An Analytical Mechanics Approach to the First Law of Thermodynamics and Construction of a Variational Hierarchy

Hamid A Said

REVIEW 4 major objections 4 minor 27 references

An Analytical Mechanics Approach to the First Law of Thermodynamics and Construction of a Variational Hierarchy

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the balance of energy with heat dissipation in one dimension can be written as a wave equation for the total energy, giving a least-action principle for the first law and an infinite hierarchy of variational…

desk verdict The isentropic trick is fine but old; the dissipative least-action claim breaks on a missing initial condition that compact support cannot supply. read the letter →

arxiv 1908.03062 v2 pith:2VFODLSD submitted 2019-08-06 physics.class-ph cond-mat.mtrl-sci

Hamid A Said This is my paper · ORCID

classification physics.class-phcond-mat.mtrl-sci MSC 37K0580M30

keywords leastactionprinciplefirstlawofthermodynamicsdissipationthermoelasticityvariationalhierarchyHamiltonianformalismNoether'stheoremheatequation

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 sets out to show that the first law of thermodynamics with heat dissipation, in one space dimension, has a genuine least-action principle of the same family as Hamilton's principle. The trick is to stop treating temperature as the fundamental field and instead treat the total energy $I=e+v^2/2$ as the variational variable, so that the energy balance becomes a second-order hyperbolic wave equation. The stationary points of the action $\int\!\int \sigma\,dx\,dt$ with $\sigma=\frac{\rho_0}{2}(\partial_t I)^2-\frac{c}{2}(\partial_x I)^2-\dot D\,\partial_x I$ are then exactly the solutions of the dissipative first law. If this is right, thermoelastic heat conduction stops being the standard counterexample to variational mechanics and acquires a Hamiltonian, brackets, Noether laws, and an infinite hierarchy of new variational principles.

What carries the argument

The central object is the total energy field $I$, promoted from a derived quantity to the independent field of a variational principle. The calculation that carries the argument is the flux-rate identity $\partial_t(vS-q)=c\,\partial_x I+\dot D$, because it makes the first law equivalent to a wave equation rather than a parabolic one and thereby gives the action a symmetric operator. The action density $\sigma=\frac{\rho_0}{2}(\partial_t I)^2-\frac{c}{2}(\partial_x I)^2-\dot D\,\partial_x I$ is the analogue of a Lagrangian, and its Legendre transform produces a Hamiltonian density $\pi$ whose integral is the conserved energy $\Pi$. The variational hierarchy is generated iteratively by $u_{i+1}=\frac{\rho_0}{2}(\partial_t u_i)^2+\frac{c}{2}(\partial_x u_i)^2$, which maps each solution of the wave equation to the next one while preserving the least-action structure.

What would settle it

Take a 1D linear thermoelastic rod with initial data for which $\rho_0\theta_0\,\partial s/\partial t+\partial q/\partial x$ is nonzero at $t=0$, for instance a temperature profile whose second spatial derivative does not vanish initially. The paper's Euler-Lagrange equation, $\partial_t(\rho_0\theta_0\,\partial s/\partial t+\partial q/\partial x)=0$, then keeps that imbalance constant in time, so the solution cannot satisfy the standard thermoelastic heat equation $\rho_0 c_0\,\partial_t\theta-\rho_0\gamma\theta_0\,\partial^2_{xt}u-k\,\partial_x^2\theta=0$. Observing such a solution would falsify the claim that the action recovers the first law without an additional initial-data restriction.

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Extended reading notes

Core claim

At the center of the paper is a calculation of the rate of change of the energy flux $G=vS-q$. For the thermoelastic constitutive relations used, this rate splits as $\partial_t G=c\,\partial_x I+\dot D$, with $c=\partial S/\partial(\partial_x u)$ and $\dot D$ collecting the entropy-production and heat-flux terms. Combined with the balance law $\rho_0\partial_t I=\partial_x(vS-q)$, this turns the first law into the nonhomogeneous wave equation $\rho_0\partial_{tt}I-\partial_x(c\,\partial_x I)-\partial_x\dot D=0$. The paper's Theorem 1 states that the actual evolution of $I$ coincides with the stationary points of the functional $\int_0^\tau\int_B \sigma\,dx\,dt$, and Section 6 extends this to the dissipative case as a least-action principle. For a linear thermoelastic body the Euler-Lagrange equation reduces to $\partial_t(\rho_0\theta_0 s_t+q_x)=0$, which the paper reads as the classic entropy balance; with compact support it recovers the standard thermoelastic diffusion equation and, without mechanical effects, the heat equation.

Load-bearing premise

The whole dissipative example depends on assuming that, at the initial time, the combination $\rho_0\theta_0\,\partial s/\partial t+\partial q/\partial x$ vanishes; compact support of the fields does not imply this, and if the combination starts nonzero the action only conserves the imbalance instead of enforcing the classic thermoelastic energy equation.

Editorial extensions

If this is right

Energy balance with heat dissipation in one dimension acquires the full formal apparatus of analytical mechanics: Hamilton's equations, a canonical Poisson bracket on the energy-power pair, and an energy-momentum tensor.
For linear thermoelasticity the same action produces the coupled thermoelastic diffusion equation, and setting the mechanical field to zero recovers the classical heat equation as a limiting case.
The iterative construction yields infinitely many Lagrangians, each with its own least-action principle and its own conserved integrals $H_i$.
Noether's theorem applies at every level of the hierarchy, so the construction implies an infinite family of conservation laws rather than a single one.

Reading between the lines

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

Editorial inference: the mechanism is essentially one-dimensional, because the flux-rate identity picks up extra terms in higher dimensions; a genuine 3D version would need an additional way to express those terms through gradients of $I$ or new fields.
Editorial inference: the same two-step pattern—write a balance law, differentiate its flux, and identify the result as a gradient of the conserved field—could generate analogous variational hierarchies for other 1D conservation laws, such as mass or momentum transport, wherever constitutive relations make the flux rate close.
Editorial inference: a direct numerical test on the linear heat equation with nonzero initial flux divergence would separate the paper's conservation-law statement from the standard heat equation; the action's stationary points should preserve the imbalance rather than dissipate it.

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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

Quick take: the isentropic derivation (eqs. 2.1–2.5) is internally consistent, and the hierarchy in Theorem 3 is a correct induction, but it repackages the known fact that the energy density of a wave-equation solution satisfies the same wave equation. The advertised dissipative least-action principle does not hold up as stated.

The problem is at Eq. (6.10). The Euler–Lagrange equation from Σ_h gives ∂t(ρ0θ0 s_t + q_x) = 0, i.e., conservation of the bracket, not the bracket itself. To recover the classic thermoelastic equation you need Φ(x,0)=0, an initial condition that is never stated. The paper says compact support on B suffices, but that only gives Φ(·,t) ∈ C_c(B), not Φ ≡ 0. The counterexample in the stress-test note is exactly right: take u=0, θ(x,t)=θ0+g(x) with g∈C_c^∞(B), g≠0. Then (6.10) holds and all fields have compact support, but θ does not satisfy the heat equation. So the central claim—'this is the least action principle in the dissipative case'—is not established.

There is also a secondary gap: the assertion that Ddot is functionally independent of I is stated without proof. In the linearized example it is finessed, but in general it looks false, since I includes v and s. This matters because the action (6.4) treats Ddot as a fixed source term.

What the paper does well: the conservative case is a clean, correct observation, and the Hamiltonian, bracket, and Noether machinery are applied carefully. The hierarchy proof is pedagogically nice, though not new in substance.

Who is this for? Someone tracking variational principles for dissipative continua, but they will need to see past the overclaim. The isentropic part could be a useful note; the dissipative part needs either a proper initial condition or a different formulation.

For peer review: I would send it to a referee, because the claim is important and the error is subtle enough that a careful reader might catch it. But my own verdict is reject as is. If the author can fix the integration-constant issue and prove the independence of Ddot, there is a salvageable core.

Referee Report

4 major / 4 minor

Summary. The paper proposes a variational formulation for the first law of thermodynamics in one space dimension. The author starts from the balance of energy, computes the rate of change of the energy flux, and shows that in the isentropic case the total energy I satisfies a second-order wave equation. An action functional Σ is written whose Euler–Lagrange equations are this wave equation, and the author develops the associated Lagrangian–Hamiltonian theory, Poisson brackets, Noether symmetries, and an infinite hierarchy of variational principles obtained by iterating the construction. In the dissipative case, the author adds a term ˙D∂xI to the action, where ˙D is treated as independent of I, and claims that the resulting stationarity condition recovers the classical thermoelastic energy equation and, as a special case, the heat equation. A modified Noether analysis for the dissipative case is also presented, together with a comparison with Onsager's least-dissipation principle.

Significance. If the central claim were correct, the paper would supply a Hamilton-type least-action principle for the dissipative first law of thermodynamics and an infinite family of interlocked variational principles, which would be a notable contribution to the variational mechanics of dissipative continua. The isentropic construction in Sections 2–5 is internally coherent: the derivation of the wave equation for I from the energy-flux rate is correct, the induction in Theorem 3 is valid, and Corollaries 1 and 2 are standard consequences of the setting. However, the advertised dissipative least-action principle is not established. In both the isentropic and dissipative formulations, the Euler–Lagrange equation is only the time derivative of the physical balance law, not the balance law itself, and the paper does not supply the additional initial condition needed to recover the first law. The specific claim that compact support resolves this gap for the thermoelastic example is false. The paper is therefore significant in its isentropic hierarchy construction, but the main dissipative claim fails.

major comments (4)

[§2 (Eq. (2.4)) and §6 (Eq. (6.10))] The action is built for a twice-differentiated form of the energy balance, not for the balance itself. In the isentropic case, the Euler–Lagrange equation (2.4) is obtained by differentiating the first law (2.1) with respect to time, so it is equivalent to ∂t[ρ0∂tI−∂x(vS)]=0; in the linear dissipative example, Eq. (6.10) states ∂t[ρ0c0θt−ρ0γθ0uxt−k∇²θ]=0. In both cases the original physical law is the statement that the bracketed quantity vanishes, while stationarity only asserts that this quantity is time-independent. Consequently the action admits solutions that are not evolutions of the first law unless the initial data satisfy the first law. This is a systematic gap: the least-action principle is for the differentiated equation, not for the balance of energy.
[§6, after Eq. (6.10)] The passage 'If we assume the fields θ and u have compact support on B then the classic evolution-diffusion equation of thermoelasticity can be readily recovered' is incorrect. From ∂tΦ=0 with Φ=ρ0c0θt−ρ0γθ0uxt−k∇²θ one obtains Φ(x,t)=Φ(x,0); compact support of θ and u implies only that Φ(·,t) is compactly supported for each t, not that Φ≡0. A concrete counterexample is u=0, θ(x,t)=θ0+g(x) with 0≠g∈C_c^∞(B). Then ∂tΦ=0 holds and the fields have compact support, but θ does not satisfy the heat equation θt=αθxx. Recovering the heat or thermoelastic equation requires the extra initial condition Φ(x,0)=0, which is not a consequence of the variational principle. This invalidates the advertised least-action principle for the dissipative first law.
[§6, Eq. (6.4) and Remark 5(i)] The dissipative functional Σ contains the term −˙D∂xI with ˙D declared functionally independent of I. The variation is taken only with respect to I, so the action does not determine the dissipative mechanism; it is an action for I with a prescribed source. The coupled thermoelastic system is then described by two separate Euler–Lagrange equations, (6.6) and (6.7), not by a single action principle. This is weaker than the paper's claim that the first law in the dissipative case has a Hamilton-type least-action principle analogous to the conservative case, where the dynamics follows from the action alone.
[§6, linear example, after Eq. (6.9)] The statement 'in the linear approximation we have in fact δRh/δI=0' is not justified as written. The term Rh contains −c(∂xI)²/2+θ0c(∂s/∂x)(∂xI), whose variation with respect to I does not vanish identically; it vanishes only after substituting the leading-order relation ∂xI≈θ0∂s/∂x and discarding terms at the appropriate order. Since this step is needed to reduce the Euler–Lagrange equations to Eq. (6.10), it should be stated and proved explicitly. Even with this step granted, the second major comment above shows that the resulting equation is still only the time derivative of the desired balance law.

minor comments (4)

[§6, Eq. (6.10)] Equation (6.10) uses ∇²θ even though the paper has restricted attention to one space dimension at the outset; it should be θxx.
[§5, Eq. (5.5)] In the displayed calculation of ∂t(∂tui c∂xui), the intermediate line is missing the factor 1/2 on the terms ρ0(∂tui)² and c(∂xui)²; the final expression for ui+1 is correct, but the intermediate formula is misleading.
[Throughout] There are several typographical errors that should be corrected: 'Theroem' in Remark 3(ii), 'charechtrized' in the lead-in to the example, 'conversation' in the heading of Table 1, and '∂u2/∂t∂x' for ∂²u/∂t∂x in the line following Eq. (6.1).
[§2, Theorem 1] Theorem 1 is stated without a precise domain for I or the boundary conditions used in the integration by parts; Remark 1(iv) mentions variations compact in space and time, but the theorem itself should make this explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the action functionals are constructed from the differentiated energy balance, and the questionable compact-support inference is a correctness gap, not a circularity.

full rationale

The paper's main derivation chain is: use the balance of momentum and the constitutive laws to compute the time derivative of the energy flux, combine that with the first law to obtain a second-order PDE for the total energy I, and then write the standard quadratic action whose Euler-Lagrange equation reproduces that PDE. This is a Lagrangian construction for an already-derived equation; no parameter is fitted to data, no empirical quantity is renamed as a prediction, and no self-cited uniqueness theorem is invoked to force the choice of action. The only self-citation, [10], appears in a historical survey and is not load-bearing. The dissipative example does contain a serious mathematical gap: the Euler-Lagrange equation of the action (6.4) is the time-differentiated thermoelastic energy balance, Eq. (6.10), and the paper's claim that compact support 'readily recovers' the classic equation is an unproved assertion about an integration constant. A time-independent compactly supported temperature field satisfies (6.10) but not the heat equation, so the advertised dissipative least-action principle is not established as stated. That is a correctness issue, however, not a circular reduction: the target equation was not used to define the action, nor was it smuggled in through fitted parameters or self-referential citations. The variational hierarchy in Theorem 3 is an algebraic induction from the previous Euler-Lagrange equation and is self-contained. Overall, no step reduces by construction to its own input, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new physical entities, forces, or particles are introduced. The assumptions are constitutive restrictions, the constancy of c, and the hidden independence and initial-data assumptions in the dissipative section. These are the load-bearing premises beyond standard calculus of variations.

assumptions (5)

domain assumption Thermoelastic constitutive relations (2.2): e=e(∂x u,s), S=ρ0 ∂e/∂(∂x u), θ=∂e/∂s, q=-k ∂θ/∂x
Defines the material class; the action and Euler-Lagrange equations are derived only for this class, not general continua.
domain assumption For Sections 4-5, elastic modulus c=∂S/∂(∂x u) is constant
Used to define the symmetric tensor B^{μη} and to prove the hierarchy with constant coefficients; the non-constant case is postponed to Appendix A and not fully worked out.
ad hoc to paper The dissipation term ˙D is functionally independent of the total energy I
Required for the Euler-Lagrange equation (6.6) to have the stated form; the paper asserts this in Remark 5-i without proof, but I depends on displacement and entropy, which also enter ˙D.
ad hoc to paper Initial data satisfy F(x,0)=0 for F=ρ0 c0 θ_t - ρ0γθ0 u_xt - k∇²θ
Needed to pass from ∂t F=0 to the classic thermoelastic heat equation. Not stated, and compact support does not imply the bracket is zero.
standard math Standard compact-support variation and symmetry of the weak operator A
Invoked in Remark 1-i and throughout for the Euler-Lagrange and Noether arguments; reasonable in this setting.

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Cite this review

Pith. "Pith review of An Analytical Mechanics Approach to the First Law of Thermodynamics and Construction of a Variational Hierarchy." pith.science (2026). https://pith.science/paper/2VFODLSD

@misc{pith2026190803062,
  author       = {Pith},
  title        = {Pith review of: An Analytical Mechanics Approach to the First Law of Thermodynamics and Construction of a Variational Hierarchy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2VFODLSD}},
  note         = {Machine review of arXiv:1908.03062}
}

read the original abstract

A simple procedure is presented to study the conservation of energy equation with dissipation in continuum mechanics in 1D. This procedure is used to transform this nonlinear evolution-diffusion equation into a hyperbolic PDE; specifically, a second order quasi-linear wave equation. An immediate implication of this procedure is the formation of a least action principle for the balance of energy with dissipation. The corresponding action functional enables us to establish a complete analytic mechanics for thermomechanical systems: a Lagrangian-Hamiltonian theory, integrals of motion, bracket formalism, and Noether's theorem. Furthermore, we apply our procedure iteratively and produce an infinite sequence of interlocked variational principles, a variational hierarchy, where at each level or iteration the full implication of the least action principle can be shown again.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 26 canonical work pages

[1]

Classical mechanics

Gregory RD. Classical mechanics. Cambridge University Press; 2 006

work page
[2]

Variational principles of continuum mechanics: I

Berdichevsky V. Variational principles of continuum mechanics: I . Fundamentals. Springer Science & Business Media; 2009

work page 2009
[3]

Theory of Sound

Rayleigh L. Theory of Sound. vol. 1 (reprinted 1945 by Dover, Ne w York). Macmillan, London; 1945

work page 1945
[4]

Reciprocal relations in irreversible processes

Onsager L. Reciprocal relations in irreversible processes. I. Ph ysical review. 1931;37(4):405

work page 1931
[5]

Thermoelasticity and irreversible thermodynamics

Biot MA. Thermoelasticity and irreversible thermodynamics. Jour nal of Applied Physics. 1956;27(3):240–253

work page 1956
[6]

Variational principles in heat transfer: a uniﬁed Lagrang ian analysis of dissipative phenomena

Biot MA. Variational principles in heat transfer: a uniﬁed Lagrang ian analysis of dissipative phenomena. BIOT (MA) NEW YORK; 1970

work page 1970
[7]

Beyond equilibrium thermodynamics

¨Ottinger HC. Beyond equilibrium thermodynamics. John Wiley & Sons; 2 005

work page
[8]

Dissipative Hamiltonian systems: a unifying principle

Kaufman AN. Dissipative Hamiltonian systems: a unifying principle. P hysics Letters A. 1984;100(8):419–422

work page 1984

Show all 27 references

[9]

A paradigm for joined Hamiltonian and dissipative syst ems

Morrison PJ. A paradigm for joined Hamiltonian and dissipative syst ems. Physica D: Nonlinear Phe- nomena. 1986;18(1-3):410–419

1986
[10]

A Lagrangian–Hamiltonian uniﬁed formalism for a class of dis sipative systems

Said H. A Lagrangian–Hamiltonian uniﬁed formalism for a class of dis sipative systems. Mathematics and Mechanics of Solids. 2019;24(4):1221–1240

2019
[11]

Variational principles for linear initial-value problems

Gurtin M. Variational principles for linear initial-value problems. Qu arterly of Applied Mathematics. 1964;22(3):252–256

1964
[12]

A variational formulation of the coup led thermo-mechanical boundary- value problem for general dissipative solids

Yang Q, Stainier L, Ortiz M. A variational formulation of the coup led thermo-mechanical boundary- value problem for general dissipative solids. Journal of the Mechan ics and Physics of Solids. 2006;54(2):401–424

2006
[13]

Variational integrators a nd the Newmark algorithm for conser- vative and dissipative mechanical systems

Kane C, Marsden JE, Ortiz M, West M. Variational integrators a nd the Newmark algorithm for conser- vative and dissipative mechanical systems. International Journa l for Numerical Methods in Engineering. 2000;49(10):1295–1325

2000
[14]

The Euler-Poin car´ e equations and double bracket dissipation

Bloch A, Krishnaprasad P, Marsden JE, Ratiu TS. The Euler-Poin car´ e equations and double bracket dissipation. Communications in mathematical physics. 1996;175(1):1 –42

1996
[15]

A Lagrangian variational formulatio n for nonequilibrium thermodynamics

Gay-Balmaz F, Yoshimura H. A Lagrangian variational formulatio n for nonequilibrium thermodynamics. Part II: continuum systems. Journal of Geometry and Physics. 2 017;111:194–212
[16]

Classical Mechanics of Nonconservative Systems

Galley CR. Classical Mechanics of Nonconservative Systems. Ph ys Rev Lett. 2013 Apr;110:174301. Available from: https://link.aps.org/doi/10.1103/PhysRevLett.110.174301

2013 doi
[17]

Mechanics with fractional derivatives

Riewe F. Mechanics with fractional derivatives. Physical Review E. 1997;55(3):3581

1997
[18]

Canonical formulation and conservation laws of thermoelasticity without dissi- pation

Kalpakides V, Maugin G. Canonical formulation and conservation laws of thermoelasticity without dissi- pation. Reports on Mathematical Physics. 2004;53(3):371–391

2004
[19]

The method of weighted residuals and variational p rinciples

Finlayson BA. The method of weighted residuals and variational p rinciples. vol. 73. SIAM; 2013

2013
[20]

Thermoelasticity without energy dissipation

Green A, Naghdi P. Thermoelasticity without energy dissipation . Journal of elasticity. 1993;31(3):189– 208

1993
[21]

Heat waves

Straughan B. Heat waves. vol. 177. Springer Science & Busines s Media; 2011

2011
[22]

Mathematical foundations of elasticit y

Marsden JE, Hughes TJ. Mathematical foundations of elasticit y. Courier Corporation; 1994

1994
[23]

Mathematical analysis: foundations and a dvanced techniques for functions of several variables

Giaquinta M, Modica G. Mathematical analysis: foundations and a dvanced techniques for functions of several variables. Springer Science & Business Media; 2011

2011
[24]

Linearized elasticity as Γ-limit of ﬁnite elasticity

Dal Maso G, Negri M, Percivale D. Linearized elasticity as Γ-limit of ﬁnite elasticity. Set-Valued Analy- sis. 2002;10(2-3):165–183

2002
[25]

Extended irreversible the rmodynamics

Jou D, Casas-V´ azquez J, Lebon G. Extended irreversible the rmodynamics. In: Extended Irreversible Thermodynamics. Springer; 1996. p. 41–74. 20

1996
[26]

Non-equilibrium thermodynamics

Gyarmati I, et al. Non-equilibrium thermodynamics. Springer; 1 970
[27]

Hyperbolic conservation laws in continuum physics

Dafermos C. Hyperbolic conservation laws in continuum physics. vol. 3. Springer; 2005. 21

2005

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Classical mechanics

Gregory RD. Classical mechanics. Cambridge University Press; 2 006

work page

[2] [2]

Variational principles of continuum mechanics: I

Berdichevsky V. Variational principles of continuum mechanics: I . Fundamentals. Springer Science & Business Media; 2009

work page 2009

[3] [3]

Theory of Sound

Rayleigh L. Theory of Sound. vol. 1 (reprinted 1945 by Dover, Ne w York). Macmillan, London; 1945

work page 1945

[4] [4]

Reciprocal relations in irreversible processes

Onsager L. Reciprocal relations in irreversible processes. I. Ph ysical review. 1931;37(4):405

work page 1931

[5] [5]

Thermoelasticity and irreversible thermodynamics

Biot MA. Thermoelasticity and irreversible thermodynamics. Jour nal of Applied Physics. 1956;27(3):240–253

work page 1956

[6] [6]

Variational principles in heat transfer: a uniﬁed Lagrang ian analysis of dissipative phenomena

Biot MA. Variational principles in heat transfer: a uniﬁed Lagrang ian analysis of dissipative phenomena. BIOT (MA) NEW YORK; 1970

work page 1970

[7] [7]

Beyond equilibrium thermodynamics

¨Ottinger HC. Beyond equilibrium thermodynamics. John Wiley & Sons; 2 005

work page

[8] [8]

Dissipative Hamiltonian systems: a unifying principle

Kaufman AN. Dissipative Hamiltonian systems: a unifying principle. P hysics Letters A. 1984;100(8):419–422

work page 1984