standard summed reading of Eq. (5), but the paper as printed is internally inconsistent and needs a careful revision. read the letter →

arxiv 2506.11873 v1 pith:T24QG7GW submitted 2025-06-13 math-ph math.MP

S. Vilariño This is my paper · ORCID

classification math-phmath.MP MSC 53D1053D0570S05

keywords k-symplecticmanifoldsk-contactHamiltoniansystemsHamilton–DeDonder–Weylequationscontactificationfieldtheoryvibratingstringwaveequation

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 precise bridge between two geometric frameworks for systems of PDEs in classical field theory: k-symplectic and k-contact Hamiltonian systems. It shows that every polarized exact k-symplectic Hamiltonian system can be extended by adding k extra coordinates (the contactification) to obtain a k-contact Hamiltonian system whose Hamiltonian is simply the pullback of the original one. The central result is that any projectable solution of the k-contact Hamilton–De Donder–Weyl equations projects down to a solution of the original k-symplectic Hamiltonian equation. This gives a constructive way to recover k-symplectic dynamics from k-contact dynamics, illustrated on the vibrating string and the wave equation.

What carries the argument

The contactification of a polarized exact k-symplectic manifold: given $(P,\omega=d\theta,V)$, form $M=P\times\mathbb{R}^k$ and define the $\mathbb{R}^k$-valued one-form $\eta=\sum_\alpha(dz^\alpha+\theta^\alpha_M)\otimes e_\alpha$. This produces a polarised k-contact manifold whose Reeb vector fields are the coordinate vector fields $\partial/\partial z^\alpha$. Because the lifted Hamiltonian $h_M=\mathrm{pr}_1^*h$ is independent of the $z^\alpha$'s, the Reeb terms in the k-contact equations drop out, and the projection of any projectable solution satisfies the original k-symplectic equation.

What would settle it

Construct a polarized exact k-symplectic Hamiltonian system and its contactification, then find a $\mathrm{pr}_1$-projectable k-contact Hamiltonian vector field solving (5) whose projection fails to satisfy the k-symplectic equation (2). Such a counterexample would refute Proposition 2. Alternatively, check that the vibrating string example's projected fields genuinely satisfy the Hamilton–De Donder–Weyl equations (8) as claimed.

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Extended reading notes

Core claim

Proposition 2: Let $(P,\omega,h)$ be a k-symplectic Hamiltonian system on a polarized exact k-symplectic manifold $(P,\omega=d\theta,V)$. Let $M=P\times\mathbb{R}^k$ be the polarised k-contact manifold obtained by contactification, with $\eta_M=\sum_\alpha(dz^\alpha+\theta^\alpha_M)\otimes e_\alpha$. Then $(M,\eta_M,h_M=\mathrm{pr}_1^*h)$ is a k-contact Hamiltonian system. Moreover, if $X^M$ is a $\mathrm{pr}_1$-projectable solution of the k-contact Hamilton–De Donder–Weyl equations, then its projection $X^P$ is a solution of the k-symplectic Hamiltonian equation. The proof uses the fact that $h_M$ is constant along the Reeb vector fields, so the term $(R_\alpha h_M)\eta^\alpha$ vanishes, and that $d\eta^\alpha=d\theta^\alpha_M$, letting the pullback structure carry the equation through.

Load-bearing premise

The lifted k-contact Hamiltonian must be the pullback of the k-symplectic Hamiltonian, meaning it is independent of the extra $z^\alpha$ coordinates; if it depended on those coordinates, the Reeb terms would not vanish and the projection would not satisfy the original equation.

Editorial extensions

If this is right

Any projectable solution of the k-contact Hamilton–De Donder–Weyl equations on the contactification yields a solution of the original k-symplectic Hamilton–De Donder–Weyl equations.
The construction gives a systematic way to build k-symplectic solutions from k-contact solutions, as demonstrated by the vibrating string example where the wave equation is recovered.
The k-contact formalism can therefore be used as a computational or geometric tool to study k-symplectic field theories, provided the lifted solutions are projectable.
The result suggests that the geometrical relation between k-symplectic and k-contact manifolds extends meaningfully to the dynamics, not just the underlying structures.

Reading between the lines

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

The converse question raised in the paper—whether all k-contact solutions arise this way—is likely false, as the author notes, because many k-contact Hamiltonian vector fields are not $\mathrm{pr}_1$-projectable; this is an inherent limitation of the projection method.
The construction could be applied to numerical methods or geometric reduction: one could solve the k-contact system in one more dimension and discard the extra coordinates to obtain k-symplectic solutions, potentially simplifying integration or preserving structure.
The requirement that the base be exact and polarized is essential; for a general k-symplectic manifold without an exact primitive, the contactification construction would not be available, suggesting a hierarchy of field theories where only exact ones admit this lift.
The vibrating string example hints at a broader class of wave-type equations that fit both frameworks; extending the relation to nonconservative (damped) cases may require relaxing the pullback condition on the Hamiltonian.

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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

Colleague —

The short version: the paper is a modest but real contribution. Proposition 2 — that a k-contact Hamiltonian system obtained by contactification with the pullback Hamiltonian projects to the original k-symplectic Hamiltonian system — is new at the Hamiltonian level, and the computation is correct once you read Eq. (5) with the summation convention that is standard in the k-contact literature. The vibrating string example is a nice illustration. So the core is fine.

The soft spots are not in the mathematics but in the presentation. As printed, Eq. (5) writes the k-contact Hamilton–de Donder–Weyl equations with α as a free index, no sum. Read literally, the proof of Proposition 2 would give ι_{Tpr1 X_α} dθ^α = dh for each α, so the summed k-symplectic equation would become k dh = dh and fail for k > 1. The example, on the other hand, only works if you sum over α: the condition A1_t + A2_x = 0 is exactly the summed p-coordinate constraint. So the paper has an internal inconsistency: Definition 8, Eq. (6), and the proof all need explicit sums on the left-hand sides. Eq. (6) as printed is also wrong under the summed reading — the second and third relations need a sum over α on the left. These are fixable by adding Sigma symbols, but they are not trivial typos; a reader cannot verify Proposition 2 from the text as it stands.

The example also has some slack: the B functions are undefined, A^2_x appears out of nowhere, and the local expression for the 2-vector field is taken from the author's own previous paper rather than derived. That is acceptable if the reference is correct, but it makes the example hard to check.

My verdict: the central claim is correct and useful, and the paper deserves a serious referee. It needs a revision that makes the summation convention explicit, fixes Eq. (6), and cleans up the example. I would not desk-reject it; I would send it back for revision. The underlying mathematics is sound, but the current text is not something you can verify as written. This is a paper for people who work in k-contact and k-symplectic Hamiltonian formulations; they will get a small but workable transfer result.

Referee Report

2 major / 5 minor

Summary. The manuscript studies the relation between k-symplectic Hamiltonian systems and k-contact Hamiltonian systems through the contactification construction for an exact polarised k-symplectic manifold. The main result, Proposition 2, states that the pullback of a k-symplectic Hamiltonian to the contactified product manifold defines a k-contact Hamiltonian system, and that every projectable k-contact Hamiltonian k-vector field projects to a solution of the k-symplectic Hamilton–De Donder–Weyl equation (2). The paper also applies this result to the vibrating string in Example 4. The proof strategy is natural: the pullback Hamiltonian is constant along the Reeb vector fields, so the Reeb terms disappear, and naturality of the pullback and interior product transfers the first k-contact HDW equation to the k-symplectic equation on P.

Significance. Once the summation issue in Eq. (5) is resolved, the result supplies a clean and useful bridge between two active geometric formalisms for first-order classical field theories: k-contact solutions of the HDW equations project to k-symplectic solutions. The construction is explicit, the argument is elementary and checkable, and the paper appropriately limits the claim to projectable solutions while explicitly declining to assert a converse. The central proof depends on the Darboux theorems and contactification results of prior work, which are properly acknowledged and are not self-citations. The principal weaknesses are the ambiguous summation convention in Eq. (5), an inconsistent coordinate characterization in Eq. (6), and a non-self-contained illustrative example; these are local and fixable, so the central claim is defensible after a substantial revision.

major comments (2)

[Section 2.5, Eq. (5); Section 3, proof of Proposition 2] Eq. (5) prints the first k-contact HDW equation without a summation over alpha, as i_{X_alpha} d eta^alpha = dh - (R_alpha h) eta^alpha, and the proof of Proposition 2 applies it as a per-alpha statement. Under that literal reading the proof yields i_{T pr1 X_alpha} d theta^alpha = dh for each alpha, and since Eq. (2) is the summed equation sum_alpha i_{X^P_alpha} omega^alpha = dh, the chain would give k dh = dh and fails for k > 1; the conclusion that T pr1(X^M) is a solution of (2) does not follow. Under the alternative summed reading the proof is correct, but then (5) and (6) are mutually inconsistent: the second line of (6), (X_alpha)_i^alpha = -(dh/dq^i + p_i^alpha dh/dz^alpha), can only be derived from a per-alpha first equation, and that per-alpha equation in turn forces dh/dp_i^beta = 0 for beta different from alpha, which contradicts the Hamiltonian (7) of Example 4. The paper must state the summation convention explicitly, display the sum over alpha in the first equation of (5), repeat this displayed sum in the proof of Proposition 2, and replace the second line of (6) by the summed relation sum_alpha (X_alpha)_i^alpha = -(dh/dq^i + sum_alpha p_i^alpha dh/dz^alpha). The proof should also mention that pr1^* is injective on forms because pr1 is a surjective submersion, as this is needed when passing from the pulled-back equality on M to the equality on P.
[Section 3, Example 4] Example 4 is not reproducible as written. The sentence that A^1_t, A^1_x, B^1_t and B^1_x are arbitrary functions on M satisfying A^1_t + A^2_x = 0 refers to functions that do not appear in the displayed vector fields: the displayed fields contain A^1_t, A^1_x, A^2_t and -A^1_t, while A^2_x, B^1_t and B^1_x are never defined. Moreover, the condition needed for the projected k-vector field to satisfy Eq. (2) is (T pr1 X_1)^{pt} + (T pr1 X_2)^{px} = 0, which with the displayed components holds identically, so the example does not illustrate the mechanism of Proposition 2 and does not verify that the displayed fields are projections of a solution of (5). The phrase 'in the particular case k=0, where k denoted the damped constant' reuses the symbol k for the damping constant in a section where k=2 is already fixed as the number of tangent copies; a different symbol should be used for the damping constant.

minor comments (5)

[Introduction] The sentence 'Summation over crossed repeated indices is comprehensible' is vague and, given the role of summation in Eq. (5), the convention should be stated explicitly rather than left to an informal phrase.
[Definition 4] The phrase 'a closed nondegenerate 1 R^k-valued two-form' is garbled and should read 'a closed nondegenerate R^k-valued two-form', and 'namely' should be 'called'.
[Abstract and Acknowledgements] There are typos: 'previoustly' should be 'previously' in the abstract, and 'finacial' should be 'financial' in the acknowledgements.
[Proposition 1, Eq. (3)] The expression dh/dq^i = -d psi^alpha_i / d x^alpha should carry an explicit summation over alpha to match the summed equation (2).
[Example 3] The phrase 'ker eta differs from 0 has corank k' should be rephrased as 'ker eta is a regular distribution of corank k'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main implication is proved from the definitions; the only self-citations are minor and non-load-bearing.

full rationale

The central claim, Proposition 2, is a direct theorem rather than a circular reduction. Given a polarised exact k-symplectic Hamiltonian system (P, omega=dtheta, h), the paper constructs M=P x R^k with eta=sum_alpha (dz^alpha + theta^alpha_M) otimes e_alpha and chooses h_M=pr1^*h. The proof then uses the k-contact Hamilton-De Donder-Weyl equations (5) and the paper's stated convention that summation over crossed repeated indices is understood; it obtains sum_alpha iota_{Tpr1 X_alpha} dtheta^alpha = dh, which is exactly the k-symplectic equation (2). This is an implication from one set of geometric equations to another, not an equivalence forced by definition. The choice h_M=pr1^*h is an explicit hypothesis, not a fitted parameter, and the conclusion is not assumed in the input. The apparent per-alpha mismatch noted by a literal reading of equation (5) is a notational subtlety: the paper announces at the outset that summation over crossed repeated indices is comprehensible, and the Darboux-coordinate expression (6) is consistent with the summed reading. No data are fitted and no quantity is predicted from fitted values. The contactification construction (Example 3) and the Darboux theorems are imported from [1] and [3], neither of which involves the present author. The paper cites its own prior book [5] for standard k-symplectic background and its own paper [2] in Example 4 for a local expression of a pr1-projectable k-contact vector field, but the wave-equation example is explicitly written out and the citation is not needed for the proof of Proposition 2. These self-citations are therefore minor and non-load-bearing; the overall circularity score is 2.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard Darboux theorems and the contactification construction from prior literature. The main domain assumption is that the lifted Hamiltonian is a pullback, which makes the Reeb term vanish. The example assumes an existence result from a self-cited reference. There are no free parameters and no new entities.

assumptions (4)

standard math Darboux theorems for polarised k-symplectic and polarised k-contact manifolds (Theorems 1 and 3).
Invoked to have local coordinates (q^i, p^α_i) and (q^i, p^α_i, z^α); these are background results from [5] and [3].
domain assumption The contactification construction of Example 3 turns an exact k-symplectic manifold into a k-contact manifold.
The paper takes this construction from [1] as the basis for the Hamiltonian-level result.
domain assumption The k-contact Hamiltonian function is a pullback h_M = pr1^* h, independent of the extra coordinates.
Used in the proof of Proposition 2 to make the Reeb term (R_α h_M) η^α vanish; this restricts the class of k-contact systems covered.
domain assumption In Example 4, existence of an integrable pr1-projectable 2-vector field solution of (5) is assumed, following [2].
The example relies on the local expression from [2] rather than constructing the solution explicitly.

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Cite this review

Pith. "Pith review of A relation between k-symplectic and k-contact Hamiltonian systems." pith.science (2026). https://pith.science/paper/T24QG7GW

@misc{pith2026250611873,
  author       = {Pith},
  title        = {Pith review of: A relation between k-symplectic and k-contact Hamiltonian systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T24QG7GW}},
  note         = {Machine review of arXiv:2506.11873}
}

read the original abstract

Systems of partial differential equations which appear in classical field theories can be studied geometrically using different geometrical structures, for example, k-symplectic geometry, k-cosymplectic geometry, multisymplectic geometry, etc. In recent years, there has been a notable increase in the study of k-contact Hamiltonian systems. These are based on the description of the dynamics of field theories using the so-called k-contact manifolds. Such structures are generalizations of contact structures and k-symplectic structures. The relation between k-symplectic manifolds and k-contact manifolds was previoustly established. In light of the above relation, this work seeks to explore the relationship between k-symplectic Hamiltonian systems and k-contact Hamiltonian systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

5 extracted references · 3 canonical work pages

[1]

ArXiv:2409.11001 [math.DG] (2024)

de Lucas, J., Rivas, X., Sobczak, T.: Foundations onk-contact geometry. ArXiv:2409.11001 [math.DG] (2024)

arXiv 2024
[2]

ArXiv:2505.05462[math.DG] (2025)

de Lucas, J., Rivas, X., Vilariño, S., Zawora, B.M.: Marden–Meyer–Weinstein re- duction fork-contact field theories. ArXiv:2505.05462[math.DG] (2025)

arXiv 2025
[3]

Gaset, J.,Grácia, X., Muñoz-Lecanda, M.C., Rivas, X, Román-Roy, N.: A contact geometry framework for field theories with dissipation. Ann. Phys., 414:168092, (2020)

work page 2020
[4]

Gaset, J.,Grácia, X., Muñoz-Lecanda, M.C., Rivas, X, Román-Roy, N.:k-contact Lagrangian formulation for nonconservative field theories. Rep. Math. Phys., 87(3):347– 368, (2021)

work page 2021
[5]

Vilariño, S.: Methods of Differential Geometry in Classical Field Theories

León, M, Salgado, M. Vilariño, S.: Methods of Differential Geometry in Classical Field Theories. World Scientific (2015)

work page 2015

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

ArXiv:2409.11001 [math.DG] (2024)

de Lucas, J., Rivas, X., Sobczak, T.: Foundations onk-contact geometry. ArXiv:2409.11001 [math.DG] (2024)

arXiv 2024

[2] [2]

ArXiv:2505.05462[math.DG] (2025)

de Lucas, J., Rivas, X., Vilariño, S., Zawora, B.M.: Marden–Meyer–Weinstein re- duction fork-contact field theories. ArXiv:2505.05462[math.DG] (2025)

arXiv 2025

[3] [3]

Gaset, J.,Grácia, X., Muñoz-Lecanda, M.C., Rivas, X, Román-Roy, N.: A contact geometry framework for field theories with dissipation. Ann. Phys., 414:168092, (2020)

work page 2020

[4] [4]

Gaset, J.,Grácia, X., Muñoz-Lecanda, M.C., Rivas, X, Román-Roy, N.:k-contact Lagrangian formulation for nonconservative field theories. Rep. Math. Phys., 87(3):347– 368, (2021)

work page 2021

[5] [5]

Vilariño, S.: Methods of Differential Geometry in Classical Field Theories

León, M, Salgado, M. Vilariño, S.: Methods of Differential Geometry in Classical Field Theories. World Scientific (2015)

work page 2015