Dirac structures and port-Lagrangian systems in thermodynamics

Fran\c{c}ois Gay-Balmaz; Hiroaki Yoshimura

arxiv: 1907.00373 · v1 · pith:BPZ4QMQQnew · submitted 2019-06-30 · 🧮 math-ph · math.MP

Dirac structures and port-Lagrangian systems in thermodynamics

Hiroaki Yoshimura , Fran\c{c}ois Gay-Balmaz This is my paper

Pith reviewed 2026-05-25 12:27 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Dirac structuresport-Lagrangian systemsnonequilibrium thermodynamicsvariational constraintsPontryagin bundlephenomenological constraintsLagrange-d'Alembert-Pontryagin principle

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The pith

Interconnections in thermodynamics for simple constrained systems are described by Dirac structures on the Pontryagin bundle and cotangent bundle induced from variational constraints.

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

The paper introduces port-Lagrangian systems in nonequilibrium thermodynamics by extending the concept from nonholonomic mechanics, where interconnections are encoded via Dirac structures. It derives this from a variational formulation that incorporates phenomenological constraints tied to the entropy production equation along with associated variational constraints. For simple thermodynamic systems with constraints, the approach shows that these interconnections arise as Dirac structures on the Pontryagin bundle and on the cotangent bundle of the thermodynamic configuration space. The structures are induced directly from the variational constraint, and the associated variational principle is the Lagrange-d'Alembert-Pontryagin principle. Concrete illustrations include a cylinder-piston system with ideal gas and an LCR circuit exhibiting resistor entropy production.

Core claim

The interconnections in thermodynamics can be described by Dirac structures on the Pontryagin bundle as well as on the cotangent bundle of the thermodynamic configuration space, each induced from the variational constraint. This formulation arises from a Lagrange-d'Alembert principle for a class of nonlinear nonholonomic constraints called phenomenological constraints, which are tied to the entropy production equation, with variational constraints imposed on the allowed variations.

What carries the argument

Dirac structures induced from the variational constraint on the Pontryagin bundle and the cotangent bundle of the thermodynamic configuration space.

If this is right

Phenomenological constraints linked to entropy production are treated as a specific class of nonlinear nonholonomic constraints within the port-Lagrangian framework.
Variational constraints are systematically derived from the phenomenological constraints and imposed on variations in the principle.
The same interconnections admit equivalent descriptions as Dirac structures on both the Pontryagin bundle and the cotangent bundle.
The variational structure of the system is recovered through the Lagrange-d'Alembert-Pontryagin principle.
The construction applies directly to examples including a cylinder-piston with ideal gas and an LCR circuit with resistive entropy production.

Where Pith is reading between the lines

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

Composing multiple such Dirac structures could model networks of interacting thermodynamic subsystems without additional ad-hoc junction conditions.
The geometric encoding might allow direct transfer of stability or passivity results from port-Hamiltonian mechanics to thermodynamic port-Lagrangian systems.
The same induction procedure from variational constraints could be tested on systems with time-dependent constraints or with explicit external ports.

Load-bearing premise

The variational formulation of nonequilibrium thermodynamics directly produces valid port-Lagrangian systems with phenomenological and variational constraints that generalize to Dirac structures.

What would settle it

A concrete calculation for the cylinder-piston example in which the Dirac structure induced from the variational constraint does not reproduce the entropy production equation or the correct power balance under the imposed constraints.

read the original abstract

In this paper, we introduce the notion of port-Lagrangian systems in nonequilibrium thermodynamics, which is constructed by generalizing the notion of port-Lagrangian systems for nonholonomic mechanics proposed in Yoshimura and Marsden [2006c], where the notion of interconnections is described in terms of Dirac structures. The notion of port-Lagrangian systems in nonequilibrium thermodynamics is deduced from the variational formulation of nonequilibrium thermodynamics developed in Gay-Balmaz and Yoshimura [2017a,2017b]. It is a type of Lagrange-d'Alembert principle associated to a specific class of nonlinear nonholonomic constraints, called phenomenological constraints, which are associated to the entropy production equation of the system. To these phenomenological constraints are systematically associated variational constraints, which need to be imposed on the variations considered in the principle. In this paper, by specifically focusing on the cases of simple thermodynamic systems with constraints, we show how the interconnections in thermodynamics can be also described by Dirac structures on the Pontryagin bundle as well as on the cotangent bundle of the thermodynamic configuration space. Each of these Dirac structures is induced from the variational constraint. Furthermore, the variational structure associated to this Dirac formulation is presented in the context of the Lagrange-d'Alembert-Pontryagin principle. We illustrate our theory with some examples such as a cylinder-piston with ideal gas as well as an LCR circuit with entropy production due to a resistor.

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 extends the authors' 2017 variational setup to define port-Lagrangian systems in thermodynamics via Dirac structures on Pontryagin and cotangent bundles, with two worked examples.

read the letter

The core move is to take the phenomenological constraints from nonequilibrium thermo, associate variational constraints to them, and induce Dirac structures that encode the interconnections. They do this for simple systems and spell out the Lagrange-d'Alembert-Pontryagin principle that goes with it. The cylinder-piston and LCR-circuit examples make the abstract construction concrete and show how entropy production enters the picture through the resistor or the piston friction term. That part is clear and follows the logic they laid out earlier. The work sits squarely inside their own line of papers, so the new piece is really the thermodynamic application rather than a standalone result. The abstract and stress-test give no sign of internal contradictions or broken Dirac axioms, and the examples are used to illustrate rather than claim full generality. The main limitation is the dependence on the 2017 variational foundation; anyone not already familiar with those papers will need to read them first. No independent checks against other thermodynamic formalisms appear. This is for people already working in geometric mechanics and port-Hamiltonian or Dirac approaches to thermo. It is narrow but the construction is consistent enough that a serious editor should send it out for refereeing rather than desk-reject it.

Referee Report

0 major / 3 minor

Summary. The paper introduces port-Lagrangian systems in nonequilibrium thermodynamics by generalizing the port-Lagrangian framework of Yoshimura and Marsden (2006) for nonholonomic mechanics. The construction is deduced from the variational formulation of nonequilibrium thermodynamics in the authors' prior work (Gay-Balmaz and Yoshimura, 2017a,b), employing a Lagrange-d'Alembert principle with phenomenological constraints (tied to entropy production) and associated variational constraints. For simple thermodynamic systems with constraints, the paper shows that interconnections are described by Dirac structures on the Pontryagin bundle and on the cotangent bundle of the thermodynamic configuration space, each induced from the variational constraint. The associated variational structure is formulated via the Lagrange-d'Alembert-Pontryagin principle. The theory is illustrated with examples of a cylinder-piston system with ideal gas and an LCR circuit with resistor entropy production.

Significance. If the constructions are valid, the work supplies a geometric port-based framework that unifies the description of constrained thermodynamic systems and their interconnections via Dirac structures on natural bundles, extending mechanical port-Lagrangian theory to nonequilibrium thermodynamics. This could support systematic modeling of networks with entropy production. The paper explicitly builds on and cites the 2017 variational results and supplies two concrete examples that instantiate the Dirac structures and the principle for both mechanical-thermodynamic and electro-thermodynamic cases.

minor comments (3)

The introduction would benefit from a short paragraph contrasting the proposed Lagrangian-Dirac approach with existing port-Hamiltonian formulations of thermodynamics (e.g., those based on contact or symplectic structures) to clarify the specific contribution of the variational-constraint induction.
[Examples] In the LCR-circuit example, the explicit coordinate expression linking the resistor's entropy-production term to the phenomenological constraint (and thence to the variational constraint) should be written out with an equation number, as the abstract only states the association without the local formula.
Notation for the thermodynamic configuration space and its bundles is introduced without a preliminary table or diagram; adding one would improve readability when the Pontryagin and cotangent constructions are compared.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The recommendation for minor revision is noted. No specific major comments were provided in the report, so we have no individual points to address.

Circularity Check

1 steps flagged

Port-Lagrangian systems and Dirac structures deduced from authors' prior 2017 variational formulation

specific steps

self citation load bearing [Abstract]
"The notion of port-Lagrangian systems in nonequilibrium thermodynamics is deduced from the variational formulation of nonequilibrium thermodynamics developed in Gay-Balmaz and Yoshimura [2017a,2017b]."

The central object (port-Lagrangian systems with phenomenological/variational constraints) is defined by direct deduction from the authors' overlapping prior work rather than derived from first principles or external data in this paper; the Dirac structure induction is then built on top of this self-referential base.

full rationale

The paper's core construction of port-Lagrangian systems in thermodynamics is explicitly stated to be deduced from the variational formulation in the authors' own prior papers [Gay-Balmaz and Yoshimura 2017a,2017b]. This creates a self-citation load-bearing step for the foundational premise. The subsequent induction of Dirac structures on Pontryagin/cotangent bundles from variational constraints is presented as a generalization, with examples supplied, but the central claim reduces to extending their own prior definitions without external benchmarks or independent verification shown. This warrants a moderate circularity score but not a full reduction to definition, as the Dirac induction step adds specific content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the authors' prior variational formulation as a domain assumption and introduces a new modeling entity without external falsifiable evidence.

axioms (1)

domain assumption Variational formulation of nonequilibrium thermodynamics from Gay-Balmaz and Yoshimura [2017a,2017b]
Port-Lagrangian systems are deduced from this formulation as stated in the abstract.

invented entities (1)

port-Lagrangian systems in nonequilibrium thermodynamics no independent evidence
purpose: To describe interconnections via Dirac structures on thermodynamic bundles
New notion introduced by generalizing the mechanics version to thermodynamics with entropy constraints.

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discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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

Dirac structures on the Pontryagin bundle as well as on the cotangent bundle ... induced from the variational constraint

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