An Analytical Mechanics Approach to the First Law of Thermodynamics and Construction of a Variational Hierarchy
aa r X i v : . [ phy s i c s . c l a ss - ph ] A ug A Principle of Least Action for the First Law ofThermodynamics in Rational Mechanics
HAMID SAID ∗ Department of Mathematics, Kuwait University, PO Box 5969, Safat 13060, Kuwait
Abstract
A new theoretical procedure is presented to study the conservationof energy equation with dissipation in continuum mechanics in 1D. Thisprocedure is used to transform this nonlinear evolution-diffusion equationinto a hyperbolic PDE; specifically, a second order quasi-linear wave equa-tion. An immediate implication of this procedure is the formation of aleast action principle for the balance of energy with dissipation. The cor-responding action functional enables us to establish a complete analyticalmechanics for thermomechanical systems: a Lagrangian-Hamiltonian the-ory, bracket formalism, and Noether’s theorem. Furthermore, we applyour procedure iteratively and produce an infinite sequence of interlockedvariational principles, a variational hierarchy , where at each level or iter-ation the full implication of the least action principle can be again shown.Finally we offer comments on the implication of our work to field theoriesin general.
Keywords:
Continuum mechanics, energy conservation, first law ofthermodynamics, least action principle, dissipation, variational hierarchy
Hamilton’s principle is, undoubtedly, one of the great insights of physics. Whilehistorically it was formulated in the context of classical mechanics [1], it has beenremarkably extended to other field theories such as fluid mechanics, electromag-netism, general relativity and various quantum field theories. It is well-knownhowever that the many consequences of this principle–such as Lagrangian me-chanics, Hamilton’s equations, and Noehter’s theorem– apply only to conserva-tive systems and can not capture the irreversible effects of a general dissipativesystem, such as the diffusion of heat.The purpose of this work is to construct an action functional whose station-ary points satisfies the (nonlinear) conservation of energy equation with heatdissipation in one spatial dimension. In other words, we look to formulate astationary principle for the first law of thermodynamics (as formulated in ratio-nal mechanics) analogous to Hamilton’s principle of stationary action. This will ∗ e-mail: [email protected] B ⊆ R n ∂ u∂t − c ∇ u = , (⃗ x, t ) ∈ B × ( , ∞) (1.1)where the scalar c denotes the speed of propagation of the wave. The La-grangian L for the above PDE is the difference between the kinetic energy andthe (potential) strain energy L = ∫ B ( ∂u∂t ) − c ∣ ∇ u ∣ d B (1.2)Then formally by Hamilton’s principle of least action δ ∫ τ L dt = u to (1.1) in [ , τ ] corresponds to the stationary points of ∫ τ L dt . Once L is defined we canrewrite the wave equation in Euler-Lagrange form ∂∂t ( δ L δ ˙ u ) − δ L δu = u has compact support on B (or decays sufficiently fast when B = R n )then the total energy of the system is conserved: ∂ t H =
H = ∫ B ( ∂u∂t ) + c ∣ ∇ u ∣ d B (1.5)These classical results, however, do not have a counterpart for the classic heatequation ∂u∂t − α ∇ u = α is the thermal diffusivity, and the function u here representsthe temperature field. In fact, it is shown [2] that no action exists in the formof ∫ B L (⃗ x, t, ∂ t u, ∇ u ) d B (1.7)such that (1.6) can be deduced as the Euler-Lagrange equations of funcionalhaving the form give in equation (1.7).As far as we can tell, the first successful attempt to include dissipative effectsinto the classsic variational framework dates back to Rayleigh in the end ofthe 19th century [3]. Rayleigh introduced, in addition to the Lagrangain, a dissipation function –a positive quadratic function in the velocities–to account2or friction in the system; this allowed him to extend Lagrange’s equationsof motion. In the 1950s M.A. Biot developed a variational formulation forthe equations of classical thermoelasticity by means of a modified free energy(referred to as Biot’s potential) and a dissipation function. However, Biot’svariational formulation was not given in terms of a single action such as inHamilton’s principle; rather he formulated a quasi-variational principle in-whichhe does not consider the total variation of the dissipation function–only theproduct of its derivatives with the appropriate infinitesimal variation [4, 5].Since then many have sought to uncover variaitonal formulations for dissipativecontinua. For instance in certain cases [6, 7, 8, 9] researchers were successful inextending the classic Lagrangian and/or Hamiltonian formulations to include ef-fects of entropy production, while falling-short of constructing a unified action.Others [10, 11] were able to formulate action integrals for evolution-diffusionequations. However, these functionals do not have the simple form of a densityfunction as does the Lagrangian, rather they are complicated expressions givenin terms of convolutions of a one parameter integral, and it is not obvious howthey would fit a Lagrangian-Hamiltonian framework. Dissipation has also beenincorporated into various variational schemes via the Lagrange-d’Alembert prin-ciple (see for example [12, 13, 14]). For other possible extensions the reader isreferred to [15, 16]. Moreover, a Noether’s theorem was advanced in [17] for the-ory of nonlinear thermoelasticity without dissipation. But despite of progress inthis area, a unified extension of the variational formalism of analytical mechanicsto general dissipative systems remains still out of reach.In this paper we construct a new least action principle analogous to Hamilton’sprinciple by calculating the rate of change in the energy flux. This allows us towrite the conservation of energy equation as a second order hyperbolic PDE forthe total energy of the system in one space dimension. A myriad of consequencewill then follow: the hyperbolic PDE can be re-written as the Euler-Lagrangeequations in a new action we denote Σ. This produces a natural way of reveal-ing the symmetry that exists between the balance of energy and momentum,and as such the new least action principle follows without extraneous physicsassumptions. A Hamiltonian and bracket formalisms also follow from the Euler-Lagrange equations in a similar fashion to analytical mechanics. Furthermore,the symmetries leaving Σ invariant correspond to new conservation laws.A noteworthy consequence of the above procedure is that it gives ground forproducing a third functional, this time form the energy equation associatedwith Σ (i.e. Noether’s theorem under time invariance applied to Σ). In fact,we can carry this procedure indefinitely, that is we prove that our procedureis iterative giving rise to a hierarchy of variatoinal principles each constructedfrom the previous iteration. We, hence, obtain an infinite number of functionals(i.e. Lagrangians) and an (infinite) iterative scheme, and advance a completeanalytic mechanics at each iteration.The paper is organized as follows. We consider isentropic systems in Section2–5, which will set the foundation for tackling the dissipative case and construct-ing the variational hierarchy. We begin in Section 2 by considering the rate ofchange in the energy flux. We show that the total energy propagates accordingto a homogeneous wave equation, which can be derived as stationary points of a3ew functional Σ. In Section 3, we establish a unified Lagrangian-Hamiltonianformulation through the newly construed functional Σ and its Legendre trans-formation Π, respectively. Hamilton’s equations for the total energy are shownto be equivalent to a bracket formulation over a new phase space that consists ofthe energy-power pair. We establish in Section 4 Noether’s theorem: transfor-mations that leave functional Σ invariant result in conservation laws. As suchwe examine two groups of transformations: arbitrary translations in the totalenergy, and space-time translations. Invariance under the first corresponds tothe balance of energy. The second is the basis for constructing a second or-der tensor comparable to the energy-momentum tensor in classical field theory.Subsequently we obtain a number of new conservation laws, one of which gov-erns the evolution of density function π . One of our main results is containedin Section 4: the formulation of infinite hierarchy of variational principles (i.e.least action principles). This result is made possible because the basic procedureused in constructing Σ is iterative in nature; the Lagrangian at each iteration isformulated based on the field variables of the previous iteration. Therefore, ateach iteration we produce a complete variational analysis for the relevant fields.In Section 6 we look to apply our procedure to non-conservative systems, namelyto dissipative thermoelastic material. The effects of entropy production due toheat flow manifest itself as a non-homogeneous term in the hyperbolic PDEfor the total energy. Hence, a set of two coupled Euler-Lagrange equationsdetermines the complete evolution of thermoelastic materials. Specifically, thefunctional Σ for a linear thermoelastic body produces the classic energy equationin the theory of thermoelasticity, and as a special case the heat equation. More-over, a modified Noether’s theorem is given for the dissipative case. While wecan not produce conservation laws–because of the inhomogeneity in the balanceof energy–we obtain auxiliary equations which are fundamental for obtainingthe next iteration in the variational hierarchy. We comment, in Section 7, onthe implication of our procedure to general field theories which depend on freeinternal variables/fields. Finally, we offer concluding remarks in Section 8. In this section we show that the balance of energy equation in 1D corresponds tothe stationary points of a functional Σ. We commeence our construction by firstconsidering the isentropic problem. The scheme we employ for the constructionof said functional will naturally suggest a more general calculation, which weapply to the full (dissipative) problem in Section 6.The point of departure for us is to consider the first law of thermodynamics (i.e.the balance of energy) in one spatial dimension, which in material coordinatesreads ρ ∂I∂t = ∂∂x ( vS − q ) in B × [ , τ ] (2.1)Here, B ≐ ( a , a ) represents the reference configuration, and time τ >
0. Thequantity ρ is the material density; I = e + v S is the scalar-valued Piola-Kirchhoffstress; and q represents the heat flux across the boundary of B . For simplicity,4e have considered the balance of energy in the absence of body forces and heatsources (see Remark 2 and Appendix A).We further assume that our constitutive laws determine a classic thermoelasticmedium, that is e = e ( ∂ x u, s ) , S = ρ ∂e∂ ( ∂ x u ) , θ = ∂e∂s , q = q ( ∂ x θ ) (2.2)Here s is the entropy density, and θ is the absolute temperature (i.e. θ = θ + T where θ is constant temperature in the undeformed configuration). Equation(2.1) holds for solids as well as fluids. In case of fluids the pressure is given as S = − p , and the internal energy e (hence the pressure) is a state function of thedensity ρ associated with the motion through the relation ρ + ∂ x u = ρ .It is known that hyperbolic PDEs such as the wave equation posses a variationalstructure because they are associated with symmetric operators [18]. The bal-ance of energy equation on the other hand, supplemented by Fourier’s law forheat conduction, usually results in a nonlinear parabolic PDE in the tempera-ture field . In fact, even upon linearization, this equation does not posess anobvious variational form since it will ultimately correspond to a non-symmetricoperator [11]. We circumvent this impediment by showing that the total energy I satisfies a wave equation.We begin by considering the isentropic problem: q = , e = e ( ∂ x u ) , S = S ( ∂ x u ) , ∂ t s =
0. The rate of change of the energy flux F = vS reads ∂F∂t = S ∂v∂t + v ∂S∂t We substitute for the conservation of momentum equation (in the absence ofbody forces) to obtain ∂F∂t = S ρ ∂S∂x + v ∂S∂t = Sρ ∂S∂ ( ∂ x u ) ∂ ( ∂ x u ) ∂x + v ∂S∂ ( ∂ x u ) ∂ ( ∂ x u ) ∂t = ∂S∂ ( ∂ x u ) ( ∂e∂ ( ∂ x u ) ∂ ( ∂ x u ) ∂x + v ∂v∂x ) = ∂S∂ ( ∂ x u ) ∂∂x ( e + v ) = c ∂I∂x (2.3)where c ≐ ∂S∂ ( ∂ x u ) is the nonlinear elastic modulus. A departure from the classic constitutive laws can result in a wave-like equation for thetemperature field. See for example [19, 20]. ρ ∂ I∂t − ∂∂x ( c ∂I∂x ) = c ∂ I∂t − cρ ∂ I∂x = I , which wechoose to replace the classic balance of energy equation (2.1). The speed ofpropagation is well-defined and finite (for constant c ) and is equal to the speedof propagation of an elastic wave √ c / ρ . That the total energy is governedby a wave equation should not come as a surprise. The propagating elasticwave packet has an associated energy, which propagates with the wave into thematerial. So as long as u ≠ I )Σ ( I ) = ∫ B σ ( ∂ t I, ∂ x I ) dx ≐ ∫ B ( ρ ( ∂I∂t ) − c ( ∂I∂x ) ) dx (2.6)Therefore have obtained a least action principle analogous to Hamilton’s prin-ciple Theorem 1 (A least action principle) . The actual evolution of the total energy I in [ , τ ] coincides with the stationary points of the functional ∫ τ Σ dt (2.7) Remark 1. i A direct implication of Theorem 1 is that equations (2.4) and (2.5) can beput in Euler-Lagrange form, which we present in the next section.ii The symmetric operator associated with (2.4) or (2.5) (in weak form) canbe constructed as follows. Let V be some appropriate Hilbert space suchthat I ∈ V solves (2.4) or (2.5) (e.g. V = L ( , τ ; H ( B )) ), and define theoperator A ∶ V × V Ð → R A ( I , I ) = ∫ τ ∫ B ρ ∂I ∂t ∂I ∂t dxdt − ∫ τ ∫ B c ∂I ∂x ∂I ∂x dxdt with ∂ t I ( ⋅ , ) = ∂ t I ( ⋅ , τ ) = for all I , I ∈ V . Clearly in this construction A is symmetric.iii While I is related to the other state variables through the constitutive laws,taking the variation of I independently is required for the stationary principleto hold. In fact, equations (2.4) and (2.5) reveal the significance of treating he total energy as an independent quantity. In other words, the main insightwe have obtained is that the total energy I is the natural field to considerif one is looking to formulate a least action principle for the first law ofthermodynamics in 1D; it is the correct conduit for obtaining a symmetricoperator, hence a variational principle. Another consequence of (2.5) is that we can obtain an analogous result to con-servation of energy in wave mechanics
Corollary 1 (Constant of motion) . If I satisfies (2.5) and has compact supporton B . Then Π ( t ) = ∫ B ( ρ ( ∂I∂t ) + c ( ∂I∂x ) ) dx is a constant of motion, that is Π ( t ) = Π ( ) . Remark 2.
We have considered the isentropic energy equation in its simplestform, that is we have omitted the effects of body forces and heat sources. How-ever, this doesn’t hinder the construction of functional Σ . We consider the mostgeneral formulation for the conservation of energy equation in Appendix A, anddemonstrate how Σ can be deduced from a calculation similar to the one in thissection. In this section we formulate the Lagrangian-Hamiltonian theory correspondingto the balance of energy, now rewritten in hyperbolic form (2.4) or (2.5).Since Theorem 1 is analogous to Hamilton’s principle of least action, the Euler-Lagrange equations directly follow in terms of the functional Σ and the totalenergy
I ddt ( δ Σ δ ( ∂ t I ) ) − δ Σ δI = δδu denotes the functional derivative. Clearly, the Euler-Lagrange equa-tion is equivalent to (2.4) (or to (2.5) for constant c ).The Hamiltonian formulation corresponding to the new least action principlefollows analogously to classical mechanics. We first introduce the variable J defined analogously to the momentum variable in mechanics J = δ Σ δ ( ∂ t I ) (3.2)By taking the Legendre transformation of the function σ , the density functionof Σ, with respect to the change of variable ( I, ∂ t I ) Ð→ ( I, J ) , we obtain ananalogous quantity to the Hamiltonian density π ( ∇ I, J ) ≐ J ⋅ ∂I∂t − σ (3.3)A simple calculation gives the total quantity of π inside B Π ≐ ∫ B πd B = ∫ B ( ρ J + c ( ∂I∂x ) ) dx (3.4)7he physical interpretation of J becomes evident once we carry-out the calcu-lations in (3.1) J = δ Σ δ ( ∂ t I ) = ρ ∂I∂t The quantity J is the total power density of the system. Therefore, the phasespace associated with Π specifies the energy-power pair of the system at eachinstant time.Equipped with he Euler-Lagrange equation (3.1) together with functional Π, wecan rewrite the nonlinear evolution equation for the total energy I as Hamilton’sequations in Π ∂I∂t = δ Π δJ (3.5a) ∂J∂t = − δ Π δI (3.5b)For arbitrary functionals Φ and Φ over the (infinite dimensional) phase spaceassociated with the functional Π: {( I, J ) ∣ I = e + v , J = δ Σ δ ( ∂ t I ) } , we canintroduce the canonical Poisson bracket { , }{ Φ , Φ } = ∫ B δ Φ δI δ Φ δJ − δ Φ δI δ Φ δJ d B (3.6)The evolution of an arbitrary functional Φ over the phase space is governed byΠ through the Poisson structure˙Φ ( I, J ) = ∫ B ( δ Φ δI , δ Φ δJ ) ⋅ ( ∂I∂t , ∂J∂t ) T d B = ∫ B δ Φ δI δ Π δJ − δ Φ δJ δ Π δI d B = { Φ , Π } Specifically, we can rewrite Hamilton’s equations (3.3) in Poisson form ∂I∂t = { I, Π } (3.7a) ∂J∂t = { J, Π } (3.7b)Finally, if we assume that Π is strictly a function over the phase space, that is c is constant, we reproduce the result of Corollary 1 ∂ Π ∂t = { Π , Π } = In this section we demonstrate that invariance of the quantity Σ under grouptransformations (up to a full divergence of a field) result in conservation laws,one of which is the balance of energy. We shall state a version of Noether’stheorem Σ convenient for our setting, and examine the consequences; namelyby constructing a quantity analogous to the energy-momentum tensor in fieldtheory. 8et z µ = { t, x } ∈ B τ ≐ [ , τ ] × B for arbitrary τ ∈ R + , and µ = ,
1. This notation(often used in field theory) proves more convenient in stating Noether’s theorem.Under this notation, the Euler-Lagrange equation (3.1) reads ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ) = c is constant. Noether’s Theorem
We introduce a one-parameter smooth transformation λ ∈ [ , ∞ ) z→ I λ = I ( z µ ; λ ) with I λ ∣ λ = = I . Similarly, we define σ λ ≐ σ ( ∂ µ I λ ) .We say that function σ is invariant under the one-parameter group of transfor-mations λ z→ I λ if ddλ ∫ B T σ λ ∣ λ = d z = ∫ B T ∂K µ ∂z µ d z (4.2)for some (possibly zero) four-vector field ⃗ K = ⃗ K ( I, ∂ µ I ) . Theorem 2 (Noether’s theorem for σ ) . Assume σ is invariant under the one-parameter group of transformations λ z→ I λ , then the Euler-Lagrange systemcorresponding to σ admits the conservation law ∂P µ ∂z µ = where the conserved current P µ is defined as P µ = ∂σ∂ ( ∂ µ I ) ⋅ ∂I λ ∂λ ∣ λ = − K µ (4.4) Proof.
We begin by computing the LHS of (4.2) ddλ ∫ B T σ λ ∣ λ = d z = ∫ B T ∂σ∂ ( ∂ µ I ) ⋅ ∂ ( ∂ µ I λ ) ∂λ ∣ λ = d z = − ∫ B T ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ) ⋅ ∂I λ ∂λ ∣ λ = d z + ∫ B T ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ⋅ ∂I λ ∂λ ∣ λ = ) d z = ∫ B T ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ⋅ ∂I λ ∂λ ∣ λ = ) d z (4.5)where we have used the fact the I satisfies the Euler-Lagrange equations (4.1).By comparing equations (4.2) and (4.5), we obtain the conservation law for thecurrent P µ . 9 emark 3. i In terms of coordinates ( t, x ) , the conservation law (4.3) can be rewritten as ∂P ∂t + ∂P ∂x = ii For simplicity we have chosen λ to be a scalar parameter. Theroem 1 isequally valid for a µ − dimensional parameter ⃗ λ [17].iii A more basic diffeomorphism can be defined with respect to the independentvariable, that is, λ Ð→ z λµ ≐ z µ ( λ ) . Then the group of transformations I λ is defined in terms of z λµ , namely, I λ ≐ I ( z λµ ) . Therefore, invariance of σ with respect to the diffeomorphism λ Ð→ z µ ( λ ) still produces the result ofTheorem 1 [21]. An immediate implication of Theorem 2 is that the conservation of energy, writ-ten in hyperbolic form (2.4), can be derived as a conservation law correspondingto invariance under energy translations .To see this, we define the family of energy translations I λ = I + λI for someconstant energy scalar I . Then, under this group of transformations, we have σ λ = σ ( ∂ µ I λ ) = σ ( ∂ µ I ) = σ Therefore, σ is invariant under the transformation λ Ð→ I λ ; in other words,equation (4.2) is satisfied with ⃗ K =
0. The conserved current P µ , in this case,reads P µ = ∂σ∂ ( ∂ µ I ) ⋅ I (4.7)and the corresponding conservation law holds ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ) = total charge Q . Integrating both sides of (4.6) over the some large region (interval) Ω yields: ∫ Ω ∂P ∂t d Ω = − ∫ Ω ∂P ∂x d Ω = − P ∣ ∂ Ω By assuming P has compact support on Ω, we conclude that the total charge Q ≐ ∫ Ω P d Ω is conserved ddt Q = nergy-Momentum Tensor Among the conservation laws associated with a field Lagrangian, those thatare derived from the energy-momentum tensor are the most significant from aphysics perspective. Here again we can construct a quantity analogous to thatof classical field theory, namely the energy-momentum tensor, and obtain a hostof conservation laws as a result of Theorem 2.We begin the construction by considering the following space-time translation(see Remark 2-ii) z λµ ≐ z µ − λ µ = z µ − λ η δ µη (4.10)where µ, η = , I satisfying the Euler-Lagrange equations canbe written I λ = I ( z µ − λ µ ) = I ( z µ ) + λ µ ∂I λ ∂z µ ∣ λ = + o (∣ λ µ ∣) (4.11)Next, we show σ λ satisfies (4.2) Proposition 1.
Function σ is invariant under the space-time translation (4.10) .Proof. Function σ λ is defined as σ ( ∂ α I λ ) . Expanding this expression in λ µ yields σ λ = σ ( ∂ α I ) + λ µ ddλ µ σ λ ∣ λ = + o (∣ λ µ ∣) = σ + λ µ ∂σ∂ ( ∂ α I ) ∂ ( ∂ α I ) ∂z µ + o (∣ λ µ ∣) = σ + λ µ ∂σ∂z µ + o (∣ λ µ ∣) (4.12)where α, µ = ,
1. Hence ddλ η σ λ ∣ λ = = ∂∂z µ ( δ µη σ ) (4.13)Since we have assumed a diffeomorphism with respect to a two dimensionalparameter λ µ , the vector field in (4.2) is augmented to a second order tensor.Therefore, according to (4.13), we take K µη = δ µη σ to satisfy the condition (4.2).As a result of Proposition 1 (together with Theorem 2), we have the followingsystem of conservation laws ∂T µη ∂z µ = T µη , defined by T µη = ∂σ∂ ( ∂ µ I ) ⋅ ∂I∂z η − δ µη σ (4.15) We shall suppress the index η, µ on λ if the parameter λ appears as a subscript.
11s a quantity analogous to the energy-momentum tensor in classical (and quan-tum) field theory. The system of conservation laws (4.14) are accompanied,again, by the conservation of their global counterparts ∫ Ω T d Ω, and ∫ Ω T d Ωas discussed earlier. Moreover, as one expects, T = ∂σ∂ ( ∂ I ) ⋅ ∂I∂z − σ = ρ ∂I∂t ∂I∂t − σ = π and the evolution of π is governed by ∂π∂t + ∂∂x ( ∂I∂t ∂σ∂ ( ∂ x I ) ) = T µη is symmetric by writing σ as σ = B µη ∂I∂z µ ∂I∂z η (4.17)where B µη = [ ρ − c ] (4.18)Since B µη is symmetric, the tensor T µη in this case: T µη = B µα ∂I∂z α ∂I∂z η − δ µη σ (4.19)must also be symmetric. In this section we show that the process sketched thus far is iterative in nature.At each iteration, a corresponding “Lagrangian” can be constructed. As a result,we can formulate a least action principle at the i th iteration, and all the resultsof Sections 2–4 can be produced once again. Moreover, the constituents of eachnew variational principle depend on the preceding level of analysis. Therefore,we can visualize a hierarchy comprising of an infinite number of interrelatedLagrangians and their resulting variational principles.To clearly illustrate the procedure for obtaining the general iteration, we firstconsider the following. In Section 2, the rate of energy flux (in the isentropiccase) ∂ t ( vS ) was computed and was shown to be proportional to the gradientof the total energy (i.e. equation (2.3)). In the same spirit, we can view theterm ∂ t I ∂σ∂ ( ∂ x I ) as the “energy flux” associated with π in (4.16). In fact, thisequation is completely analogous to the isentropic balance of energy equation ρ ∂I∂t − ∂∂x ( vS ) = ∂I∂t + ∂∂x ( ∂u∂t ∂L∂ ( ∂ x u ) ) = ∂ t I ∂σ∂ ( ∂ x I ) : ∂∂t ( ∂I∂t ∂σ∂ ( ∂ x I ) ) = − ∂∂t ( c ∂I∂x ∂I∂t ) = − c ∂ I∂x∂t ∂I∂t − c ∂ I∂t ∂I∂x = − c ∂ I∂x∂t ∂I∂t − c ρ ∂ I∂x ∂I∂x = − cρ ∂∂x ( ρ ( ∂I∂t ) ) − cρ ∂∂x ( c ( ∂I∂x ) ) = − cρ ∂∂x ( ρ ( ∂I∂t ) + c ( ∂I∂x ) ) = − cρ ∂∂x π (5.1)Equation (5.1) together with (4.16) gives ∂ π∂t − cρ ∂ π∂x = π . Hence, bytreating π as an independent quantity–as we did with the total energy I –all theresults proven in Sections 2–4 hold too once we replace I with π , and treat π asthe new field.We, now, present the main result of this section: the preceding calculation canbe put into an iterative scheme Theorem 3.
Let L = ρ ( ∂u ∂t ) − c ( ∂u ∂x ) denote the classic Lagrangianwith displacement field u ( x, t ) = u ( x, t ) . Assume c ≐ ∂ L∂ ( ∂ x u ) > is constant.Then there exist infinitely many density functionals L i , i = , ... satisfying theleast action principle δ i ∫ τ ∫ B L i dxdt = where L i = ρ ( ∂u i ∂t ) − c ( ∂u i ∂x ) , u i + = ρ ( ∂u i ∂t ) + c ( ∂u i ∂x ) , and δ i is thevariation taken with respect to u i , i = , , ... . Proof.
We prove by induction.For i =
0, we obtain Hamilton’s principle of least action.Assume theorem holds for the i -th iteration. We look to construct the ( i + ) thiteration. Assume that u i is a solution to the i th variational problem. First wecompute the Euler-Lagrange equation at this iteration ddt ( ∂L i ∂ ( ∂ t u i ) ) + ∂∂x ( ∂L i ∂ ( ∂ x u i ) ) = ρ ∂ u i ∂t − c ∂ u i ∂x = ∂u i + ∂t = ρ ∂u i ∂t ∂ u i ∂t + c ∂u i ∂x ∂ u i ∂t∂x = c ∂u i ∂t ∂ u i ∂x + c ∂u i ∂x ∂ u i ∂t∂x = ∂∂x ( ∂u i ∂t c ∂u i ∂x ) = − ∂∂x ( ∂u i ∂t ∂L i ∂ ( ∂ x u i ) ) Thus, we have the local balance of energy ∂u i + ∂t + ∂∂x ( ∂u i ∂t ∂L i ∂ ( ∂ x u i ) ) = ∂∂t ( ∂u i ∂t ∂L i ∂ ( ∂ x u i ) ) = − ∂∂t ( ∂u i ∂t c ∂u i ∂x ) = − c ∂ u i ∂t ∂u i ∂x − c ∂u i ∂t ∂ u i ∂t∂x = − c ρ ∂ u i ∂x ∂u i ∂x − c ∂u i ∂t ∂ u i ∂t∂x = − c ∂∂x ⎛⎝( ∂u i ∂t ) ⎞⎠ − c ρ ∂∂x ⎛⎝( ∂u i ∂x ) ⎞⎠ = − cρ ∂∂x ⎛⎝ ρ ( ∂u i ∂t ) + c ( ∂u i ∂x ) ⎞⎠ = − cρ ∂∂x u i + (5.6)Combining (5.6) with (5.5) we obtain ρ ∂ u i + ∂t − c ∂ u i + ∂x = ddt ( ∂L i + ∂ ( ∂ t u i + ) ) + ∂∂x ( ∂L i + ∂ ( ∂ x u i + ) ) = u i + is a stationary point for the ( i + ) th variational problem. Remark 4.
A more general variational hierarchy theorem holds for non-constant c . In Appendix A, we sketch the general method of obtaining the second iteration(i.e. u = π ), which then can be extended to higher iterations. Since at each iteration the corresponding scalar field u i satisfies the wave equa-tion (5.2), we have an infinite number of constants of motion14 orollary 2 (Constants of motion) . If u i solves the variational problem (5.3) for i = , , , ... and has compact support on B . Then for every i ∈ N H i ( t ) = ∫ B ⎛⎝ ρ ( ∂u i ∂t ) + c ( ∂u i ∂x ) ⎞⎠ dx is a constant of motion, that is H i ( t ) = H i ( ) . Since at each level of analysis we have a corresponding Lagrangian L i , we candefine the i -th Hamiltonian H i via the Legendre transform (3.2) by first definingthe i -th momentum p i = ∂L i ∂ ( ∂ t u i ) (5.9)Then H i will have the form H i ( p i , ∂ x u i ) = ρ ( p i ) + c ( ∂u i ∂x ) (5.10)Clearly for i = p = J and H = π given in Section 3.Therefore, at the i -th iteration, Hamilton’s equations are given ∂u i ∂t = δ H i δp i (5.11a) ∂p i ∂t = − δ H i δu i (5.11b)And in a similar fashion to Section 3, we can define the Poisson brackets { , } i for each i .Finally, we touch on the key results of Section 4. We begin by rewriting theEuler-Lagrange equations at the i -th level in the space-time coordinates z µ : ∂∂z µ ( ∂L i ∂ ( ∂ µ u i ) ) = i fixed we obtain a conserved current P iµ associated with L i by Noether’stheorem (Theorem 2): ∂P iµ ∂z µ = P iµ = ∂L i ∂ ( ∂ µ u i ) ⋅ ∂u iλ ∂λ ∣ λ = − K iµ (5.14)for i = , , ... . L i is invariant under space-time translation: ddλ η L iλ ∣ λ = = ∂∂z µ ( δ µη L i ) (5.15)Therefore, the i -th energy-momentum tensor: T iµη = ∂L i ∂ ( ∂ µ u i ) ⋅ ∂u i ∂z η − δ µη L i (5.16)satisfies ∂T iµη ∂z µ = i = , , ... . So in addition to the constants of motion obtained in Corollary 2, we havea host of other conservation laws (both global and local) at each emanatingfrom Noether’s theorem. Hence, in aggregate, our hierarchy contains an infinitenumber of conservation laws.
We now turn our attention to the original problem of heat flow given by thefirst law of thermodynamics (2.1). Our main goal here is to show that the totalenergy field I solves a hyperbolic PDE as well; specifically a non-homogeneouswave equation in the total energy. The dissipative effects appear as additionalterms independent of the field I . A more general action functional can thenbe constructed that includes dissipation. Therefore, a least action principle,Lagrangian-Hamiltonian formalism, and a (modified) Noether’s theorem will allfollow as well in the dissipative case.We proceed as in Section 2 and consider the rate of change in the total energyflux: G = vS − q : ∂G∂t = S ∂v∂t + v ∂S∂t − ∂q∂t = ρ S ∂S∂x + v ∂S∂t − ∂q∂t (6.1)Since the internal energy and the stress are functions of both ∂ x u and s , weobtain terms in (6.1) involving the entropy s in addition to those obtained in162.3) ∂G∂t = Sρ ∂S∂ ( ∂ x u ) ∂ u∂x + v ∂S∂ ( ∂ x u ) ∂u ∂t∂x + Sρ ∂S∂s ∂s∂x + v ∂S∂s ∂s∂t − ∂q∂t = ∂S∂ ( ∂ x u ) ( ∂e∂ ( ∂ x u ) ∂ u∂x + v ∂v∂x ) + ∂S∂s ( ∂e∂ ( ∂ x u ) ∂s∂x + v ∂s∂t ) − ∂q∂t = ∂S∂ ( ∂ x u ) ( ∂e∂x + ∂∂x ( v )) + ∂S∂s ( ∂e∂ ( ∂ x u ) ∂s∂x + v ∂s∂t ) − ∂S∂ ( ∂ x u ) ∂e∂s ∂s∂x − ∂q∂t = c ∂I∂x + ∂S∂s v ∂s∂t + ∂s∂x S ∂∂ ( ∂ x u ) ( θS ) − ∂q∂t = c ∂I∂x + ˙ D (6.2)where we have defined˙ D = ∂S∂s v ∂s∂t + ∂s∂x S ∂∂ ( ∂ x u ) ( θS ) − ∂q∂t (6.3)Clearly, for a conservative system ˙ D is identically zero and we recover (2.3) asexpected. We also note that ˙ D is independent of I .Now by equation (2.1) and (6.2) we obtain ρ ∂I ∂t − ∂∂x ( c ∂I∂x ) − ∂ ˙ D∂x = ( I, ˙ D ) = ∫ B σ ( ∂ t I, ∂ x I, ˙ D ) dx ≐ ∫ B ( ρ ( ∂I∂t ) − c ( ∂I∂x ) − ˙ D ∂I∂x ) dx (6.5)Therefore, the evolution of the energy component I in an arbitrary interval [ , τ ] coincides with the stationary points of ∫ τ Σ dt (6.6)provided the boundary conditions δI ∣ t = = δI ∣ τ = = Remark 5.
The Euler-Lagrange equations for (6.5) share the same form withthe conservative case since the variation is taken only with respect to the totalenergy I ∂∂t ( δ Σ δ ( ∂ t I ) ) − δ Σ δI = . (6.7) Furthermore, the evolution of the dissipative material is entirely governed by aset of two coupled Euler-Lagrange equations: (6.7) coupled with ∂∂t ( δ L δ ( ∂ t u ) ) − δ L δu = , here the Lagrangian L is L = ∫ B ρ ( ∂u∂t ) − e ( ∂ x u, s ) dx . Therefore, the functionals Σ and L completely determine the state of the dissi-pative system at each instant in time, and each must be varied with respect to u and I , respectively, to obtain the evolution equations. In the example below, we obtain a special form for functional Σ in the case ofthe heat equation. Throughout the example we assume that the reference con-figuration is stress free i.e. S ( ∂ x u, s ) ∣ =
0, and the temperature at the referenceconfiguration θ ∣ = θ , where we have written Φ ∣ to mean that the function Φis evaluated at ∂ x u = s = Example (heat flow in a thermoelastic medium)
We begin first by writing σ as the sum of two quantities: σ = σ h + R h , where wehave defined σ h = ρ ( ∂I∂t ) + ∂q∂t ∂I∂x (6.8) R h = − c ( ∂I∂x ) + ( ∂S∂s v ∂s∂t + ∂s∂x S ∂∂ ( ∂ x u ) ( θS )) ∂I∂x (6.9)We examine the linear theory of thermoelasticity which is charechtrized by“small” thermomechanical deformations. Thereofore, it is appropriate to rescalethe displacement field ǫu for suitable non-dimensional positive small parameter ǫ [22], and consider the temperature θ to be everywhere close to θ . Since boththe displacement field and the temperature undergo small changes, the entropywould also be rescaled to ǫs . Linearizing the theory entails keeping terms up tothe order of ǫ in the functional Σ, while terms up to the order of ǫ alone areconsidered in the Euler-Lagrange equations.By expanding the constitutive equations for the stress and temperture around ∂ x u = s =
0, while recalling S ∣ =
0, we obtain S ( ∂ x u, s ; ǫ ) = ∂S∂ ( ∂ x u ) ∣ ǫ ∂u∂x + ∂S∂s ∣ ǫs + O ( ǫ ) (6.10) θ ( ∂ x u, s ; ǫ ) = θ + ∂θ∂ ( ∂ x u ) ∣ ǫ ∂u∂x + ∂θ∂s ∣ ǫs + O ( ǫ ) (6.11)By substituting the above linearizations together with the rescaled displacementand entropy fields into R h , the second order approximation (for constant c ) givesus: R h = − c ( ∂I∂x ) + θ c ∂s∂x ∂I∂x (6.12)While we have not rescaled the total energy explicitly, a simple calculationreveals that up to the order of ǫ , we have ∂I∂x = θ ∂s∂x . Therefore, it is justified toretain the terms involving the total energy in (6.12). However, for the stationaryprinciple to hold (see equation (6.6)), we maintain the basic form of R h in termsof total energy field I and not the displacement and entropy fields.18he Euler-Lagrange equations for the system are ∂∂t ( δ ( Σ h + R h ) δ ˙ I ) − δ ( Σ h + R h ) δI = ∂∂t ( δ Σ h δ ˙ I ) − δ ( Σ h + R h ) δI = ǫ ) we have in fact δ R h δI = ∂∂t ( δ Σ h δ ˙ I ) − δ Σ h δI = h determines the evolution of the energy for a dissipative linearthermoelastic medium. Equation (6.15) reads ∂∂t ( ρ ∂I∂t + ∂q∂x ) = ∂∂t ( ρ θ ∂s∂t + ∂q∂x ) = ǫ ) together withFourier’s law q = − k ∇ θ , we produce the classic evolution-diffusion equation ofthermoelasticity ∂∂t ( ρ c ∂θ∂t − ρ γθ ∂ u∂x∂t − k ∇ θ ) = c = θ ∂θ / ∂s ∣ , γ = c θ ∂θ∂ ( ∂ x u ) ∣ , and k are the heat capacity, stress-temperature modulus, and conductivity constant, respectively. If we assumethe fields θ and u have compact support in B then the classic evolution-diffusionequation of thermoelasticity can be readily recovered.In the absence of mechanical processes, that is u =
0, equation (6.17) reducesto the heat equation. In fact, in this case c =
0, ˙ D = − ∂ t q , which gives R h = h . Thisconcludes our example.The Hamiltonian and bracket formalisms for dissipative systems follow similarlyas in Section 3. Define the Legendre transform of σ : π ( ∇ I, J, ˙ D ) ≐ J ⋅ ˙ I − σ (6.18)which gives Π = ∫ B ( ρ J + c ( ∂I∂x ) + ˙ D ∂I∂x ) dx (6.19)19nd the evolution equations can be re-written once again in terms of Π:˙ I = δ Π δJ (6.20a)˙ J = − δ Π δI (6.20b)or in terms of the Poisson bracket: ˙ I = { I, Π } (6.21a)˙ J = { J, Π } (6.21b)where the { , } is defined (3.6) and the phase space is given in Section 3.Lastly, we consider some variational symmetries. If we consider one-parametertransformation in terms of the total energy: λ z→ I λ = I ( z µ ; λ ) with σ λ = σ ( ∂ µ I λ , ˙ D ) , then it is not hard to see that Neother’s theorem, as presentedin Section 4, holds . Hence, the simple transformation: I λ = I + λI leaves σ invariant and so, by equation (4.3)–(4.4), we obtain ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ) = c constant–see Appendix A for the non-constant c ) λ z→ z λµ = z µ − λ µ = z µ − λ η δ µη (6.22)The question here is whether or not σ remains invariant under the action of thistransformation now that σ depends not only on the parameterized field I λ butalso on ˙ D λ . In principle we could have also required that function σ be invariantunder the transformation λ Ð→ ˙ D λ . However, for the conservation law (4.4) tohold, the calculation in (4.5) must be justified. This is not true for arbitrarysmooth transformations λ Ð→ ˙ D λ , even for the basic space-time translations(6.22). This fact will result, as is shown below, in non-homogeneous conserva-tion laws, which are still valuable in the context of constructing a variationalhierarchy (see Section 7 and Appendix A).Under space-time translations, functions I and ˙ D satisfying the Euler-Lagrangeequations become I λ = I ( z µ − λ µ ) = I ( z µ ) + λ µ ∂I λ ∂z µ ∣ λ = + O (∣ λ µ ∣ ) (6.23a)˙ D λ = ˙ D ( z µ − λ µ ) = ˙ D ( z µ ) + λ µ ∂ ˙ D λ ∂z µ ∣ λ = + O (∣ λ µ ∣ ) (6.23b) Notice that according to our definition σ λ = σ ( ∂ µ I λ , ˙ D ) we must have ∂ ˙ D∂λ =
0, and sothe calculation in (4.5) is still valid σ λ satisfies (4.2) Proposition 2.
Function σ is invariant under the space-time translation (4.2) .Proof. Function σ λ is defined as σ ( ∂ α I λ , ˙ D λ ) . Expanding this expression in λ µ yields σ λ = σ ( ∂ α I, ˙ D ) + λ µ ddλ µ σ λ ∣ λ = + O (∣ λ µ ∣ ) = σ + λ µ ( ∂σ∂ ( ∂ α I ) ∂ ( ∂ α I ) ∂z µ + ∂σ∂ ˙ D ∂ ˙ D∂z µ ) + O (∣ λ µ ∣ ) = σ + λ µ ∂σ∂z µ + O (∣ λ µ ∣ ) (6.24)Hence, ddλ η σ λ ∣ λ = = ∂∂z µ ( δ µη σ ) (6.25)Here agian since we have assumed a diffeomorphism with respect to a two di-mensional parameter λ η , the vector field in (4.2) is augmented to a secondorder tensor. Therefore, according to (6.25), we take K µη = δ µη σ to satisfy thecondition (4.2).However, we can not apply Noether’s theorem to conclude the conservationlaws associated with the energy-momentum tensor since the LHS of (6.25) doesnot result in (4.5) in the presence of ˙ D λ . Neverthelss, we can establish a non-homogeneous version of equation (4.3), that is with additional terms on theRHS.We start out calculation with ddλ η σ λ ∣ λ = = ∂σ∂ ( ∂ µ I λ ) ∂ ( ∂ µ I λ ) ∂λ η ∣ λ = + ∂σ∂ ˙ D λ ∂ ˙ D λ ∂λ η ∣ λ = = − ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ) ∂I λ ∂λ η ∣ λ = + ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ∂I λ ∂λ η ∣ λ = ) + ∂σ∂ ˙ D ∂ ˙ D λ ∂λ η ∣ λ = = ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ∂I∂z η ) + ∂σ∂ ˙ D ∂ ˙ D∂z η (6.26)Equations (6.25) and (6.26) give us the following equation for the energy mo-mentum tensor T µη (defined in (4.15)) ∂T µη ∂z µ = − ∂σ∂ ˙ D ∂ ˙ D∂z η (6.27)In particular, the quantity T = π is governed by ∂π∂t + ∂∂x ( ∂I∂t ∂σ∂ ( ∂ x I ) ) = ∂I∂x ∂ ˙ D∂t (6.28)It is also clear that the tensor T µη is no longer symmetric.21he quantity π in the dissipative case is no longer governed by a conservationlaw as evident by equation (6.28). Nevertheless, a similar procedure outlinedin the calculations (6.1) and (6.2) applied to the flux ∂I∂t ∂σ∂ ( ∂ x I ) can produce thefunctional L for the dissipative case. So while we no longer have conservationlaws given by Noether’s theorem in the presence of dissipation, we still manageto retain an infinite number of variational principles together with their corre-sponding Lagrangian-Hamiltonian formalism associated with functional L i (seeRemark 4). A closer look at the results above reveals a direction for obtaining more complexfield theories from simpler ones. It is well knows that in classical field theories,a Lagrangian L is specified to capture the dynamics of some field φ . Usually L depends on φ and its derivatives. The question we are interested in is: “if wewere to introduce a field Γ–independent of φ –into our Lagrangian, how can wedetermine the evolution of this additional degree of freedom?”Before we tackle this question in its general form, we turn to the interplaybetween the theories of elasticity and thermoelasticity for insight. For elasticity,the field φ is identified with the displacement u , and the Lagrangian density L is given as L = L ( ∂ t u, ∂ x u ) = ρ ( ∂u∂t ) − e ( ∂ x u ) (7.1)where the internal energy e can be identified with is the stored strain energy W . The evolution of the displacement field u is governed by the balance ofmomentum, which is equivalent to the Euler-Lagrange equations for L withrespect to u .In the theory of thermoelasticity we add the dependence of the entropy density s to the internal energy e , hence L = L ( ∂ t u, ∂ x u, s ) , and assume the presence ofheat flow through the boundary given by the flux q . For example, in linear ther-moelasticity e is a quadratic form in ∂ x u and s as implied in Example in Section6. The balance of momentum is still equivalent to the Euler-Lagrange equationsof L with respect to u . However, the evolution of s is not governed by the Euler-Lagrange equations of L with respect to s , but rather by a physical principle: thefirst law of thermodynamics (2.1). However, as we know from Noehter’s theoremthe conservation of energy can be deduced through time symmetry invariance(i.e. the conservation of the zeroth component of the energy-momentum tensor,which corresponds to the total energy or Hamiltonian). This conversation lawreads ρ ∂I∂t = ∂∂x ( vS ) (7.2)While both I and S depend on the field s , we still do not completely recoverthe first law of thermodynamics; we do not obtain the heat flux q which is therequisite source of dissipation. In fact, in the absence of dissipation, equation(7.2) is equivalent to ∂ t s = (i.e the first law). Secondly, a variational analysis for the field u alone canneither produce the physics of the problem nor the correct equation governingthe evolution of s , even though it correctly predicts the balance of momen-tum equation for thermoelasticity. In order to obtain the energy equation forthermoelasticity we need to introduce a more sophisticated variational principleobtained (ultimately) from the classic Hamilton’s principle, after we have for-mulated the first law of thermodynamics. In other words, to find a variationalderivation for the equations of thermoelasticity, that is to determine the evolu-tion of u and s in a dissipative material, we need two functionals: L and Σ (or L and σ ).The addition of a new independent field Γ to the Lagrangian L = L , in additionto the fields u and s , necessarily implies that L = L ( ∂ x u, s, Γ ) . However, weneed an additional equation to close the system; this equation must be, byTheorem 3, the Euler-Lagrange equations for L with respect to u (includingthe additional non-homogeneous terms coming from Noether’s theorem for L ).And while the three coupled Euler-Lagrange equations for L , L , L are writtenin terms of unknowns u , u , u , the constitutive laws reveal that in actualitywe have three coupled PDEs in u, s, Γ. Yet, the addition of Γ can not be tooarbitrary, it must be accompanied by some physical principle that can be addedsymbolically to Noether’s theorem for L or equivalently to the Euler-Lagrangeequations for L . This is necessary. A physical field can not be solely governedby a mathematical formalism. It is the field’s distinct response when cominginto contact with physical reality that distinguishes it, and is manifested in somecharacterizing law or equation. The variational hierarchy has the flexibility toaccommodate these physical principles as extra terms in the Lagrangian L andNoether’s theorem, and at the same time allows for a formalism to produce thecorrect number of equations for any number of additional fields.We can go a step further. Given a general field φ (not necessarily the displace-ment field) and a Lagrangian L = L = L ( φ, ∂ µ φ ) in the form L = ( ∂φ∂t ) − ( ∂φ∂x ) − W ( φ ) (7.3)then we can apply the same procedure outlined in the proof of Theorem 3 (andAppendix A) to obtain L since we have the following additional equation (byNoether’s theorem) [24]: ∂h∂t + ∂∂x ( ∂φ∂t ∂L∂ ( ∂ x φ ) ) = W is some nonlinear function in φ and h = ∂ t φ ∂L∂ ( ∂ t φ ) − L . So while we havefocused in this work on φ = u , the same theoretical procedure can be applied tofield theories with Lagrangian having the form written in equation (7.3). Thatis to say, equation (7.4) can be written in hyperbolic form, which allows for theconstruction of the next Lagrangian L in the hierarchy.The addition of an some other independent field Γ to the Lagrangian L entailsthat the evolution of Γ is governed by the coupled Euler-Lagrange for L and L with respect to φ and h , respectively. Again, it is not enough to add Γ to23 alone, a physical law governing Γ must also be specified, which can enter theformalism through Noehter’s theorem for L . In general, it seems we need asmany Lagrangian densities L i ’s as there are independents fields.Finally, it should be emphasized that the Lagrangians L i , i = , , ... are auxiliaryfunctions, in the sense that they are predetermined once the original Lagrangian L = L is specified. Therefore, the new physics associated with an additionalfield is not only mediated by a certain physical principle, but also by L . Thevariational hierarchy provides us with a systematic approach to fit all new de-grees of freedom into one framework. In 1969 C. Truesdell wrote [24]: ”The difference [between mechanics and ther-modynamics] is that thermodynamics never grew up”. While the theory ofrational thermodynamics has been developed significantly since the late 60s,largley due to Truesdell himself, a variational formulation akin to Hamilton’sprinciple had still remained out of reach. In this context, we view our workas an effort to establish a foundation for the variational treatment of the firstlaw of thermodynamics (in one space dimension), which aims to be analogousto mechanics. This is only the first step. To claim a comprehensive variationaltheory of thermodynamics is to present the second law of thermodynamics interms of some variation of the functional Σ. This is an area for future research.However, the fact that a least action principle exists for the first law of ther-modynamics is an indication that Σ is fundamental for both a conservative anddissipative systems.We also showed that infinite new variational principles follow from the classicHamilton’s principle by recasting the conservation of energy equation, obtainedby Noether’s theorem, into hyperbolic form. This hierarchical structure has asimple iterative form for isentropic (conservative) systems, and is more involvedfor dissiptaive and forced systems. We also commented on the physics impli-cation of the variational hierarchy. We established a systematic approach forincreasing the complexity of field theories provided that the original Lagrangian L has the form in equation (7.3). The addition of a new degree of freedom isgoverned by two factors: the first is given by the next iteration in the hierarchy,and the second is by some physical principle similar to heat transfer in the firstlaw of thermodynamics. The implications considered here will be a topic offurther study. Acknowledgments.
The author thanks James Glimm for helpful discus-sions and encouragement. This work was supported and funded by KuwaitUniversity Research Grant No. [ZS02/19].24
Appendix
Throughout the paper we have asserted that the scheme for constructing thefunctional Σ can accomedate external sources and the procedure is similiar tothat of Section 6. We, therefore, consider the most general case of the first lawof thermodyanics applied to a continuum, that is the non-linear conservationof energy equation with body forces b ( x, t ) , heat sources r ( x, t ) , and heat flux q ( ∂ x θ ) [25] ρ ∂I∂t = ∂∂x ( vS − q ) + ρ vb + ρ r in B × [ , τ ] (A.1)where the constitutive equations are given in (2.2).In addition to the construction of L = Σ for the above problem (together withits corresponding Hamiltonian formulation and Noether’s Theorem), we alsodemonstrate the construction of L in this context.By the calculations in (6.1) and (6.2) we have ∂∂t ( vS − q ) = c ∂I∂x + ˙ D (A.2)The above two equations thus give ρ ∂ I∂t = ∂∂x ( c ∂I∂x ) + ∂ ˙ D∂x − ˙ B (A.3)where we have defined ˙ B = − ρ ∂∂t ( vb − r ) .As such we choose functional Σ to beΣ = ∫ B ( ρ ( ∂I∂t ) − c ( ∂I∂x ) − ˙ D ∂I∂x − ˙ BI ) dx (A.4)Hence, equation (A.3) is equivalent to the standard Euler-Lagrange equations ddt ( δ Σ δ ( ∂ t I ) ) − δ Σ δI = π can be fashioned similarly to Sections 3 & 6. Indeed,we can combine ˙ B and ∂ x ˙ D into one term and simply apply the procedure inSection 6 to obtainΠ = ∫ B ( ρ J + c ( ∂I∂x ) + ˙ D ∂I∂x + ˙ BI ) dx (A.6)and ∂I∂t = δ Π δJ (A.7a) ∂J∂t = − δ Π δI (A.7b)where π is defined as in equation (6.18).25quation (A.3) (or equivalently equation (A.5)) follows from Noehter’s theoremif σ remains invariant with respect to λ Ð→ I λ = I + λI . However, the appli-cation of Noether’s theorem is not straightforward with respect to space-timetranslations (4.10) because of the presence of ˙ D (as we have seen in Section 6),and now ˙ B . We explore this below.Under space-time translations (4.10) we obtain two more equations in additionto (6.23) c λ = c ( z µ − λ µ ) = c ( z µ ) + λ µ ∂c λ ∂z µ ∣ λ = + O (∣ λ µ ∣ ) (A.8a) B λ = B ( z µ − λ µ ) = B ( z µ ) + λ µ ∂B λ ∂z µ ∣ λ = + O (∣ λ µ ∣ ) (A.8b)A similar calculation to the proof Proposition (2) reveals that σ λ = σ ( I λ , ∂ µ I λ , c λ , ˙ D λ , ˙ B λ ) obeys ddλ η σ λ ∣ λ = = ∂∂z µ ( δ µη σ ) (A.9)On the other hand ddλ η σ λ ∣ λ = = ∂σ∂I λ ∂I λ ∂λ η ∣ λ = + ∂σ∂ ( ∂ µ I λ ) ∂ ( ∂ µ I λ ) ∂λ η ∣ λ = + ∂σ∂c λ ∂c λ ∂λ η ∣ λ = + ∂σ∂ ˙ D λ ∂ ˙ D λ ∂λ η ∣ λ = + ∂σ∂ ˙ B λ ∂ ˙ B λ ∂λ η ∣ λ = = ( ∂σ∂I − ∂∂z µ ( ∂σ∂ ( ∂ µ I ) )) ∂I λ ∂λ η ∣ λ = + ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ∂I λ ∂λ η ∣ λ = ) + ⋯ = ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ∂I λ ∂λ η ∣ λ = ) + ∂σ∂c ∂c λ ∂λ η ∣ λ = + ∂σ∂ ˙ D ∂ ˙ D λ ∂λ η ∣ λ = + ∂σ∂ ˙ B ∂ ˙ B λ ∂λ η ∣ λ = = ∂∂z µ ( ∂σ∂ ( ∂ µ I ) ∂I∂z η ) + ∂σ∂c ∂c∂z η + ∂σ∂ ˙ D ∂ ˙ D∂z η + ∂σ∂ ˙ B ∂ ˙ B∂z η (A.10)Therefore, the second order tensor T µη = ∂σ∂ ( ∂ µ I ) ∂I∂z η − σδ µη satisfies the equation ∂T µη ∂z µ − F η = F η = ∂σ∂c ∂c∂z η + ∂σ∂ ˙ D ∂ ˙ D∂z η + ∂σ∂ ˙ B ∂ ˙ B∂z η .And so T = π satisfies ∂π∂t + ∂∂x ( ∂I∂t ∂σ∂ ( ∂ x I ) ) = F (A.12)The presence of F does not impede the construction of L (i.e. the seconditeration). We proceed as we did in the beginning of Section 5, and computethe rate of change of the flux in (A.12) ∂∂t ( ∂I∂t ∂σ∂ ( ∂ x I ) ) = − ∂∂t ( ∂I∂t ( c ∂I∂x + ˙ D )) = − ∂ I∂t ( ˙ D + c ∂I∂x ) − ∂I∂t ( ∂c∂t ∂I∂x + c ∂ I∂x∂t + ∂ ˙ D∂t )
26y using the evolution equation for the total energy (A.3) and rearranging theterms we obtain ∂∂t ( ∂I∂t ∂σ∂ ( ∂ x I ) ) = − cρ ∂∂x ( π ) + Q (A.13)where Q = − ρ ∂ x ( c∂ x I ) ˙ D − c ρ ∂ x c ( ∂ x I ) Dρ ∂ x ˙ D + cρ ∂ xx I ˙ D + ˙ D ˙ Bρ − cρ I∂ x ˙ B + cρ ∂ x ( I ˙ B ) − ∂ t I∂ t c∂ x I − ∂ t ˙ D∂ t I .Equations (A.13) and (A.12) give us ρ ∂ π∂t − ∂∂x ( c ∂π∂x ) = ρ ( ∂F ∂t − ∂Q∂x ) (A.14)One possible Lagrangian L to derive (A.12) from is L = ρ ( ∂π∂t ) − c ( ∂π∂x ) − ˙ Cπ (A.15)where ˙ C = − ρ ( ∂ t F − ∂ x Q ) .The Lagrangian L is the basis for formulating the rest of the variational frame-work in the second iteration. While, as we see, the detailed expressions becomeincreasingly more complex, they can be organized in an expected pattern as wemove up from one iteration to the other. B Appendix
For a linear elastic medium the displacement u satisfies the conservation of linearmomentum: ρ ∂ u∂t = ∂S∂x = ∂∂x ( c ∂u∂x ) (B.1)In the presence of thermal effects (B.1) can be extended to ρ ∂ u∂t = ∂∂x ( c ∂u∂x ) + γ ∂θ∂x (B.2)where γ is the stress-temperature modulus.By comparing equations (B.1) and (B.2) with (2.4) and (6.4), respectively, aone-to-one correspondence between the balance of momentum and energy canbe established. We summarize this correspondence in the table below.27 able 1: A one-to-one correspondence is illustrated between the balance of momentum andenergy in linear elastic medium with the presence of thermal effects. For simplicity we havetaken constant γ = Identity Conservation of Momentum Conservation of Energy
Density Function L = ρ ( ∂u∂t ) − c ( ∂u∂x ) − θ ∂u∂x σ = ρ ( ∂I∂t ) − c ( ∂I∂x ) − ˙ D ∂I∂x
Governing Equation ρ ∂ u∂t − c ∂ u∂x − ∂θ∂x = ρ ∂ I∂t − c ∂ I∂x − ∂ ˙ D∂x = ∫ τ ∫ B L ( ∂ t u, ∂ x u, θ ) dxdt ∫ τ ∫ B σ ( ∂ t I, ∂ x I, ˙ D ) dxdt The above table establishes the following correspondence u ←→ I (B.3a) θ ←→ ˙ D (B.3b)28 eferences [1] Gregory RD. Classical mechanics. Cambridge University Press; 2006.[2] Berdichevsky V. Variational principles of continuum mechanics: I. Fun-damentals. Springer Science & Business Media; 2009.[3] Rayleigh L. Theory of Sound. vol. 1 (reprinted 1945 by Dover, NewYork). Macmillan, London; 1945.[4] Biot MA. Thermoelasticity and irreversible thermodynamics. Journal ofApplied Physics. 1956;27(3):240–253.[5] Biot MA. Variational principles in heat transfer: a unified Lagrangiananalysis of dissipative phenomena. BIOT (MA) NEW YORK; 1970.[6] ¨Ottinger HC. Beyond equilibrium thermodynamics. John Wiley & Sons;2005.[7] Kaufman AN. Dissipative Hamiltonian systems: a unifying principle.Physics Letters A. 1984;100(8):419–422.[8] Morrison PJ. A paradigm for joined Hamiltonian and dissipative systems.Physica D: Nonlinear Phenomena. 1986;18(1-3):410–419.[9] Said H. A Lagrangian–Hamiltonian unified formalism for a class of dissi-pative systems. Mathematics and Mechanics of Solids. 2019;24(4):1221–1240.[10] Gurtin M. Variational principles for linear initial-value problems. Quar-terly of Applied Mathematics. 1964;22(3):252–256.[11] Yang Q, Stainier L, Ortiz M. A variational formulation of the coupledthermo-mechanical boundary-value problem for general dissipative solids.Journal of the Mechanics and Physics of Solids. 2006;54(2):401–424.[12] Kane C, Marsden JE, Ortiz M, West M. Variational integrators andthe Newmark algorithm for conservative and dissipative mechanical sys-tems. International Journal for Numerical Methods in Engineering.2000;49(10):1295–1325.[13] Bloch A, Krishnaprasad P, Marsden JE, Ratiu TS. The Euler-Poincar´eequations and double bracket dissipation. Communications in mathemati-cal physics. 1996;175(1):1–42.[14] Gay-Balmaz F, Yoshimura H. A Lagrangian variational formulation fornonequilibrium thermodynamics. Part II: continuum systems. Journal ofGeometry and Physics. 2017;111:194–212.[15] Galley CR. Classical Mechanics of Nonconservative Sys-tems. Phys Rev Lett. 2013 Apr;110:174301. Available from: https://link.aps.org/doi/10.1103/PhysRevLett.110.174301https://link.aps.org/doi/10.1103/PhysRevLett.110.174301