Irreversible dynamics on Poisson manifolds

Erwin Luesink; Erwin Luesink

doi:10.3934/cam.2026031

Communications in Analysis and Mechanics

2026, Volume 18, Issue 4: 749-782. doi: 10.3934/cam.2026031

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Research article

Irreversible dynamics on Poisson manifolds

Erwin Luesink ^{1,2
,
,}

1.
Korteweg-de Vries Institute of Mathematics, University of Amsterdam, Amsterdam, The Netherlands
2.
Mathematical Institute, Utrecht University, Utrecht, The Netherlands

Received: 10 September 2025 Revised: 22 May 2026 Accepted: 15 September 2026 Published: 12 October 2026
37J06, 53D17, 70G45, 70H33, 82C35

We present a geometric construction of irreversible dynamics on Poisson manifolds that satisfies the axioms of metriplectic mechanics and the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework. Our approach relies solely on the underlying Poisson structure and its deformation theory, without requiring any additional metric structure. Specifically, we show that if the second Lichnerowicz–Poisson cohomology group of a Poisson manifold is nontrivial, one can construct a symmetric bracket that generates irreversible dynamics compatible with energy conservation and entropy production. This bracket is derived from a 2-cocycle that deforms the original Poisson structure, thereby modifying the associated Casimir foliation. We illustrate the construction with two finite-dimensional examples and one infinite-dimensional example: the duals of the Lie algebras of the special Euclidean group $ SE(2) $, the Galilei group $ SGal(3) $, and the group of diffeomorphisms over the circle. These examples demonstrate the applicability of the method in classical mechanics, control theory, and mathematical physics.
- irreversible dynamics,
- geometric mechanics,
- metriplectic mechanics,
- GENERIC,
- Poisson manifolds
Citation: Erwin Luesink. Irreversible dynamics on Poisson manifolds[J]. Communications in Analysis and Mechanics, 2026, 18(4): 749-782. doi: 10.3934/cam.2026031

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Abstract

We present a geometric construction of irreversible dynamics on Poisson manifolds that satisfies the axioms of metriplectic mechanics and the general equation for nonequilibrium reversible-irreversible coupling (GENERIC) framework. Our approach relies solely on the underlying Poisson structure and its deformation theory, without requiring any additional metric structure. Specifically, we show that if the second Lichnerowicz–Poisson cohomology group of a Poisson manifold is nontrivial, one can construct a symmetric bracket that generates irreversible dynamics compatible with energy conservation and entropy production. This bracket is derived from a 2-cocycle that deforms the original Poisson structure, thereby modifying the associated Casimir foliation. We illustrate the construction with two finite-dimensional examples and one infinite-dimensional example: the duals of the Lie algebras of the special Euclidean group $ SE(2) $, the Galilei group $ SGal(3) $, and the group of diffeomorphisms over the circle. These examples demonstrate the applicability of the method in classical mechanics, control theory, and mathematical physics.

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