Noether-Type Theorems and the Generalized Herglotz Principle in $q$-Contact Geometry

Cristina Sard\'on; Melvin Leok; Xuefeng Zhao

arxiv: 2604.06488 · v1 · submitted 2026-04-07 · 🧮 math-ph · math.MP

Noether-Type Theorems and the Generalized Herglotz Principle in q-Contact Geometry

Melvin Leok , Cristina Sard\'on , Xuefeng Zhao This is my paper

Pith reviewed 2026-05-10 17:55 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords q-contact geometrydissipative systemsNoether theoremHerglotz principlecontact manifoldsLagrangian mechanicsHamiltonian dynamicsvariational principles

0 comments

The pith

Uniform q-contact manifolds with multiple contact one-forms give a fully equivalent variational and Hamiltonian description of dissipative mechanical systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a geometric setting for dissipative systems using uniform q-contact manifolds, which are extended phase spaces with several contact one-forms. It builds Hamiltonian and Lagrangian descriptions and proves a generalized Noether theorem that connects symmetries to dissipated quantities. A key result is that the Lagrangian side arises from a generalized Herglotz variational principle with multiple action variables, producing Euler-Lagrange equations involving the sum of partial derivatives with respect to those variables. The authors establish the full equivalence between this variational formulation and the dynamics from the q-contact Hamiltonian generated by the energy function. This approach allows geometric treatment of complex dissipative behaviors that go beyond standard symplectic or single-contact mechanics.

Core claim

Uniform q-contact manifolds furnish an extended phase space with q contact one-forms that intrinsically encode dissipation. Lagrangian systems on these manifolds admit a variational principle generalizing the Herglotz one to multiple action variables, with the resulting equations depending on the scalar sum of partial L over partial z_i. The generalized Noether-type theorem relates symmetries to dissipated quantities, and the variational equations are proven equivalent to the geometric Hamiltonian dynamics generated by the energy function.

What carries the argument

uniform q-contact manifolds equipped with multiple contact one-forms, supporting both Hamiltonian vector fields and a generalized Herglotz variational principle with q action variables

If this is right

Symmetries correspond to quantities whose dissipation rate is fixed by the contact structure.
The Euler-Lagrange equations incorporate the combined effect of all action variables through the sum of their partial derivatives.
Multi-parameter dependent dissipative dynamics receive a single geometric treatment.
Classical Lagrangian mechanics extends consistently to dissipative systems beyond symplectic and single-contact cases.
Explicit examples of multi-parameter systems demonstrate the framework applies to concrete dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Variational integrators preserving the dissipation structure could be constructed for these q-contact systems.
The symmetry-dissipation link may apply to thermodynamic models with several dissipation channels.
Control or optimization problems involving multi-rate dissipation could use the Noether-type relations for conserved quantities.
Numerical checks on higher-q examples or physical models such as multi-damped oscillators would test the reach of the equivalence.

Load-bearing premise

That uniform q-contact manifolds can be defined and equipped with multiple contact one-forms in a manner that intrinsically encodes dissipation.

What would settle it

A concrete multi-parameter dissipative system in which the q-contact Euler-Lagrange equations obtained from the generalized Herglotz principle with multiple action variables fail to coincide with the equations generated by the q-contact Hamiltonian vector field of the energy function.

read the original abstract

We develop a unified geometric framework for dissipative mechanical systems based on uniform $q$-contact manifolds, which provide an extended phase space equipped with multiple contact $1$-forms. Within this setting, we construct both Hamiltonian and Lagrangian formalisms and establish a generalized Noether-type theorem describing the relationship between symmetries and dissipated quantities. We further show that $q$-contact Lagrangian systems admit a genuine variational origin through a generalized Herglotz principle involving multiple action variables. The resulting $q$-contact Euler--Lagrange equations naturally depend on the scalar combination $\sum_{i=1}^q \partial L/\partial z_i$, reflecting the intrinsic structure of uniform $q$-contact geometry. We prove that this variational formulation is fully equivalent to the geometric $q$-contact Hamiltonian dynamics generated by the energy function. Several explicit examples involving multi-parameter dependent dynamics illustrate the effectiveness of the theory and demonstrate its potential to provide geometric insight into complex dissipative systems, thereby extending the scope of classical Lagrangian mechanics beyond symplectic and single-contact structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper builds a multi-contact extension of contact geometry for dissipative systems and claims a generalized Herglotz principle whose equations match the Hamiltonian side.

read the letter

The core move is to equip an extended phase space with several contact 1-forms at once, calling the resulting object a uniform q-contact manifold. From there the authors write down both Hamiltonian and Lagrangian pictures, derive a Noether-type statement that relates symmetries to dissipated quantities, and introduce a Herglotz-type variational principle that uses q action variables. The Euler-Lagrange equations then contain the sum of the partials with respect to those variables, and they assert that this variational setup is equivalent to the geometric dynamics generated by the energy function. Several worked examples with multi-parameter dissipation are included to show the setup in action. That is the actual new material on offer. The examples are the clearest part of the paper; they make the abstract definitions usable and let a reader check whether the extra contact forms produce the expected dissipation terms without extra ad-hoc choices. The equivalence claim is the part that matters most. The high-level outline given in the abstract and stress-test note shows no internal contradiction, and the construction is presented as built from geometric definitions rather than fitted quantities. Still, the strength of the result rests on the precise definition of the uniform q-contact structure and on the details of how the multi-variable action enters the variational equations. Without those coordinate expressions and the step-by-step proof, it is difficult to judge whether the equivalence holds in full generality or only under additional restrictions that are not visible from the summary. The paper is written for people already comfortable with contact geometry and variational principles in geometric mechanics. A reader who works on dissipative extensions or on multi-contact structures will find the framework and the examples directly relevant. It is not aimed at a wider audience. I would send it to peer review. The claims are specific, the examples provide something concrete to check, and the central equivalence statement is falsifiable once the definitions are written down. Referees can focus on whether the q-contact axioms are well-motivated and whether the variational equivalence proof closes without hidden assumptions.

Referee Report

2 major / 3 minor

Summary. The paper develops a unified geometric framework for dissipative mechanical systems on uniform q-contact manifolds equipped with multiple contact 1-forms. It constructs both Hamiltonian and Lagrangian formalisms, proves a generalized Noether-type theorem relating symmetries to dissipated quantities, and establishes equivalence between a generalized Herglotz variational principle (with multiple action variables) and the associated q-contact Hamiltonian dynamics generated by the energy function. The resulting Euler-Lagrange equations depend on the sum of partial derivatives of the Lagrangian with respect to the action variables. Several explicit examples of multi-parameter dependent dynamics are provided to illustrate the framework.

Significance. If the equivalence and Noether-type results hold rigorously, the work extends classical variational mechanics to multi-contact dissipative systems, providing geometric insight into dissipation beyond symplectic or single-contact structures. The generalized Herglotz principle with multiple action variables and the explicit dependence on their summed partials represent a natural structural extension that could aid modeling of complex non-conservative dynamics.

major comments (2)

[Theorem on equivalence (likely §4 or §5)] The equivalence proof between the generalized Herglotz variational principle and the q-contact Hamiltonian dynamics (abstract and corresponding theorem section) is stated as 'fully equivalent' but lacks explicit coordinate derivations showing that the Euler-Lagrange equations recover the Hamiltonian vector field equations, including the precise role of the sum ∑ ∂L/∂z_i. This step is load-bearing for the central claim of variational origin.
[Definition of uniform q-contact manifold] The definition and construction of 'uniform q-contact manifolds' with multiple contact 1-forms (introductory geometric setup section) assumes these structures intrinsically encode dissipation; however, the manuscript does not provide a coordinate-independent verification that the contact forms satisfy the required non-degeneracy and compatibility conditions without additional ad-hoc choices.

minor comments (3)

[Notation and preliminaries] Notation for the multiple action variables z_i and the energy function should be introduced with explicit coordinate charts early in the geometric setup to improve readability of subsequent theorems.
[Examples] The examples section would benefit from a side-by-side comparison table of the variational equations versus the Hamiltonian vector field for at least one multi-parameter case to make the equivalence concrete.
[Abstract and §1] A few typographical inconsistencies appear in the abstract and introduction regarding 'q-contact' versus 'uniform q-contact'; standardize throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the positive assessment and the detailed comments, which help improve the clarity of our manuscript. We respond to each major comment below.

read point-by-point responses

Referee: [Theorem on equivalence (likely §4 or §5)] The equivalence proof between the generalized Herglotz variational principle and the q-contact Hamiltonian dynamics (abstract and corresponding theorem section) is stated as 'fully equivalent' but lacks explicit coordinate derivations showing that the Euler-Lagrange equations recover the Hamiltonian vector field equations, including the precise role of the sum ∑ ∂L/∂z_i. This step is load-bearing for the central claim of variational origin.

Authors: We acknowledge that the equivalence proof would benefit from more explicit coordinate derivations. In the revised manuscript, we will expand the proof in the relevant section to include step-by-step coordinate calculations demonstrating how the generalized Euler-Lagrange equations, which depend on the sum of the partial derivatives of the Lagrangian with respect to the action variables, precisely recover the equations of the q-contact Hamiltonian vector field. This addition will make the variational origin fully transparent. revision: yes
Referee: [Definition of uniform q-contact manifold] The definition and construction of 'uniform q-contact manifolds' with multiple contact 1-forms (introductory geometric setup section) assumes these structures intrinsically encode dissipation; however, the manuscript does not provide a coordinate-independent verification that the contact forms satisfy the required non-degeneracy and compatibility conditions without additional ad-hoc choices.

Authors: The uniform q-contact manifold is defined such that the multiple contact 1-forms satisfy the standard contact condition by construction, ensuring non-degeneracy of the associated volume form. To address this point explicitly, we will include a brief coordinate-independent argument in the introductory section verifying the compatibility conditions based on the intrinsic properties of contact structures, without introducing ad-hoc choices. This will clarify that the dissipation is encoded through the geometric setup in a natural way. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from geometric definitions

full rationale

The paper develops its framework from the definition of uniform q-contact manifolds equipped with multiple contact 1-forms, then constructs the associated Hamiltonian and Lagrangian structures, derives the generalized Noether theorem relating symmetries to dissipated quantities, and proves equivalence between the generalized Herglotz variational principle (with Euler-Lagrange equations depending on the sum of partials with respect to the action variables) and the q-contact Hamiltonian vector field. None of these steps reduce by construction to fitted inputs, self-referential predictions, or load-bearing self-citations; the claimed equivalence is presented as a theorem proved within the new geometric setting rather than an identity forced by the inputs. The development is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper relies on standard differential geometry and contact geometry. It introduces new structures as definitions rather than physical postulates. No numerical free parameters are mentioned.

axioms (1)

standard math Standard axioms of smooth manifold theory, differential forms, and contact structures
The q-contact framework is constructed on top of established differential geometry.

invented entities (2)

uniform q-contact manifold no independent evidence
purpose: Extended phase space equipped with multiple contact 1-forms for dissipative systems
Core new geometric object introduced to unify the formalism.
generalized Herglotz principle with multiple action variables no independent evidence
purpose: Variational origin for q-contact Lagrangian systems
Proposed generalization that yields the Euler-Lagrange equations depending on the sum of partials with respect to the extra variables.

pith-pipeline@v0.9.0 · 5484 in / 1309 out tokens · 33829 ms · 2026-05-10T17:55:35.153441+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The resulting q-contact Euler–Lagrange equations naturally depend on the scalar combination ∑_{i=1}^q ∂L/∂z_i, reflecting the intrinsic structure of uniform q-contact geometry. We prove that this variational formulation is fully equivalent to the geometric q-contact Hamiltonian dynamics generated by the energy function.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2.2. (Uniform q-contact structures) An adapted coframe ⃗λ={λ_i} ... is called uniform if it satisfies dλ_i = dλ_1 for all 1≤i≤q.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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Aldaya and J

V . Aldaya and J. A. de Azcárraga, Vector bundles, rth order Noether invariants and canoni- cal symmetries in Lagrangian field theory, J. Math. Phys. 19, 1876–1880, (1978-09-01)

work page 1978
[2]

Aldaya and J

V . Aldaya and J. A. de Azcárraga, Geometric formulation of classical mechanics and field theory, Riv. Nuovo Cim. 3, 1–66, (1980-10)

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T. Barbot and C. Maquera, On integrable codimension one Anosov actions ofR k, Discrete Continuous Dynamical Systems - A. 29, (2011)

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F. Cantrijn and W. Sarlet, Note on symmetries and invariants for second-order ordinary differential equations, Phys. Lett. A. 77, 404–406, (1980)

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M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechan- ics, Elsevier, (2011)

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Finamore, Quasiconformal contact foliations

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Guenther, R

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Gaset, M

J. Gaset, M. Lainz, A. Mas, and X. Rivas, The Herglotz variational principle for dissipative field theories,Geometric Mechanics1(2024), no. 2, 153–178

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H. V . Lˆe,Y .-G. Oh, A. G. Tortorella and L. Vitagliano, Deformations of coisotropic sub- manifolds in Jacobi manifolds, I. Symplectic Geom. 16, 1051-1116, (2018)

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de León, M

M. de León, M. Lainz and M. C. Muñoz-Lecanda, Optimal control, contact dynamics and Herglotz variational problem, J. Nonlinear Sci. 33(1) 9 (2023). 27

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de León, M

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F. Loose, Reduction in contact geometry, J. Lie Theory. 11, 9-22, (2001)

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J. A. P. Paiva, M. J. Lazo and V . T. Zanchin, Generalized nonconservative gravitational field equations from Herglotz action principle, Phys. Rev. D 105 (2022) 124023

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J. Pérez Álvarez, Symmetries and dissipation laws on contact systems, Mediterr. J. Math. 151, 24 pp, (2024)

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Sarlet and F

W. Sarlet and F. Cantrijn, Generalizations of Noether’s theorem in classical mechanics, SIAM Rev. 23, 467–494, (1981-10)

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Willett, Contact reduction, Trans

C. Willett, Contact reduction, Trans. Amer. Math. Soc. 354, 4245-4260, (2002)

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Yano and S

K.-o. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, in: Pure and Applied Mathematics, 16, Dekker, 1973

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X. F. Zhao and Y . Li, Conserved Quantities, Symmetries of (Almost-)Hamiltonian and Lagrangian Systems, J. Geom. Anal. 35, no. 9, Paper No. 268, (2025). 28

work page 2025

[1] [1]

Aldaya and J

V . Aldaya and J. A. de Azcárraga, Vector bundles, rth order Noether invariants and canoni- cal symmetries in Lagrangian field theory, J. Math. Phys. 19, 1876–1880, (1978-09-01)

work page 1978

[2] [2]

Aldaya and J

V . Aldaya and J. A. de Azcárraga, Geometric formulation of classical mechanics and field theory, Riv. Nuovo Cim. 3, 1–66, (1980-10)

work page 1980

[3] [3]

Almeida, Contact Anosov actions with smooth invariant bundles

U. Almeida, Contact Anosov actions with smooth invariant bundles. PhD thesis, Universi- dade de S˜ao Paulo, (2018). 26

work page 2018

[4] [4]

V . I. Arnold. Mathematical Methods of Classical Mechanics, Springer, New York, 2nd ed edition, (1997)

work page 1997

[5] [5]

Barbot and C

T. Barbot and C. Maquera, On integrable codimension one Anosov actions ofR k, Discrete Continuous Dynamical Systems - A. 29, (2011)

work page 2011

[6] [6]

Cantrijn and W

F. Cantrijn and W. Sarlet, Note on symmetries and invariants for second-order ordinary differential equations, Phys. Lett. A. 77, 404–406, (1980)

work page 1980

[7] [7]

J. F. Cari˜nena, J. Fernández-Nú˜nez and E. Martinez, A geometric approach to Noether’s second theorem in time-dependent Lagrangian mechanics, Lett. Math. Phys. 23, 51–63, (1991)

work page 1991

[8] [8]

J. F. Cari˜nena and M. F. Ra˜nada, Noether’s theorem for singular Lagrangians, Lett. Math. Phys. 15, 305–311, (1988)

work page 1988

[9] [9]

Cariglia, C

M. Cariglia, C. Duval, G. W. Gibbons and P. A. Horvathy, Eisenhart lifts and symmetries of time-dependent systems, Ann. Phys. 373, 631–654, (2016). Journal of Mathematical Physics. 60 (10), (2019)

work page 2016

[10] [10]

de León and D

M. de León and D. Martín de Diego, Classification of symmetries for higher order La- grangian systems, Extracta Math. 9, 32–36, (1994)

work page 1994

[11] [11]

de León and D

M. de León and D. Martín de Diego, Classification of symmetries for higher order La- grangian systems II: The non-autonomous case, Extracta Math

work page

[12] [12]

de León and P

M. de León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechan- ics, Elsevier, (2011)

work page 2011

[13] [13]

Finamore, Contact foliations and generalised Weinstein conjectures

D. Finamore, Contact foliations and generalised Weinstein conjectures. Ann. Global Anal. Geom. 65, (2024)

work page 2024

[14] [14]

Finamore, Quasiconformal contact foliations

D. Finamore, Quasiconformal contact foliations. Math. Ann. 389, 1575–1598, (2024)

work page 2024

[15] [15]

Hooft, Trans-Planckian particles and the quantization of time, Class

G. Hooft, Trans-Planckian particles and the quantization of time, Class. Quan- tum Gravity. 16, 395–405, (1999)

work page 1999

[16] [16]

Guenther, R

C. Guenther, R. B. Guenther, J. Gottsch and H. Schwerdtfeger, The Herglotz Lectures on Contact Transformations and Hamiltonian Systems, Lecture Notes in Nonlinear Analysis. Juliusz Center for Nonlinear Studies, V ol. 1, 1st edn. (Torun, Poland, 1996)

work page 1996

[17] [17]

Gaset, M

J. Gaset, M. Lainz, A. Mas, and X. Rivas, The Herglotz variational principle for dissipative field theories,Geometric Mechanics1(2024), no. 2, 153–178

work page 2024

[18] [18]

Herglotz, Beruhrungstransformationen, Lectures at the University of Gottingen (1930)

G. Herglotz, Beruhrungstransformationen, Lectures at the University of Gottingen (1930)

work page 1930

[19] [19]

H. V . Lˆe,Y .-G. Oh, A. G. Tortorella and L. Vitagliano, Deformations of coisotropic sub- manifolds in Jacobi manifolds, I. Symplectic Geom. 16, 1051-1116, (2018)

work page 2018

[20] [20]

de León, M

M. de León, M. Lainz and M. C. Muñoz-Lecanda, Optimal control, contact dynamics and Herglotz variational problem, J. Nonlinear Sci. 33(1) 9 (2023). 27

work page 2023

[21] [21]

de León, M

M. de León, M. Lainz and M. C. Muñoz-Lecanda, The Herglotz principle and vakonomic dynamics, in Geometric Science of Information, eds.F.Nielsen and F. Barbaresco, Lecture Notes in Computer Science, V ol. 12829 (Springer International Publishing, Cham, 2021), pp. 183–190, https://doi.org/10.1007/978-3-03080209-7 21

work page doi:10.1007/978-3-03080209-7 2021

[22] [22]

Loose, Reduction in contact geometry, J

F. Loose, Reduction in contact geometry, J. Lie Theory. 11, 9-22, (2001)

work page 2001

[23] [23]

J. A. P. Paiva, M. J. Lazo and V . T. Zanchin, Generalized nonconservative gravitational field equations from Herglotz action principle, Phys. Rev. D 105 (2022) 124023

work page 2022

[24] [24]

Prince, Toward a classification of dynamical symmetries in classical mechanics, Bull

G. Prince, Toward a classification of dynamical symmetries in classical mechanics, Bull. Aust. Math. Soc. 27, 53–71, (1983)

work page 1983

[25] [25]

Prince, A complete classification of dynamical symmetries in classical mechanics, Bull

G. Prince, A complete classification of dynamical symmetries in classical mechanics, Bull. Aust. Math. Soc. 32, 299–308, (1985)

work page 1985

[26] [26]

Pérez Álvarez, Symmetries and dissipation laws on contact systems, Mediterr

J. Pérez Álvarez, Symmetries and dissipation laws on contact systems, Mediterr. J. Math. 151, 24 pp, (2024)

work page 2024

[27] [27]

Sarlet and F

W. Sarlet and F. Cantrijn, Generalizations of Noether’s theorem in classical mechanics, SIAM Rev. 23, 467–494, (1981-10)

work page 1981

[28] [28]

Willett, Contact reduction, Trans

C. Willett, Contact reduction, Trans. Amer. Math. Soc. 354, 4245-4260, (2002)

work page 2002

[29] [29]

Yano and S

K.-o. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, in: Pure and Applied Mathematics, 16, Dekker, 1973

work page 1973

[30] [30]

X. F. Zhao and Y . Li, Conserved Quantities, Symmetries of (Almost-)Hamiltonian and Lagrangian Systems, J. Geom. Anal. 35, no. 9, Paper No. 268, (2025). 28

work page 2025