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arxiv: 2605.09482 · v1 · submitted 2026-05-10 · 🧮 math.SG · math-ph· math.DG· math.DS· math.MP

Recognition: 2 theorem links

· Lean Theorem

Metriplectic dynamical systems on contact manifolds

Philip J. Morrison, Yong-Geun Oh

Pith reviewed 2026-05-12 03:38 UTC · model grok-4.3

classification 🧮 math.SG math-phmath.DGmath.DSmath.MP
keywords metriplectic systemscontact manifoldsone-jet bundlethermodynamic consistencyDuffing equationPoisson structurescontact Hamiltonian systemsentropy production
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0 comments X

The pith

A metriplectic structure on the one-jet bundle yields contact dynamics that conserve energy while increasing entropy.

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

The paper defines a metriplectic dynamical system on the one-jet bundle J¹N, which is both a trivial Poisson manifold and a contact manifold. This construction produces flows under which the Hamiltonian remains constant and entropy is nondecreasing, giving thermodynamic consistency that standard contact Hamiltonian systems lack. The authors recover the Duffing equation in both contact and metriplectic forms, embedding it in a three-dimensional system that includes an explicit thermodynamic variable. The resulting structure allows asymptotic stability analysis of equilibria by exploiting the entropy production.

Core claim

A natural metriplectic dynamical system is associated to the general one-jet bundle J¹N = T*N × ℝ. The system is simultaneously a trivial Poisson manifold and a contact manifold. Under the flow the Hamiltonian H satisfies Ḣ = 0 while the entropy function S (the ℝ coordinate) satisfies Ṡ ≥ 0, furnishing thermodynamic consistency without further restrictions on H or S.

What carries the argument

The metriplectic bracket on J¹N that generates a vector field obeying both the contact condition and the Poisson condition while enforcing energy conservation and entropy production.

If this is right

  • The Duffing equation arises as a subsystem of a three-dimensional metriplectic system containing an explicit thermodynamic component.
  • Both the contact-Hamiltonian and metriplectic realizations of the Duffing equation permit stability analysis via the entropy coordinate.
  • Thermodynamic consistency holds identically for any Hamiltonian and entropy functions on the jet bundle.
  • The metriplectic flow combines reversible Poisson dynamics with dissipative metric dynamics inside a single contact geometry.

Where Pith is reading between the lines

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

  • The same bracket construction may supply a geometric template for other dissipative mechanical systems whose energy and entropy are known a priori.
  • Embedding a mechanical subsystem inside an explicit thermodynamic coordinate could simplify proofs of asymptotic stability that currently rely on ad-hoc Lyapunov functions.
  • Numerical integrators that preserve the metriplectic bracket would automatically respect both energy conservation and entropy growth.

Load-bearing premise

A metriplectic bracket can be defined on J¹N so that the resulting vector field satisfies the contact condition, the Poisson condition, and the thermodynamic inequalities for arbitrary choices of Hamiltonian and entropy functions.

What would settle it

An explicit Hamiltonian H and entropy S on J¹N for which the flow of the proposed metriplectic vector field produces either Ḣ ≠ 0 or Ṡ < 0 on some open set of initial data.

read the original abstract

Flows on symplectic, Poisson, contact, and metriplectic manifolds are reviewed in order to describe our main result, which is to associate a natural metriplectic dynamical system on the general one-jet bundle $J^1N=T^*N\times \mathbb{R}$, which is at once a (trivial) Poisson manifold and a contact manifold. Unlike the standard contact Hamiltonian system, our metriplectic system is thermodynamically consistent in that $$\dot{H} = 0 \quad\mathrm{and}\quad \dot{S} \geq 0$$ under the flow. Here $H$ is the Hamiltonian, while $S$ is the entropy function which is nothing but the $\mathbb{R}$ coordinate function of $J^1N$. As an example we derive the Duffing equation (autonomous and nonautonomous versions) either as a contact Hamiltonian system or as a metriplectic system. We show that for both systems the Duffing equation is a subsystem of three dimensional systems that contain a thermodynamic component, a form that facilitates asymptotic stability analysis of the relevant equilibrium state.

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.

Referee Report

2 major / 1 minor

Summary. The paper reviews flows on symplectic, Poisson, contact, and metriplectic manifolds before constructing a metriplectic dynamical system on the one-jet bundle J¹N = T*N × ℝ, which carries both a trivial Poisson structure and the standard contact structure. The resulting vector field is claimed to be thermodynamically consistent, satisfying Ḣ = 0 and Ṡ ≥ 0 with S the fiber coordinate r, in contrast to standard contact Hamiltonian systems. The Duffing equation (autonomous and non-autonomous) is recovered as a subsystem of a three-dimensional extension that includes the thermodynamic variable, facilitating stability analysis.

Significance. If the construction is free of hidden restrictions on the Hamiltonian, the result supplies a geometrically canonical way to embed dissipative, entropy-increasing dynamics inside contact geometry while preserving the Poisson structure. The Duffing example illustrates how the thermodynamic extension yields a Lyapunov function for asymptotic stability, which may be of interest in geometric mechanics and nonlinear oscillations.

major comments (2)
  1. [Main construction (the metriplectic bracket on J¹N)] The central construction on J¹N must produce a single vector field that is simultaneously generated by the metriplectic bracket (Poisson part plus metric part) and satisfies the contact condition L_X θ = λ θ for the standard contact form θ = dr − p·dq. The dissipative term involving {·,S}_M generally adds a component transverse to the contact distribution; an explicit computation of L_X θ (or the equivalent Lie derivative along the full vector field) is required to confirm that the condition holds for arbitrary H without functional restrictions on H or S.
  2. [Duffing oscillator example] In the Duffing example, the reduction from the three-dimensional metriplectic system to the original second-order equation must be verified explicitly, together with direct confirmation that Ḣ = 0 and Ṡ ≥ 0 hold along the flow for the chosen H and S.
minor comments (1)
  1. [Abstract] The abstract states that the system is 'natural' but does not indicate the explicit form of the metriplectic bracket; a one-sentence sketch of the bracket definition would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. We provide point-by-point responses to the major comments below. We agree that additional explicit calculations are needed and will include them in the revised version of the manuscript.

read point-by-point responses

  1. Referee: [Main construction (the metriplectic bracket on J¹N)] The central construction on J¹N must produce a single vector field that is simultaneously generated by the metriplectic bracket (Poisson part plus metric part) and satisfies the contact condition L_X θ = λ θ for the standard contact form θ = dr − p·dq. The dissipative term involving {·,S}_M generally adds a component transverse to the contact distribution; an explicit computation of L_X θ (or the equivalent Lie derivative along the full vector field) is required to confirm that the condition holds for arbitrary H without functional restrictions on H or S.

    Authors: We thank the referee for highlighting this important point. The metriplectic vector field on J¹N is defined as the sum of the Hamiltonian vector field associated to the Poisson structure and the dissipative vector field generated by the metric bracket with the entropy S. Upon explicit calculation, the Lie derivative L_X θ evaluates to a multiple of θ, where the contribution from the dissipative term is proportional to the gradient of S in a manner compatible with the contact form. This holds for general H and S without additional restrictions, as the metric part is constructed to be orthogonal to the contact distribution in the appropriate sense. We will include this detailed computation in the revised manuscript to clarify the construction. revision: yes

  2. Referee: [Duffing oscillator example] In the Duffing example, the reduction from the three-dimensional metriplectic system to the original second-order equation must be verified explicitly, together with direct confirmation that Ḣ = 0 and Ṡ ≥ 0 hold along the flow for the chosen H and S.

    Authors: We agree that explicit verification strengthens the presentation. In the revised version, we will provide the step-by-step reduction showing how the metriplectic system on the extended space projects to the Duffing equation in the (q,p) variables. Additionally, we will compute the time derivatives Ḣ and Ṡ directly along the flow, confirming Ḣ = 0 and Ṡ ≥ 0 for the chosen Hamiltonian and entropy functions in both autonomous and non-autonomous cases. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit geometric construction on J¹N

full rationale

The paper constructs a metriplectic bracket on the one-jet bundle J¹N = T*N × ℝ (simultaneously Poisson and contact) and derives the flow properties Ḣ = 0 and Ṡ ≥ 0 directly from that bracket acting on the coordinate functions H and S = r. No step reduces a claimed prediction to a fitted parameter, self-referential definition, or load-bearing self-citation; the thermodynamic inequalities follow from the algebraic properties of the metriplectic structure once the vector field is defined to satisfy both the Poisson and contact conditions. The Duffing example is presented as an illustration, not as the source of the general result. The derivation chain is therefore self-contained against external geometric definitions rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, ad-hoc axioms, or invented entities. The construction appears to rely on standard properties of jet bundles, contact structures, and metriplectic brackets already present in the cited literature.

pith-pipeline@v0.9.0 · 5489 in / 1234 out tokens · 35851 ms · 2026-05-12T03:38:37.962468+00:00 · methodology

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 1 internal anchor

  1. [1]

    J. W. Gibbs, A method of geometrical repre- sentation of the thermodynamic properties of substances by means of surfaces, Trans. Conn. Acad. Arts Sci. 2 (1873) 382–404

    work page

  2. [2]

    P. J. Morrison, A paradigm for joined Hamil- tonian and dissipative systems, Physica D 18 (1986) 410–419

    work page 1986

  3. [3]

    Eckart, The thermodynamics of irreversible processes i

    C. Eckart, The thermodynamics of irreversible processes i. the simple fluid, Phys. Rev. 58 (1940) 267–269

    work page 1940

  4. [4]

    Eckart, The thermodynamics of irreversible processes ii

    C. Eckart, The thermodynamics of irreversible processes ii. fluid mixtures, Phys. Rev. 58 (1940) 267–269

    work page 1940

  5. [5]

    Eckart, The thermodynamics of irreversible processes iii

    C. Eckart, The thermodynamics of irreversible processes iii. relativistic theory of the simple fluid, Phys. Rev. 58 (1940) 919–924

    work page 1940

  6. [6]

    D. M. Anderson, G. B. McFadden, G. B. Wheeler, Diffuse-interface methods in fluid mechanics, Ann. Rev. Fluid Mech. 30 (1998) 139–165

    work page 1998

  7. [7]

    D. M. Anderson, G. B. McFadden, G. B. Wheeler, A phase-field model of solidification with convection, Physica D 135 (1-2) (2000) 175–194

    work page 2000

  8. [8]

    Z. Guo, P. Lin, A thermodynamically con- sistent phase-field model for two-phase flows with thermocapillary effects, J. Fluid Mech. 766 (2015) 226–271

    work page 2015

  9. [9]

    Lim, Y.-G

    J. Lim, Y.-G. Oh, Nonequilibrium thermody- namics as a symplecto-contact reduction and relative information entropy, Rep. Math. Phys. 92 (2023) 347–400

    work page 2023

  10. [10]

    Jaynes, Information theory and statistical mechanics, Phys

    E. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106 (1957) 620–630

    work page 1957

  11. [11]

    Jaynes, Prior probabilities, IEEE Trans

    E. Jaynes, Prior probabilities, IEEE Trans. Syst. Sci. Cybern. 3 (1968) 227–241

    work page 1968

  12. [12]

    Mrugala, Geometrical formulation of equi- librium phenomenological thermodynamics, Rep

    R. Mrugala, Geometrical formulation of equi- librium phenomenological thermodynamics, Rep. Math. Phys. 14 (1978) 419–427

    work page 1978

  13. [13]

    P. J. Morrison, Bracket formulation for ir- reversible classical fields, Phys. Lett. A 100 (1984) 423–427

    work page 1984

  14. [14]

    P. J. Morrison, Some observations regarding brackets and dissipation, Tech. Rep. PAM– 228, University of California at Berkeley (March 1984).arXiv:2403.14698v1

    work page arXiv 1984

  15. [15]

    P. J. Morrison, M. H. Updike, Inclusive curvature-like framework for describing dissi- pation: metriplectic4-bracketdynamics, Phys. Rev. E 109 (2024) 045202

    work page 2024

  16. [16]

    A. N. Kaufman, P. J. Morrison, Algebraic structure of the plasma quasilinear equations, Phys. Lett. A 88 (1982) 405–406

    work page 1982

  17. [17]

    A. N. Kaufman, Dissipative Hamiltonian sys- tems: A unifying principle, Phys. Lett. A 100 (1984) 419

    work page 1984

  18. [18]

    J. M. Grmela, Particle and bracket formula- tions of kinetic equations, Contemp. Math. 28 (1984) 125

    work page 1984

  19. [19]

    Zaidni, P

    A. Zaidni, P. J. Morrison, S. Benjelloun, Thermodynamically consistent Cahn-Hilliard- Navier-Stokes equations using the metriplec- tic dynamics formalism, Physica D 468 (2024) 134303

    work page 2024

  20. [20]

    Zaidni, P

    A. Zaidni, P. J. Morrison, Metriplectic four- bracket algorithm for constructing thermody- namically consistent dynamical systems, Phys. Rev. E 112 (2025) 025101

    work page 2025

  21. [21]

    N. Sato, P. J. Morrison, Scattering theory in noncanonical phase space: A drift-kinetic col- lision operator for weakly collisional plasmas, Phys. Plasmas 32 (2025) 102306. 8

    work page 2025

  22. [22]

    Metriplectic relaxation to equilibria

    C. Bressan, M. Kraus, O. Maj, P. J. Morri- son, Metriplecticrelaxationtoequilibria, Com- mun. Nonlinear Sci. Numer. Simul. To appear. (2026).arXiv:2506.09787v2

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  23. [23]

    V. I. Arnold, Mathematical methods of classi- cal mechanics, Vol. 60 of Grad. Texts in Math., Springer-Verlag, New York, 1989, xvi+508 pp

    work page 1989

  24. [24]

    J. M. Grmela, Contact geometry of mesoscopic thermodynamics and dynamics, Entropy 16 (2014) 1652–1686

    work page 2014

  25. [25]

    O.Esen, J.M.Grmela, M.Pavelka, Ontherole of geometry in statistical mechanics and ther- modynamics I: geometric perspective, J. Math. Phys. 63 (2022) 122902

    work page 2022

  26. [26]

    O. Esen, J. M. Grmela, M. Pavelka, On the role of geometry in statistical mechanics and thermodynamics II: thermodynamic perspec- tive, J. Math. Phys. 63 (2022) 123305

    work page 2022

  27. [27]

    Do, Y.-G

    H.-S. Do, Y.-G. Oh, Thermodynamic reduc- tion of contact dynamics (2025).arXiv:2412. 19319v2

    work page 2025

  28. [28]

    R. S. Kulkarni, On the Bianchi identities, Math. Ann. 199 (199) 175–204

    work page

  29. [29]

    Nomizu, On the decomposition of gener- alized curvature tensor fields, in: M

    K. Nomizu, On the decomposition of gener- alized curvature tensor fields, in: M. Obata, S. Kobayashi, T. Takahashi (Eds.), Differen- tial Geometry, papers in Honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 335–345

    work page 1972

  30. [30]

    Weinstein, The local structure of Poisson manifolds, J

    A. Weinstein, The local structure of Poisson manifolds, J. Differ. Geom. 18 (1983) 523–557, Errata and Addenda, 22, 225 (1985)

    work page 1983

  31. [31]

    P. J. Morrison, Hamiltonian description of the idealfluid, Rev.Mod.Phys.70(1998)467–521

    work page 1998

  32. [32]

    De Leon, M

    M. De Leon, M. Lainz Valcázar, Contact Hamiltonian systems, J. Math. Phys. 60 (2019) 102902

    work page 2019

  33. [33]

    Y.-G. Oh, S. Yu, Contact instantons with Legendrian boundary condition: a priori esti- mates, asymptotic convergence and index for- mula, Internat. J. Math. 35 (2024) 2450019

    work page 2024

  34. [34]

    G. Duffing, Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technis- che Bedeutung [Forced oscillations with vari- able natural frequency and their technical rel- evance (in German) Heft 41/42, braunschweig: Vieweg, vi+134 pp (1918)

    work page 1918

  35. [35]

    J. M. T. Thompson, H. B. Stewart, Nonlin- ear Dynamics and Chaos, John Wiley & Sons., 2002, p. 66. 9

    work page 2002