aa r X i v : . [ m a t h - ph ] F e b Liouville geometry of classicalthermodynamics
Arjan van der Schaft ∗ February 11, 2021
Starting from Gibbs’ fundamental thermodynamic relation, contact geometry has been recognized as a natural framework for the geometric formulation ofclassical thermodynamics since the early 1970s [21]. This spurred a series ofpapers; see e.g. [29, 30, 31, 32, 33, 34, 4, 19, 11, 13, 16, 8, 26, 6, 17, 35, 23,39, 10, 12], and [7] for a recent introduction and survey. Other geometric workemphasizing the variational formulation of thermodynamics includes [28, 15].On the other hand, as discussed in [5], the contact-geometric formulationof thermodynamics makes a distinction between the energy and the entropyrepresentation of the same thermodynamic system. By itself this need not beconsidered as a major flaw since the two representations are conformally equiva-lent. Nevertheless, it was shown in [5], and later in [36, 27, 37], that an attractivepoint of view that is merging the energy and entropy representation is offered bythe extension of contact manifolds to symplectic manifolds. Compared with theodd-dimensional contact manifold this even-dimensional symplectic manifoldhas one more degree of freedom, called a gauge variable in [5]. From a thermo-dynamics perspective it amounts to replacing the intensive variables by their homogeneous coordinates . In fact, this symplectization of contact manifolds israther well-known in differential geometry [2, 25]; dating back to [20]. As arguedin [37], the extension of contact manifolds to symplectic manifolds, in fact tocotangent bundles without zero section, has additional advantages for the geo-metric formulation of thermodynamics as well. First, it yields a clear distinctionbetween the extensive and intensive variables of the thermodynamic system.Secondly, it enables the definition of port-thermodynamic systems , which arethermodynamic systems that interact with their environment via either power or entropy flow ports. Finally, symplectization has computational benefits; aswas already argued within differential geometry by Arnold [2, 3]. ∗ A.J. van der Schaft is with the Bernoulli Institute for Mathematics, Computer Scienceand AI, and the Jan C. Willems Center for Systems and Control, University of Groningen,PO Box 407, 9700 AK, the Netherlands, [email protected] he present paper aims at providing an in-depth treatment of the resultinggeometry of thermodynamic systems, continuing the earlier investigations in[36, 37] and building upon [2, 3, 25]. Starting point are cotangent bundles with-out zero section, endowed with their natural one-form; also called the
Liouvilleform . Instead of considering the symplectic geometry derived from the sym-plectic form ω = dα , where α is the Liouville form, a smaller set of geometricobjects will be defined solely based on this Liouville form. The resulting geome-try is called Liouville geometry . In particular, it will be shown how a particularclass of Lagrangian submanifolds (called Liouville submanifolds) can be definedas maximal submanifolds on which the Liouville form is zero. Furthermore, aparticular type of Hamiltonian vector fields is defined consisting of vector fieldswhich leave the Liouville form invariant. All these geometric objects have theproperty that they are homogeneous in the cotangent variables. As a resultthey are in one-to-one correspondence with objects on the underlying contactmanifold (of dimension one less). We will study in detail the generating func-tions of Liouville submanifolds and the homogeneous Hamiltonian functions ofthis special type of Hamiltonian vector fields, and relate them to their contactgeometry counterparts. Continuing upon [37] it will be shown how this leads tothe definition of a port-thermodynamic system, and its projection to the contactmanifold. Finally we will focus on an additional homogeneity structure, presentin some thermodynamic systems, corresponding to homogeneity in the exten-sive variables. This leads to a new geometric view on the classical Gibbs-Duhemrelation, and a subsequent projection to an even-dimensional space.The rest of the paper is structured as follows. In Section 2 it is discussed,using the example of a simple gas, how thermodynamics leads to the study ofcotangent bundles over the base space of extensive variables, with cotangentvariables being the homogeneous coordinates for the intensive variables. Theresulting Liouville geometry of a general cotangent bundle without zero section,and its projection to contact geometry, is studied in Section 3. Then Section4 provides the definition of port-thermodynamic systems using Liouville geom-etry, and its projection to a contact-geometric description. Section 5 discusseshomogeneity with respect to the extensive variables, the Gibbs-Duhem relation,and its geometric formalization. Finally, Section 6 contains the conclusions.
In this section we will motivate how classical thermodynamics, starting fromGibbs’ thermodynamic relation, naturally leads to contact geometry, and howby considering homogeneous coordinates for the intensive variables this resultsin Liouville geometry. Sometimes also called the
Poincar´e-Liouville form , or tautological form. .1 From Gibbs’ fundamental thermodynamic relation tocontact geometry
Consider a simple thermodynamic system such as a mono-phase, single con-stituent, gas in a confined compartment with volume V and pressure P at tem-perature T . It is well-known that the state properties of the gas are describedby a 2-dimensional submanifold of the ambient space R (the thermodynamicphase space ) with coordinates E (energy), S (entropy), V , P , and T . Such asubmanifold characterizes the properties of the gas (e.g., an ideal gas, or a Vander Waals gas), and all of them share the following property. Define the Gibbsone-form on the thermodynamic phase space R as θ := dE − T dS + P dV (1)Then θ is zero restricted to the submanifold characterizing the state properties.This is called Gibbs’ fundamental thermodynamic relation . It implies that the extensive variables
E, S, V and the intensive variables
T, P are related in aspecific way. Geometrically this is formalized by noting that the Gibbs one-form θ defines a contact form on R , and that any submanifold L capturingthe state properties of the thermodynamic system is a submanifold of maximaldimension restricted to which the contact form θ is zero. Such submanifolds arecalled Legendre submanifolds of the contact manifold ( R , θ ).By expressing the extensive variable E as a function E = E ( S, V ) of the tworemaining extensive variables S and V , Gibbs’ fundamental relation implies thatthe Legendre submanifold L specifying the state properties is given as L = { ( E, S, V, T, P ) | E = E ( S, V ) , T = ∂E∂S , − P = ∂E∂V } (2)Hence L is completely described by the energy function E ( S, V ), whence thename energy representation for (2). On the other hand, there are other ways torepresent L . If L is parametrizable by the variables T, V (instead of
S, V as in(2)), then one defines the partial Legendre transform of E ( S, V ) with respect to S as A ( T, V ) := E ( V, S ) − T S, T = ∂E∂S ( S, V ) , (3)where S is solved from T = ∂E∂S ( S, V ). Then L is also described as L = { ( E, S, V, T, P ) | E = A ( T, V ) − T ∂A∂T , S = − ∂A∂T , − P = ∂A∂V } (4) A is known as the Helmholtz free energy , and is one of the thermodynamicpotentials derivable from the energy function E ( S, V ); see e.g. [14]. Two otherpossible parametrizations of L (namely by S, P , respectively by
T, P ) correspondto two more thermodynamic potentials, namely the enthalpy H ( S, P ) and theGibbs’ free energy G ( T, P ), resulting in similar expressions for L .n general [2, 25], a contact manifold ( M, θ ) is an odd-dimensional manifoldequipped with a contact form θ . A one-form θ on a (2 n +1)-dimensional manifold M is a contact form if and only around any point in M we can find coordinates( q , q , · · · , q n , γ , · · · , γ n ) for M , called Darboux coordinates , such that θ = dq − n X j =1 γ j dq j (5)Equivalently, θ is a contact form if θ ∧ ( dθ ) n is nowhere zero on M . A Legendresubmanifold of a contact manifold (
M, θ ) is a submanifold of maximal dimensionrestricted to which the contact form θ is zero. The dimension of any Legendresubmanifold of a (2 n + 1)-dimensional contact manifold is equal to n .In fact, we will use throughout this paper the slightly generalized definition of a contact manifold as given in e.g. [2], where the contact form θ is onlyrequired to be defined locally . What counts is the contact distribution ; the 2 n -dimensional subspace of the tangent space at any point of M defined by the kernel of the contact form θ at this point. This turns out to be the appropriateconcept for the thermodynamic phase space being a contact manifold .Apart from the above parametrizations of the Legendre submanifold L , cor-responding to an energy function E ( S, V ) and its Legendre transforms, there isstill another , although very similar, way of describing L . This alternative optionis motivated from a modeling point of view. Namely, often thermodynamic sys-tems are formulated by first listing the balance laws for the extensive variablesapart from the entropy S , and then expressing S as a function S = S ( E, V ).This leads to the entropy representation of the submanifold L ⊂ R , given as L := { ( E, S, V, T, P ) | S = S ( E, V ) , T = ∂S∂E , PT = ∂S∂V } (6)Analogously the case of the energy representation E = E ( S, V ), one mayconsider thermodynamic potentials obtained by partial Legendre transform of S ( E, V ). Geometrically the entropy representation corresponds to the modified
Gibbs contact form e θ := dS − T dE − PT dV, (7)which is obtained from the original Gibbs contact form θ in (1) by division by − T (called conformal equivalence ). In this way the Gibbs fundamental relationis rewritten as e θ | L = 0, and the intensive variables become T , PT . The contact-geometric view on thermodynamics, directly motivated by Gibbs’fundamental thermodynamic relation, has two shortcomings: Contact manifolds for which the contact form θ is defined globally are sometimes called exact contact manifolds .
1) Switching from the energy representation E = E ( S, V ) to the entropy rep-resentation S = S ( E, V ) corresponds to replacing the Gibbs form θ by themodified Gibbs form e θ in (7), and thus leads to a similar, but different , contact-geometric description.(2) The contact-geometric description does not make a clear distinction between,on the one hand, the extensive variables E, S, V and, on the other hand, theintensive variables T, − P (energy representation), or T , PT (entropy represen-tation). In fact, given a contact form θ there are many Darboux coordinates q , q , q , p , p for R such that θ = dq − p dq − p dq , where q , q , q are not necessarily obtained by a transformation of only the extensive variables E, S, V .The way to remedy these shortcomings is to extend the contact manifold byone extra dimension to a symplectic manifold, in fact a cotangent bundle , withan additional homogeneity structure. This construction is rather well-known indifferential geometry [2, 25], but was advocated within a thermodynamics con-text only in [5], and followed up in [36, 37]. For a simple thermodynamic systemwith extensive variables
E, S, V and intensive variables T, − P , the constructionamounts to replacing the intensive variables T, − P by their homogeneous coor-dinates p E , p S , p V with p E = 0, i.e., T = p S − p E , − P = p V − p E (8)Equivalently, the intensive variables T , PT in the entropy representation are rep-resented as1 T = p E − p S , PT = p V − p S (9)This means that the two contact forms θ = dE − T dS + P dV and e θ = dS − T dE − PT dV are replaced by a single symmetric expression, namely by α := p E dE + p S dS + p V dV, (10)The one-form α is nothing else than the canonical Liouville one-form on thecotangent bundle T ∗ R , with R the space of extensive variables E, S, V . Thusthe thermodynamic phase space R has been replaced by T ∗ R . More precisely,by definition of homogeneous coordinates the vector ( p E , p S , p V ) is differentfrom the zero vector, and hence the space with coordinates E, S, V, p E , p S , p V isactually the cotangent bundle T ∗ R minus its zero section; denoted as T ∗ R .Any 2-dimensional Legendre submanifold L ⊂ R describing the state prop-erties is now replaced by a 3-dimensional submanifold L ⊂ T ∗ R , given as L = { ( E, S, V, p E , p S , p V ) ∈ T ∗ R | ( E, S, V, p S − p E , p V − p E ) ∈ L } (11)It turns out that L is a Lagrangian submanifold of T ∗ R with symplectic form ω := dα , with an additional property of homogeneity . Namely, whenever E, S, V, p E , p S , p V ) ∈ L , then also ( E, S, V, λp E , λp S , λp V ) ∈ L , for any non-zero λ ∈ R . Such Lagrangian submanifolds turn out to be fully character-ized as maximal manifolds restricted to which the Liouville one-form α = p E dE + p S dS + p V dV is zero, and will thus be called Liouville submanifolds of T ∗ R . As we will see in the next section the extension of contact manifoldsto cotangent bundles, replacing the intensive variables by their homogeneouscoordinates, also leads to a natural homogeneous Hamiltonian dynamics on theextended space T ∗ R . This does not only facilitate the analysis, but has clearcomputational advantages as well. In fact, all computations become standardoperations on cotangent bundles and in Hamiltonian dynamics. In the wordsof Arnold [3]: one is advised to calculate symplectically (but to think rather interms of contact geometry).All of this is immediately extended from the thermodynamic phase space R with coordinates E, S, V, T, P to general thermodynamic phase spaces. Forinstance, in the case of multiple chemical species the Gibbs form θ extends to dE − T dS + P dV − P k µ k dN k , where N k and µ k , k = 1 , · · · , s, are the molenumbers, respectively, chemical potentials of the k -th species. Correspondingly,the thermodynamic phase R × R s is replaced by the cotangent bundle withoutzero-section T ∗ R s , with extensive variables E, S, V, N , · · · , N s and Liouvilleform p E dE + p S dS + p V dV + p dN + · · · + p s dN s , (12)where µ = p − p E , · · · , µ s = p s − p E . This section is concerned with the general definition and analysis of geometricobjects on the cotangent bundle without zero section, which project to theunderlying contact manifold. Since everything is based on the Liouville formthis will be called
Liouville geometry . In particular, we will deal with Liouvillesubmanifolds and homogeneous Hamiltonian dynamics.
In the previous section it was indicated how the thermodynamic phase spacecan be extended to a cotangent bundle, without its zero section, by the use ofhomogeneous coordinates for the intensive variables. Furthermore, it was shownhow in this way the energy and entropy representation are unified, and how thisprovides a geometric definition of extensive and intensive variables. Conversely,in this subsection we will start with a general cotangent bundle without zerosection, and show how this leads to the canonical contact manifold serving asthermodynamic phase space.Consider a thermodynamic system with total space of extensive variables,including energy E and entropy S , given by the manifold Q . Then consider theotangent bundle T ∗ Q without its zero section. The Liouville one-form α on T ∗ Q is defined as follows. Consider η ∈ T ∗ Q , X ∈ T η T ∗ Q , and define α η ( X ) := η (pr ∗ X ) , (13)where pr : T ∗ Q → Q is the bundle projection. Then ω := dα , with d exteriorderivative, is the canonical symplectic form on T ∗ Q . Furthermore, the Eulervector field Z is defined as the unique vector field satisfying dα ( Z, · ) = α (14)This also implies L Z α = α , with L denoting Lie derivative.In coordinates α, ω and Z take the following simple form. Let dim Q = n + 1, with local coordinates q , · · · , q n , and let p , · · · , p n be the correspondingcoordinates for the cotangent spaces T ∗ q Q . Then α = n X i =0 p i dq i , ω = n X i =0 dp i ∧ dq i , Z = n X i =0 p i ∂∂p i (15)Based on T ∗ Q we may define a canonical contact manifold in the following way[2]. For each q ∈ Q and each cotangent space T ∗ q Q consider the projective space P ( T ∗ q Q ), given as the set of rays in T ∗ q Q , that is, all the non-zero multiples ofa non-zero cotangent vector. Thus the projective space P ( T ∗ q Q ) has dimension n , and there is a canonical projection π q : T ∗ q Q → P ( T ∗ q Q ), where T ∗ q Q denotesthe cotangent space without its zero vector. The fiber bundle of the projectivespaces P ( T ∗ q Q ), q ∈ Q , over the base manifold Q will be denoted by P ( T ∗ Q ).Furthermore, denote the bundle projection obtained by considering π q : T ∗ q Q → P ( T ∗ q Q ) for every q ∈ Q by π : T ∗ Q → P ( T ∗ Q ).As detailed in [2, 3, 37, 36], P ( T ∗ Q ) defines a canonical contact manifold ofdimension 2 n +1. The contact manifold P ( T ∗ Q ) will serve as the thermodynamicphase space for the thermodynamic system with space of external variables Q .Given natural coordinates q , · · · , q n , p , · · · , p n for T ∗ Q , we may select dif-ferent sets of local coordinates for P ( T ∗ Q ) and corresponding different expres-sions of the projection π : T ∗ q Q → P ( T ∗ q Q ). In fact, whenever p = 0 we mayexpress the projection π q : T ∗ q Q → P ( T ∗ q Q ) by the map( p , p , · · · , p n ) ( γ , · · · , γ n ) (16)where γ = p − p , · · · , γ n = p n − p (17)This means that α = p dq + p dq + · · · + p n dq n = p (cid:0) dq − γ dq · · · − γ n dq n (cid:1) =: p θ, (18) In the sense that any other (2 n + 1)-dimensional contact manifold is locally contactomor-phic to P ( T ∗ Q ) [2, 25]. ith θ a locally defined contact form on P ( T ∗ Q ). Clearly, the same can bedone for any of the other coordinates p i , defining different contact forms. Forexample, if p = 0 we may express π q : T ∗ q Q → P ( T ∗ q Q ) also by the map( p , p , · · · , p n ) ( e γ , e γ , · · · , e γ n ) , (19)where e γ = p − p , e γ = p − p , · · · , e γ n = p n − p , (20)so that α = p (cid:0) dq − e γ dq − e γ dq · · · − e γ n dq n (cid:1) =: p e θ (21)In the thermodynamics context of Section 2, with q = E, q = S , and thus p = p E , p = p S , the first option corresponds to the energy representation andthe second to the entropy representation .Importantly, there is a direct correspondence between all geometric ob-jects (functions, Legendre submanifolds, vector fields) on the contact manifold P ( T ∗ Q ) with the same objects on T ∗ Q endowed with an additional homogene-ity property in the p variables. A key element in this is Euler’s theorem onhomogeneous functions; see e.g. [37]. Definition 3.1.
Let r ∈ Z . A function K : T ∗ Q → R is called homogeneous ofdegree r in p if K ( q, λp ) = λ r K ( q, p ) , for all λ = 0 (22) Theorem 3.2 (Euler’s homogeneous function theorem) . A differentiable func-tion K : T ∗ Q → R is homogeneous of degree r in p if and only if n X i =0 p i ∂K∂p i ( q, p ) = rK ( q, p ) , for all ( q, p ) ∈ T ∗ Q (23) Moreover, if K is homogeneous of degree r in p , then all its derivatives ∂K∂p i ( q, p ) , i = 0 , , · · · , n, are homogeneous of degree r − in p .Furthemore K : T ∗ Q → R is homogeneous of degree in p if and only if L Z K = 0 , and homogeneous of degree in p if and only if L Z K = K , where Z is the Euler vector field and L denotes Lie derivation. Since until Section 5 homogeneity will always refer to homogeneity in the p -variables we will often simply talk about ’homogeneity’.Obviously, functions K : T ∗ Q → R which are homogeneous of degree 0in p are those functions which project under π to functions on P ( T ∗ Q ), i.e., K = π ∗ b K with b K : P ( T ∗ Q ) → R . In the next two subsections we will considertwo more classes of objects which project to P ( T ∗ Q ). Legendre submanifolds of the canonical thermodynamic phase space P ( T ∗ Q )are in one-to-one correspondence with Liouville submanifolds of T ∗ Q , defined Previously called homogeneous Lagrangian submanifolds in [37]. s follows.
Definition 3.3.
A submanifold
L ⊂ T ∗ Q is called a Liouville submanifold ifthe Liouville form α restricted to L is zero and dim L = dim Q . Recall that L is a Lagrangian submanifold of T ∗ Q if ω = dα is zero on L anddim L = dim Q (or, equivalently, ω is zero on L and L is maximal with respectto this property.) The following proposition shows that Liouville submanifoldsare actually Lagrangian submanifolds of T ∗ Q with an additional homogeneityproperty. Proposition 3.4.
L ⊂ T ∗ Q is a Liouville submanifold if and only if L is aLagrangian submanifold of the symplectic manifold ( T ∗ Q , ω ) with the propertythat ( q, p ) ∈ L ⇒ ( q, λp ) ∈ L (24) for every = λ ∈ R .Proof. First of all note that the homogeneity property (24) is equivalent to tangency of the Euler vector field Z to L .(Only if) By Palais’ formula (see e.g. [1], Proposition 2.4.15) dα ( X , X ) = L X ( α ( X )) − L X ( α ( X )) − α ([ X , X ]) (25)for any two vector fields X , X . Hence, for any X , X tangent to L we obtain dα ( X , X ) = 0, implying that L is a Lagrangian submanifold. Furthermore,by (14) dα ( Z, X ) = α ( X ) = 0 , (26)for all vector fields X tangent to L . Because L is a Lagrangian submanifold thisimplies that Z is tangent to L (since a Lagrangian submanifold is a maximal submanifold restricted to which ω = dα is zero.)(If). If L is Lagrangian and satisfies (24), then Z is tangent to L , and thus (26)holds for all vector fields X tangent to L , implying that α is zero restricted to L . (cid:4) Remark 3.5.
It also follows that
L ⊂ T ∗ Q is a Liouville submanifold if andonly if it is a maximal submanifold on which α is zero. Liouville submanifolds of T ∗ Q are in one-to-one correspondence with Leg-endre submanifolds of the canonical contact manifold P ( T ∗ Q ). Recall that asubmanifold of a (2 n + 1)-dimensional contact manifold is a Legendre subman-ifold [2, 25] if the locally defined contact form θ is zero restricted to it, and itsdimension is equal to n (the maximal dimension of a submanifold on which θ iszero). Proposition 3.6 ([25], Proposition 10.16, [37]) . Consider the projection π : T ∗ Q → P ( T ∗ Q ) . Then b L ⊂ P ( T ∗ Q ) is a Legendre submanifold if and onlyif L := π − ( b L ) ⊂ T ∗ Q is a Liouville submanifold. Conversely, any Liouvillesubmanifold L ⊂ T ∗ Q is of the form π − ( b L ) for some Legendre submanifold b L . his implies as well a one-to-one correspondence between generating func-tions of Legendre submanifolds b L ⊂ P ( T ∗ Q ) and generating functions of Liou-ville submanifolds L ⊂ T ∗ Q with π − ( b L ). Recall from [25, 2] that any Legendresubmanifold b L ⊂ P ( T ∗ Q ) with Darboux coordinates q , q , · · · , q n , γ , · · · , γ n can be represented as b L = { ( q , q , · · · , q n , γ , · · · , γ n ) | q = b F − γ J ∂ b F∂γ J , q J = − ∂ b F∂γ J , γ I = ∂ b F∂q I } (27)for some disjoint partitioning I ∪ J = { , · · · , n } and some function b F ( q I , γ J ),called a generating function for b L . Here γ J is the vector with elements γ ℓ = p ℓ − p , ℓ ∈ J , and γ J ∂ b F∂γ J is shorthand notation for P ℓ ∈ J γ ℓ ∂ b F∂γ ℓ . Conversely anysubmanifold b L as given in (27), for any partitioning I ∪ J = { , · · · , n } and func-tion b F ( q I , γ J ), is a Legendre submanifold. This implies that the correspondingLiouville submanifold L = π − ( b L ) is given as L = { ( q , · · · , q n , p , · · · , p n ) | q = − ∂F∂p , q J = − ∂F∂p J , p I = ∂F∂q I } , (28)where F ( q I , p , p J ) := − p b F ( q I , p J − p ) (29)This is immediately verified by exploiting the identities − ∂F∂p = b F ( q I , − p J p ) + p ∂ b F∂γ J ( q I , − p J p ) p J p = b F ( q I , γ J ) − γ J ∂ b F∂γ J ∂F∂p J = − p ∂ b F∂γ J · − p = ∂ b F∂γ J , ∂F∂q I = − p ∂ b F∂q I = − p γ I = p I (30)Thus F ( q I , p , p J ) is a generating function of L . Conversely, any Liouvillesubmanifold as in (28) for some p (possibly after renumbering the index set { , , · · · , n } ) and generating function F as given in (29) for some b F ( q I , γ J ),with I ∪ J = { , · · · , n } and γ J = − p J p defines a Liouville submanifold of T ∗ Q .Note that the generating function F ( q I , p , p J ) = − p b F ( q I , p J − p ) as in (29)for the Liouville submanifold L is homogeneous of degree p . The corre-spondence (29) between the generating function F ( q I , p , p J ) of the Liouvillesubmanifold L = π − ( b L ) and the generating function b F ( q I , γ J ) of the Legendresubmanifold b L is of a well-known type in the theory of homogeneous functions.Indeed, for any function K ( q, p ) that is homogeneous of degree 1 in p , we candefine b K ( q, γ , · · · , γ n ) := K ( q, − , γ , · · · , γ n ) , (31)implying that K ( q, p , p , · · · , p n ) = − p b K ( q, p − p , · · · , p n − p ) (32)inally note that the correspondence between the Liouville submanifold L andthe Legendre submanifold b L and their generating functions can be obtained for any numbering of the set { , , · · · , n } , and thus for any choice of p . This pro-vides other coordinatizations of the same Legendre submanifold b L ⊂ P ( T ∗ Q ).The representation of b L either in energy or in entropy representation is an ex-ample of this. For any function K : T ∗ Q → R the Hamiltonian vector field X K on T ∗ Q isdefined by the standard Hamiltonian equations˙ q i = ∂K∂p i ( q, p ) , ˙ p i = − ∂K∂q i ( q, p ) , i = 0 , · · · , n, (33)or equivalently, ω ( X K , − ) = − dK . Note that since dα ( Z, · ) = α , we have α ( X K ) = dα ( Z, X K ) = L Z K = K . Hence a Hamiltonian K is homogeneous ofdegree 1 in p if and only if α ( X K ) = K (34)Furthermore Proposition 3.7. If K : T ∗ Q → R is homogeneous of degree in p then itsHamiltonian vector field X K satisfies L X K α = 0 (35) Conversely, if the vector field X satisfies L X α = 0 , then X = X K where thefunction K := α ( X ) is homogeneous of degree in p .Proof. By Cartan’s formula, with L denoting Lie derivative and i contraction, L X α = i X dα + di X α = i X dα + d ( α ( X )) (36)If K is homogeneous of degree 1 in p then by (34) i X K dα + d ( α ( X K )) = − dK + dK = 0, implying by (36) that L X K α = 0. Conversely, if L X α = 0, then (36)yields i X dα + d ( α ( X )), implying that X = X K with K = α ( X ), which by (34)is homogeneous of degree 1 in p . (cid:4) Thus the Hamiltonian vector fields with a Hamiltonian homogeneous of de-gree 1 in p are precisely the vector fields that leave the Liouville form α invariant.For simplicity of exposition the Hamiltonians K : T ∗ Q → R that are homoge-neous of degree 1 in p , and their corresponding Hamiltonian vector fields X K ,will be simply called homogeneous in the sequel.Note that by Theorem 3.2 (Euler’s theorem) the expressions ∂K∂p i ( q, p ) , i =0 , · · · , n, are homogeneous of degree 0 in p since K is homogeneous of degree 1in p . Hence the dynamics of the extensive variables q in (33) is invariant undercaling of the p -variables, and thus expressible as a function of q and the inten-sive variables γ . In fact, any homogeneous Hamiltonian vector field projects to a contact vector field on the thermodynamic phase space P ( T ∗ Q ), and converselyany contact vector field on P ( T ∗ Q ) is the projection of a homogeneous Hamil-tonian vector field on T ∗ Q . This can be seen from the following computations.Consider a homogeneous Hamiltonian vector field X K . Since K is homogeneousof degree 1 in p we can write as in (32) K ( q, p ) = − p b K ( q, p − p , · · · , p n − p ), with b K ( q, γ ) as defined in (31). This means that the equations (33) of the Hamilto-nian vector field X K take the form˙ q = − b K ( q, γ ) − p P nℓ =1 ∂ b K∂γ ℓ ( q, γ ) · − p ℓ p = − b K ( q, γ ) + P nℓ =1 γ ℓ ∂ b K∂γ ℓ ( q, γ )˙ q j = − p ∂ b K∂γ j ( q, γ ) · − p = ∂ b K∂γ j ( q, γ ) , j = 1 , · · · , n ˙ p i = p ∂ b K∂q i ( q, γ ) , i = 0 , · · · , n (37)where γ j = p j − p , j = 1 , · · · , n . Combining with˙ γ j = 1 − p ˙ p j + p j p ˙ p , j = 1 , · · · , n, (38)this yields the following projected dynamics on the contact manifold P ( T ∗ Q )with coordinates ( q, γ )˙ q = P nℓ =1 γ ℓ ∂ b K∂γ ℓ ( q, γ ) − b K ( q, γ )˙ q j = ∂ b K∂γ j ( q, γ ) , j = 1 , · · · , n ˙ γ j = − ∂ b K∂q j ( q, γ ) − γ j ∂ b K∂q ( q, γ ) , j = 1 · · · , n (39)This is recognized as the contact vector field [25] with contact Hamiltonian b K . Indeed, given a contact form θ the contact vector field X b K with contactHamiltonian b K is defined through the relations L X c K θ = ρ b K θ, − b K = θ ( X b K ) (40)for some function ρ b K (depending on b K ). The first equation in (40) expresses thecondition that the contact vector field leaves the contact distribution (the kernelof the contact form θ ) invariant. Equations (40) for θ = dq − γ dq · · · − γ n dq n and b K ( q, γ ) can be seen to yield the same equations as in (39); see [25, 11] fordetails. Conversely, any contact vector field with contact Hamiltonian b K ( q, γ )defines a homogeneous Hamiltonian vector field on T ∗ Q with homogeneousHamiltonian − p b K ( q, p − p , · · · , p n − p ). As before, the coordinate expression (39) Here the sign convention of [7] is followed. f the contact vector field depends on the numbering of the homogeneous coor-dinates p , p , · · · , p n ; i.e., the choice of p . In the thermodynamics context thisis again illustrated by the choice of either the energy or entropy representation(corresponding to choosing p = p E or p = p S ).The projectability of any homogeneous Hamiltonian vector field X K to acontact vector field X b K on P ( T ∗ Q ) also follows from the following proposition,and the fact that the projection π : T ∗ Q → P ( T ∗ Q ) is along the Euler vectorfield Z . Proposition 3.8.
Any homogeneous Hamiltonian vector field X K satisfies [ X K , Z ] = 0 .Proof. By [1](Table 2.4-1) i [ X K ,Z ] dα = L X K i Z dα − i Z L X K dα = L X K α − i Z d L X K α = 0 − , (41)since L X K α = 0. Because ω = dα is a symplectic form this implies [ X K , Z ] = 0. (cid:4) Although homogeneous Hamiltonian vector fields are in one-to-one corre-spondence with contact vector fields, typically computations for homogeneousHamiltonian vector fields are much easier than the corresponding computationsfor their contact vector field counterparts. First note the following propertiesproved in [37, 36].
Proposition 3.9.
Consider the Poisson bracket { K , K } of functions K , K on T ∗ Q defined with respect to the symplectic form ω = dα . Then(a) If K , K are both homogeneous of degree in p , then also { K , K } ishomogeneous of degree in p .(b) If K is homogeneous of degree in p , and K is homogeneous of degree in p , then { K , K } is homogeneous of degree in p .(c) If K , K are both homogeneous of degree in p , then { K , K } is zero. Using property ( a ) we may define the following bracket { b K , b K } J := \ { K , K } (42)where b K is the contact Hamiltonian corresponding to the homogeneous Hamil-tonian K as in (40). The bracket { b K , b K } J is equal to the Jacobi bracket ofthe contact Hamiltonians b K , b K ; see e.g. [25, 7, 2] for the coordinate expres-sions of the Jacobi bracket. The Jacobi bracket is obviously bilinear and skew-symmetric. Furthermore, since the Poisson bracket satisfies the Jacobi-identity,so does the Jacobi bracket. However, the Jacobi bracket does not satisfy theLeibniz rule; i.e., in general the following equality does not hold { b K , b K · b K } J = { b K , b K } J · b K + b K · { b K , b K } J (43)See also [39] for additional information on the Jacobi bracket. .4 Hamilton-Jacobi theory of Liouville and Legendre sub-manifolds Recall that any homogeneous Hamiltonian vector field X K on T ∗ Q leaves in-variant the Liouville form α and that Liouville submanifolds are maximal sub-manifolds on which α is zero. It follows that for any Liouville submanifold L and any time t ∈ R the evolution of L along the homogeneous Hamiltonianvector field X K given as φ t ( L ) := { φ t ( z ) | z ∈ L} , (44)where φ t : T ∗ Q → T ∗ Q is the flow map at time t ≥ X K , is also a Liou-ville submanifold. Applied to the Liouville submanifold characterizing the stateproperties of a thermodynamic system this means that the flow of a homoge-neous Hamiltonian vector field transforms the Liouville submanifold to anotherLiouville submanifold at any time t ≥
0. For example, the Liouville subman-ifold corresponding to an ideal gas may be continuously transformed into theLiouville submanifold of a Van der Waals gas. This point of view was exploredin [29, 30, 32].Furthermore, cf. (29), let F ( q I , p , p J ) := − p b F ( q I , p J − p ), with I ∪ J = { , · · · , n } , be the generating function of L , then it follows that for any t ≥ G ( q I , p , p J , t ) := − p b G ( q I , p J − p , t ) of the transformedLiouville submanifold φ t ( L ) satisfies the Hamilton-Jacobi equation ∂G∂t + K ( q , q I , − ∂G∂p J , p , ∂G∂q J , p J ) = 0 G ( q I , p , p J ,
0) = F ( q I , p , p J ) (45)In case of the evolution of a general Lagrangian submanifold under the dy-namics of a general Hamiltonian vector field this is classical Hamilton-Jacobitheory (see e.g. [1, 2]), which directly specializes to Liouville submanifoldsand homogeneous Hamiltonian vector fields. Furthermore, the generating func-tions b G ( q I , γ J , t ) of the corresponding Legendre submanifolds \ φ t ( L ) satisfy theHamilton-Jacobi equation (see also [8]) ∂ b G∂t + b K ( q = b G − γ J ∂ b G∂γ J , q J = − ∂ b F∂γ J , γ I = ∂ b F∂q I ) = 0 b G ( q I , γ J ,
0) = b F ( q I , γ J ) (46)Note furthermore that \ φ t ( L ) = b φ t ( b L ), where b φ t is the flow map at time t of thecontact vector field X b K . This implies as well the following result concerning invariance of Liouville and corresponding Legendre submanifolds, which will beone of the starting points for the definition of port-thermodynamic systems inthe following section. Proposition 3.10. [31, 25, 36] Let K : T ∗ Q → R be homogeneous of degree in p , and let b K : P ( T ∗ Q ) → R be the corresponding contact Hamiltonian.urthermore let L ⊂ T ∗ Q be a Liouville submanifold, and b L ⊂ P ( T ∗ Q ) , with L = π − ( b L ) , the corresponding Legendre submanifold. Then the following state-ments are equivalent:1. The homogeneous Hamiltonian vector field X K leaves L invariant.2. The contact vector field X b K leaves b L invariant.3. K is zero on L .4. b K is zero on b L . So far the geometric description of classical thermodynamics has been concernedwith the state properties ; starting from Gibbs’ fundamental relation. Since thesestate properties are intrinsic to any thermodynamic system, they should be re-spected by any dynamics (thermodynamic processes). Hence any dynamics ofan actual thermodynamic system should leave invariant the Liouville and Legen-dre submanifold characterizing the state properties [31, 33, 6, 37]. Furthermore,desirably this should be the case for all possible state properties of the thermo-dynamic system, i.e., for all Liouville and Legendre submanifolds. This suggeststhat the dynamics on the canonical thermodynamic phase space P ( T ∗ Q ) shouldbe a contact vector field X b K , and the corresponding dynamics on T ∗ Q shouldbe a homogeneous Hamiltonian vector field X K .Because of its simplicity, we first focus on the homogeneous Hamiltoniandescription. Consider a thermodynamic system with constitutive relations (stateproperties) specified by a Liouville submanifold L ⊂ T ∗ Q . Respecting thegeometric structure means that the dynamics is a Hamiltonian vector field X K on T ∗ Q , with K homogeneous of degree 1 in the p -variables. Furthermore, sincethe state properties captured by L are intrinsic to the system, the homogeneousHamiltonian vector field X K should leave L invariant. By Proposition 3.10 thismeans that the homogeneous Hamiltonian K governing the dynamics should bezero on L . Furthermore, we will split K into two parts, i.e., K a + K c u, u ∈ R m , (47)where K a : T ∗ Z → R is the homogeneous Hamiltonian corresponding to the autonomous dynamics due to internal non-equilibrium conditions, while K c =( K c , · · · , K cm ) is a row vector of homogeneous Hamiltonians (called control or interaction Hamiltonians) corresponding to dynamics arising from interactionwith the surrounding of the system. This second part of the dynamics will besupposed to be affinely parametrized by a vector u of control or input variables(see however [37] for an example of non-affine dependency). This means that all( m + 1) functions K a , K c , · · · , K cm are homogeneous of degree 1 in p and zeroon L .y invoking Euler’s homogeneous function theorem (cf. Theorem 3.2) ho-mogeneity of degree 1 in p means K a = p ∂K a ∂p + p ∂K a ∂p + · · · + p n ∂K a ∂p n K c = p ∂K c ∂p + p ∂K c ∂p + · · · + p n ∂K c ∂p n , (48)where the functions ∂K a ∂p i , as well as the elements of the m -dimensional row vec-tors of partial derivatives ∂K c ∂p i , i = 0 , , · · · , n , are all homogeneous of degree 0in the p -variables. (Hence, as noted before, the dynamics of the extensive vari-ables can be expressed as a function of the extensive variables and the intensivevariables.)The class of allowable autonomous Hamiltonians K a is further restricted bythe First and Second Law of thermodynamics. Since the energy and entropyvariables
E, S are among the extensive variables q , q , · · · , q n , let us denote q = E, q = S . With this convention, the evolution of E in the autonomousdynamics X K a arising from non-equilibrium conditions is given by ˙ E = ∂K a ∂p .Since by the First Law the energy of the system without interaction with thesurrounding (i.e., for u = 0) should be conserved , this implies that necessarily ∂K a ∂p | L = 0. Similarly, ˙ S in the autonomous dynamics X K a is given by ∂K a ∂p .Hence by the Second Law necessarily ∂K a ∂p | L ≥ K c . In fact, the analogous terms in the control Hamiltonians may beutilized to define natural output variables. First option is to define the outputvector as the m -dimensional row vector ( p for power) y p = ∂K c ∂p (49)Then it follows that along the complete dynamics X K on L , with K = K a + K c u , ddt E = y p u (50)Thus y p is the vector of power-conjugate outputs corresponding to the inputvector u . We call the pair ( u, y p ) the power port of the system. Similarly, bydefining the output vector as the m -dimensional row vector ( e for ’entropy flow’) y e = ∂K c ∂p (51)it follows that along the dynamics X K on L ddt S ≥ y e u (52)Hence y e is the output vector which is conjugate to u in terms of entropy flow .The pair ( u, y e ) is called the flow of entropy port of the system.The above discussion is summarized in the following definition of a port-thermodynamic system . efinition 4.1 ([37]) . Consider the manifold of extensive variables Q . A port-thermodynamic system on Q is a pair ( L , K ) , where L ⊂ T ∗ Q is a Liouvillesubmanifold describing the state properties, and K = K a + K c u, u ∈ R m , isa Hamiltonian on T ∗ Q , homogeneous of degree in p , and zero restricted to L , which generates the dynamics X K . Furthermore, let q = ( q , q , · · · , q n ) with q = E (energy), and q = S (entropy). Then K a is required to satisfy ∂K a ∂p | L = 0 and ∂K a ∂p | L ≥ . The power conjugate output vector of the port-thermodynamic system is defined as y p = ∂K c ∂p , and the entropy flow conjugateoutput vector as y e = ∂K c ∂p . Note that any port-thermodynamic system on T ∗ Q immediately defines acorresponding system on the thermodynamic phase space P ( T ∗ Q ). Indeed, since L ⊂ T ∗ Q is a Liouville submanifold it projects to a Legendre submanifold b L ⊂ P ( T ∗ Q ). Furthermore, since K is homogeneous of degree 1 in p it has theform K ( q, p ) = − p b K ( q, γ ), γ j = p j − p , j = 1 , · · · , n , with b K ( q, γ ) = b K a ( q, γ ) + b K c ( q, γ ) u the contact Hamiltonian of the energy representation . This contactHamiltonian is zero on b L , and the dynamics X K projects to the contact vectorfield X b K that leaves invariant b L . Similarly, we can write K ( q, p ) = − p be K ( q, ˜ γ ),˜ γ j = p j − p , j = 0 , · · · , n , with be K ( q, ˜ γ ) the contact Hamiltonian of the entropyrepresentation . Furthermore, by Euler’s theorem both the power conjugateoutput y p and the entropy flow conjugate output y e are homogeneous of degree0, and thus project to functions on P ( T ∗ Q ). Finally, in the energy representationwe can rewrite the power conjugate output as y p = ∂K c ∂p = n X ℓ =1 γ ℓ ∂ b K c ∂γ ℓ ( q, γ ) − b K c ( q, γ ) (53)Similarly for the entropy flow conjugate output y e = ∂K c ∂p = P nℓ =0 , ˜ γ ℓ ∂ be K c ∂ ˜ γ ℓ ( q, ˜ γ ) − be K c ( q, ˜ γ ). Finally note that the constraints imposed on K a by the First and Sec-ond law can be written in contact-geometric terms as (cid:16)P nℓ =1 γ ℓ ∂ b K a ∂γ ℓ ( q, γ ) − b K a ( q, γ ) (cid:17) | b L = 0 (cid:18)P nℓ =0 , ˜ γ ℓ ∂ be K a ∂ ˜ γ ℓ ( q, γ ) − be K a ( q, ˜ γ ) (cid:19) | b L ≥ Example 4.2 (Gas-piston-damper system) . Consider a gas in a thermally iso-lated compartment closed by a piston. Assume the thermodynamic properties ofthe system to be fully covered by the properties of the gas. The extensive variablesare given by energy E , entropy S , volume V , and momentum of the piston π .The state properties of the system are described by the Liouville submanifold L with generating function (in energy representation) − p E (cid:16) U ( S, V ) + π m (cid:17) , where U ( S, V ) is the energy of the gas, and π m the kinetic energy of the piston withass m . This defines the state properties L = { ( E, S, V, π, p E , p S , p V , p π ) | E = U ( S, V ) + π m ,p S = − p E ∂U∂S ( S, V ) , p V = − p E ∂U∂V ( S, V ) , p π = − p E πm } (55) Assume the damper is linear with damping constant d . The dynamics of thegas-piston-damper system, with piston actuated by a force u , is given by X K ,where the homogeneous Hamiltonian K : T ∗ R → R is given as K = p V πm + p π (cid:18) − ∂U∂V − d πm (cid:19) + p S d ( πm ) ∂U∂S + (cid:16) p π + p E πm (cid:17) u, (56) which is zero on L . The power-conjugate output y p = πm is the velocity of thepiston. In energy representation the description projects to the thermodynamicphase space P ( T ∗ R ) = { ( E, S, V, π, T, − P, v ) } , with γ S = T (temperature), γ V = − P (pressure), and γ π = v (velocity of the piston) as follows. First notethat L projects to the Legendre submanifold b L = { ( E, S, V, π, T, − P, v ) | E = U ( S, V ) + π m , T = ∂U∂S , − P = ∂U∂V , v = πm } (57) Furthermore, K = − p E b K with b K = − P πm + v (cid:18) − ∂U∂V − d πm (cid:19) + T d ( πm ) ∂U∂S + ( v − πm ) u (58) This yields the following dynamics of the extensive variables ˙ E = πm u ˙ S = d ( πm ) / ∂U∂S ( ≥ V = πm ˙ π = − ∂U∂V − d πm + u, (59) while the intensive variables satisfy ˙ T = − ∂ b K∂S , − ˙ P = − ∂ b K∂V , ˙ v = − ∂ b K∂π . Similarlyfor the entropy representation. In composite thermodynamic systems, there is typically no single energyor entropy. In this case the sum of the energies needs to be conserved bythe autonomous dynamics, and likewise the sum of the entropies needs to beincreasing. A simple example is the following; see [37] for further information. Example 4.3 (Heat exchanger) . Consider two heat compartments, exchanginga heat flow through a conducting wall according to Fourier’s law. Each heatcompartment is described by an entropy S i and energy E i , i = 1 , , correspondingto the Liouville submanifolds L i = { ( E i , S i , p E i , p S i | E i = E i ( S i ) , p S i = − p E i E ′ i ( S i ) } , E ′ i ( S i ) ≥ aking u i as the incoming heat flow into the i -th compartment corresponds to K ci = p S i E ′ i ( S i ) + p E i , (61) while K ai = 0 . This defines the flow of entropy conjugate outputs as y ei = E ′ i ( S i ) (reciprocal temperatures). The conducting wall is described by the interconnec-tion equations (with λ Fourier’s conduction coefficient) − u = u = λ ( 1 y e − y e ) , (62) relating the incoming heat flows u i and reciprocal temperatures y i , i = 1 , , atboth sides of the conducting wall. This leads to (setting E ( S , S ) := E ( S ) + E ( S ) , p E = p E =: p E , cf. [37]) to the autonomous dynamics generated bythe homogeneous Hamiltonian K a := K c u + K c u = λ (cid:18) p S E ′ ( S ) + p S E ′ ( S ) (cid:19) ( E ′ ( S ) − E ′ ( S )) (63) Hence the total entropy on the Liouville submanifold L = { ( E, S , S , p E , p S , p S ) | E = E + E , p S = − p E E ′ ( S ) , p S = − p E E ′ ( S ) } (64) satisfies ddt ( S + S ) = λ ( 1 E ′ ( S ) − E ′ ( S ) )( E ′ ( S ) − E ′ ( S )) ≥ energy , the Hamiltonians K in the above examplesare dimensionless (in the sense of dimensional analysis). This holds in general.Furthermore, it can be verified that the contact Hamiltonian of its projecteddynamics (a contact vector field) has dimension of power in case of the energyrepresentation (with intensive variables T, − P ), and has dimension of entropyflow in case of the entropy representation (with intensive variables T , PT ). To-gether with the fact that the dynamics of a thermodynamic system is capturedby the dynamics restricted to the invariant Liouville submanifold, this empha-sizes that the interpretation of the Hamiltonian dynamics X K is rather different from the Hamiltonian formulation of mechanical (or other physical) systems.Finally, let us recall the well-known correspondence [25, 2] between Pois-son brackets of Hamiltonians K , K , and Lie brackets of their correspondingHamiltonian vector fields, i.e.,[ X K , X K ] = X { K ,K } (66)In particular, this property implies that if the homogeneous Hamiltonians K , K are zero on the Liouville submanifold L , and thus by Proposition 3.10 the ho-mogeneous Hamiltonian vector fields X K , X K are tangent to L , then also X K , X K ] is tangent to L , and therefore the Poisson bracket { K , K } is alsozero on L . Together with Proposition 3.9 this was crucially used in the control-lability and observability analysis of port-thermodynamic systems in [38]. In many thermodynamic systems, when taking into account all extensive vari-ables, there is an additional form of homogeneity; now with respect to the extensive variables q . To start with, consider a Liouville submanifold L withgenerating function − p b F ( q , · · · , q n ). Recall that if q denotes the energy vari-able, then b F ( q , · · · , q n ) equals the energy q expressed as a function of the otherextensive variables q , · · · , q n . Assume that the manifold of extensive variables Q is the linear space Q = R n +1 . Homogeneity with respect to the extensivevariables means that the function b F is homogeneous of degree 1 in q , · · · , q n .This implies by Euler’s theorem (Theorem 3.2) that b F = P nj =1 q j ∂ b F∂q j . Henceon the corresponding Legendre submanifold b L = π ( L ) we have b F = P nj =1 γ j q j ,and thus d b F = n X j =1 γ j dq j + n X j =1 q j dγ j (67)By Gibbs’ relation this implies that on b L n X j =1 q j dγ j = 0 , (68)which is known as the Gibbs-Duhem relation ; see e.g. [24, 18]. The relationimplies that the intensive variables γ j on b L are dependent .More generally this can be formulated in the following geometric way. Definition 5.1.
Let Q = R n +1 with linear coordinates q . A Liouville subman-ifold L ⊂ T ∗ R n +1 is homogeneous with respect to the extensive variables q if ( q , q , · · · , q n , p , · · · , p n ) ∈ L ⇒ ( µq , µq , · · · , µq n , p , · · · , p n ) ∈ L (69) for all = µ ∈ R . Using the same theory as exploited before for homogeneity with respectto the p -variables, cf. Proposition 3.4, homogeneity of L with respect to q isequivalent to the vector field W := P ni =0 q i ∂∂q i being tangent to L . Hence,using the same argumentation as in Proposition 3.4, not only the Liouville form α = P ni =0 p i dq i is zero on L , but also the one-form β := n X i =0 q i dp i (70) Homogeneity can be generalized to manifolds using the theory developed in [25]. his could be called the generalized
Gibbs-Duhem relation.
Proposition 5.2.
The Liouville submanifold L is homogeneous with respectto the extensive variables q if and only if β = P ni =0 q i dp i is zero on L . Let L have generating function − p b F ( q I , γ J ) for some partitioning { , · · · , n } = I ∪ J .Then L is homogeneous with respect to the extensive variables q if and only ifif I is non-empty and b F ( q I , γ J ) is homogeneous of degree in q I . Furthermore,if L is homogeneous with respect to the extensive variables q , then n X i =0 q i p i = 0 , for all ( q, p ) ∈ L (71) Proof.
As mentioned above, the first statement follows from the same reasoningas in Proposition 3.4, swapping the p and q variables. Equivalence of homo-geneity of L with respect to q to b F ( q I , γ J ) being homogeneous of degree 1 in q I directly follows from the expression of L in (27) in case I = ∅ , while clearly ho-mogeneity of L fails if I = ∅ . Finally, if both α = P ni =0 p i dq i and β = P ni =0 q i dp i are zero on L , then d ( P ni =0 q i p i ) is zero on L . Hence P ni =0 q i p i is constant on L . Since Z = P ni =0 p i ∂∂p i is tangent to L necessarily this constant is zero. (cid:4) Remark 5.3.
In a contact-geometric setting, an identity similar to (71) wasnoticed in [22]. A related scenario, explored in [9], is the case that L is a La-grangian submanifold which is non-mixing: there exists a partitioning { , , · · · n } = I ∪ J such that q J = q J ( q I ) , p I = p I ( p J ) for all ( q I , q J , p I , p J ) ∈ L . Then L being Lagrangian amounts to ∂q J ∂q I = − (cid:18) ∂p I ∂p J (cid:19) ⊤ (72) Since the left-hand side only depends on q I and the right-hand side only on p J ,this means that both sides are constant, implying that q J = Aq I , p I = − A ⊤ p J for some matrix A . Hence L is obviously satisfying (71) , and is actually theproduct of two orthogonal linear subspaces; one in Q = R n +1 and the other inthe dual space Q ∗ = R n +1 . Homogeneity of L with respect to q has the following classical implication.Consider again the case of a generating function F ( q, p ) = − p b F ( q , · · · , q n ) for L , with q being the energy variable. Since b F is homogeneous of degree 1 wemay define for q = 0¯ F ( ǫ , · · · , ǫ n ) := b F (1 , q q , · · · , q n q ) = 1 q b F ( q , · · · , q n ) , ǫ j := q j q , j = 0 , , · · · , n (73)Equivalently, b F ( q , · · · , q n ) = q ¯ F ( ǫ , · · · , ǫ n ), where the function ¯ F is knownas the specific energy [24].eometrically this means the following. By homogeneity with respect tothe p -variables the Liouville submanifold L ⊂ T ∗ R n +1 is projected to the Leg-endre submanifold b L ⊂ R n +1 × P ( R n +1 ), where P ( R n +1 ) is the n -dimensionalprojective space. Subsequently, by homogeneity with respect to the q -variables b L ⊂ R n +1 × P ( R n +1 ) is projected to a submanifold ¯ L ⊂ P ( R n +1 ) × P ( R n +1 ). Incoordinates the expression of ¯ L is given as follows. Start from the expression of b L as given in (27). Using the identities q = q ¯ F ( ǫ , · · · , ǫ n ) ⇔ ǫ = ¯ F ( ǫ , · · · , ǫ n ) γ = ∂ b F∂q = ¯ F ( ǫ , · · · , ǫ n ) − q P nℓ =2 ∂ ¯ F∂ǫ ℓ q ℓ q = ¯ F ( ǫ , · · · , ǫ n ) − P nℓ =2 ǫ ℓ ∂ ¯ F∂ǫ ℓ γ j = ∂ b F∂q j = ∂ ( q ¯ F ) ∂q j = ∂ ¯ F∂ǫ j , j = 2 , · · · , n (74)the description (27) amounts to¯ L = { ( ǫ , ǫ , · · · , ǫ n , γ , · · · , γ n ) | ǫ = ¯ F ( ǫ , · · · , ǫ n ) ,γ = ¯ F ( ǫ , · · · , ǫ n ) − P nℓ =2 ǫ ℓ ∂ ¯ F∂ǫ ℓ , γ = ∂ ¯ F∂ǫ , · · · , γ n = ∂ ¯ F∂ǫ n } , (75)where F ( q, p ) = − p b F ( q ) = − p q ¯ F ( ǫ , · · · , ǫ n ) , ǫ j := q j q , j = 0 , , · · · , n (76)Similar expressions hold in the general case that the generating function for b L is given by b F ( q I , γ J ) for some partitioning { , · · · , n } = I ∪ J .Furthermore, if the state properties captured by L are homogeneous withrespect to q , it is natural to require the dynamics to be homogeneous withrespect to q as well. Thus one requires the Hamiltonian K ( q, p ) governing thedynamics to be homogeneous of degree 1, not only with respect to p , but alsowith respect to q , i.e., K ( µq, p ) = µK ( q, p ) , for all 0 = µ ∈ R (77)Equivalently (analogously to Proposition 3.7) one requires X K to satisfy L X K β = 0 (78)Similarly to Proposition 3.8, this implies[ X K , W ] = 0 , W = n X i =0 q i ∂∂q i (79)Hence the flow of X K commutes both with the flow of the Euler vector field Z = P ni =0 p i ∂∂p i and with the vector field W = P ni =0 q i ∂∂q i .We have seen before that projection along Z yields the contact vector field X b K , with K ( q, p ) = − p b K ( q, γ ) , γ j = p j − p , j = 1 , · · · , n , where ( q, γ ) ∈ R n +1 × ( R n +1 ). Subsequent projection along W to the reduced space P ( R n +1 ) × P ( R n +1 ) can be computed as follows. First write as above b K ( q, γ ) = q ¯ K ( ǫ, γ ) , ǫ j = q j q , j = 0 , , · · · , n (80)Then compute, analogously to (30), ∂ b K∂q = ¯ K − P nℓ =0 , ǫ ℓ ∂ ¯ K∂ǫ ℓ ∂ b K∂q j = ∂ ¯ K∂ǫ j , j = 0 , · · · , n ∂ b K∂γ j = q ∂ ¯ K∂γ j , j = 1 , · · · , n (81)Combining, analogously to (38), with the expression˙ ǫ j = ˙ q j q − q j q ˙ q , (82)this yields the following 2 n -dimensional dynamics on the reduced thermodynamicphase space P ( R n +1 ) × P ( R n +1 )˙ ǫ j = ∂ ¯ K∂γ j − ǫ j (cid:16)P nℓ =1 γ ℓ ∂ ¯ K∂γ ℓ − ¯ K (cid:17) , j = 0 , , · · · , n ˙ γ j = − ∂ ¯ K∂ǫ j + γ j (cid:16)P nℓ =0 , ǫ ℓ ∂ ¯ K∂ǫ ℓ − ¯ K (cid:17) , j = 1 , · · · , n, (83)where ¯ K is determined by K ( q, p ) = − p q ¯ K ( ǫ, γ ) , ǫ = (cid:18) q q , q q · · · , q n q (cid:19) , γ = (cid:18) p − p , · · · , p n − p (cid:19) (84)Obviously, if q represents entropy the same expressions hold with differentinterpretation of ǫ , ǫ , · · · , ǫ n .Note that the 2 n -dimensional dynamics (83) consists of standard Hamilto-nian equations with respect to the Hamiltonian ¯ K , together with extra terms.In view of (54), the first part of these extra terms for the autonomous term ¯ K a ,i.e., P nℓ =1 γ ℓ ∂ ¯ K a ∂γ ℓ − ¯ K a , is zero on L .As a final remark it can be noted that while the above reduction from L and X K to ¯ L and the dynamics (83) was done via b L and X b K (the contact-geometricdescription on the thermodynamic phase space), the same outcome is obtainedby instead first projecting onto P ( R n +1 ) × R n +1 along W , and then projectingonto P ( R n +1 ) × P ( R n +1 ) along Z . Said otherwise, this alternative route involvesa different intermediate contact geometric description on the contact manifold P ( R n +1 ) × R n +1 with coordinates ǫ , ǫ , · · · , ǫ n , p , · · · , p n . The geometric formulation of classical thermodynamics gives rise to a specificbranch of symplectic geometry, coined as
Liouville geometry , which is closelyelated to contact geometry. A detailed treatment of Liouville submanifolds andtheir generating functions has been provided. The same has been done for ho-mogeneous Hamiltonian vector fields, extending the treatment in e.g. [2, 3, 25].For the formulation of the Weinhold and Ruppeiner metrics in this setting werefer to [37]. The interpretation of the resulting Hamiltonian formulation ofport-thermodynamic systems turns out to be rather different from Hamiltonianformulations of other parts of physics, such as mechanics. In particular, the stateproperties of the thermodynamic system define a Liouville submanifold, whichis left invariant by the Hamiltonian dynamics. Furthermore, the Hamiltonianis dimensionless, while its corresponding contact Hamiltonians have dimensionof power (energy representation) or entropy flow (entropy representation). Anopen modeling problem concerns the determination of the Hamiltonian govern-ing the dynamics. A partial answer is given in [37], where it is shown how theHamiltonian of a thermodynamic system can be derived from the Hamiltoniansof the constituent thermodynamic subsystems. In Section 5 another type ofhomogeneity has been considered; this time with respect to the extensive vari-ables, corresponding to the classical Gibbs-Duhem relation. It has been shownhow this gives rise to a further projected dynamics on the product of the n -dimensional projective space with itself. The precise geometric interpretationand properties of the reduced dynamics (83) deserve further study. Acknowledgements
I thank Bernhard Maschke, Universit´e de Lyon-1, France, for ongoing collabo-rations that stimulated the writing of the present paper.
References [1] R.A. Abraham, J.E. Marsden,
Foundations of Mechanics , 2nd ed., Ben-jamin/Cummings, Reading, MA, 1978.[2] V.I. Arnold,
Mathematical Methods of Classical Mechanics , Springer, 2ndedition, 1989.[3] V.I. Arnold,
Contact Geometry and Wave Propagation , Lectures at theUniversity of Oxford under the sponsorship of the International Mathe-matical Union, L’Enseignement Math´ematique, 1989.[4] V.I. Arnold, Contact geometry: the geometrical method of Gibbs’s ther-modynamics,
Gibbs Symposium , AMS, 1989.[5] R. Balian, P. Valentin, Hamiltonian structure of thermodynamics withgauge,
Eur. J. Phys. B , 21:269–282, 2001.[6] A. Bravetti, Contact Hamiltonian dynamics: The concept and its use,
Entropy , 19(12):535, 2017.7] A. Bravetti, Contact geometry and thermodynamics,
Int. J. GeometricMethods in Modern Physics , 16(1), 1940003 (51 pages), 2019.[8] A. Bravetti, C.S. Lopez-Monsalvo, F. Nettel, Contact symmetries andHamiltonian thermodynamics,
Annals of Physics , 361:377 – 400, 2017.[9] R. K. Brayton, Nonlinear reciprocal networks, pp. 1–12,
Proc. Symp. inApplied Mathematics of AMS and SIAM , eds. H.S. Wilf, F. Harary, 1969.[10] M. de Leon, M. Lainz Valcazar, Contact Hamiltonian systems,
Journal ofMathematical Physics , 60(10):102902, 2019.[11] D. Eberard, B.M. Maschke, A.J. van der Schaft, An extension of pseudo-Hamiltonian systems to the thermodynamic space: towards a geometry ofnon-equilibrium thermodynamics,
Reports in Mathematical Physics , 60(2),175–198, 2007.[12] S.C. Farantos, Hamiltonian classical thermodynamics and chemical kinet-ics,
Physica
D (2020), doi: https://doi.org/10.1016/j.physd.2020.132813.[13] A. Favache, D. Dochain, B.M. Maschke. An entropy-based formulation ofirreversible processes based on contact structures,.
Chemical EngineeringScience , 65, 5204–5216, 2010.[14] E. Fermi,
Thermodynamics , Prentice-Hall, 1937 (Dover edition, 1956).[15] F. Gay-Balmaz, H. Yoshimura, A Lagrangian variational formulation fornonequilibrium, thermodynamics. Part i: Discrete systems,
Journal ofGeometry and Physics , 111, 169 – 193, 2017.[16] M. Grmela. Contact geometry of mesoscopic thermodynamics and dynam-ics.
Entropy , 16(3), 1652, 2014.[17] D. Gromov, F. Castanos, The geometric structure of interconnectedthermo-mechanical systems, IFAC World Congress, Toulouse, France,
IFAC-Papers OnLine , 50(1), 582–587, 2017.[18] D. Gromov, A. Toikka, Towards formal analysis of thermody-namic stability: Le Chatelier-Brown principle,
Entropy , 22, 1113;doi:10.3390/e22101113, 2020.[19] H.W. Haslach, Jr., Geometric structure of the non-equilibrium thermo-dynamics of homogeneous systems,
Reports in Mathematical Physics , 39,147–162, 1997.[20] G. Herglotz,
Ber¨uhrungstransformationen . In Lectures at the Universityof G¨ottingen, G¨ottingen, 1930. English edition: The Herglotz Lectures onContact Transformations and Hamiltonian Systems, by R. B. Guenther, H.Schwerdtfeger, G. Herglotz, C.M. Guenther, J.A. Gottsch, Julius SchauderCenter for Nonlinear Studies, Nicholas Copernicus University, Torun, 1996.21] R. Hermann,
Geometry, physics and systems , Marcel Dekker, New York,1973.[22] N.H. Hoang, T.K. Phung, T.T. Hong Phan, D. Dochain, On contact Hamil-tonian functions in open irreversible thermodynamic systems, preprint2020.[23] N. Hudon, M. Guay, D. Dochain, Control design for thermodynamic sys-tems on contact manifolds,
IFAC-Papers OnLine , 50(1), 588–593, 2017.[24] D. Kondepudi, I. Prigogine,
Modern Thermodynamics; From Heat Enginesto Dissipative Structures , 2nd edition, Wiley, 2015.[25] P. Libermann, C.-M. Marle,
Symplectic geometry and analytical mechan-ics , D. Reidel Publishing Company, Dordrecht, Holland, 1987.[26] B. Maschke, About the lift of irreversible thermodynamic systems to thethermodynamic phase space,
IFAC-Papers OnLine , 49(24), 40–45, 2016.[27] B. Maschke, A. van der Schaft, Homogeneous Hamiltonian control systems,Part II: Applications to thermodynamic systems,
IFAC-Papers OnLine
Con-tinuum Mechanics and Thermodynamics , 25(6), 779–793, 2013.[29] R. Mruga la, Geometric formulation of equilibrium phenomenological ther-modynamics.
Reports in Mathematical Physics , 14(3), 419–427, 1978.[30] R. Mruga la. Submanifolds in the thermodynamic phase space.
Reports inMathematical Physics
21, 197, 1985.[31] R. Mruga la, J.D. Nulton, J.C. Sch¨on, P. Salamon, Contact structure inthermodynamic theory,
Reports in Mathematical Physics , 29(1), 109–121,1991.[32] R. Mruga la, Continuous contact transformations in Thermodynamics.
Re-ports in Mathematical Physics , 33(1/2), 149–154, 1993.[33] R. Mruga la, On a special family of thermodynamic processes and theirinvariants,
Reports in Mathematical Physics , 46(3), 461–468, 2000.[34] R. Mruga la, On contact and metric structures on thermodynamic spaces,
RIMS, Kokyuroku , 1142, 167–181, 2000.[35] H. Ramirez, B. Maschke, D. Sbarbaro, Partial stabilization of input-outputcontact systems on a Legendre submanifold,
IEEE Trans. Aut. Contr. ,62(3), 1431–1437, 2017.[36] A. van der Schaft, B. Maschke, Homogeneous Hamiltonian control systems,Part I: Geometric formulation,
IFAC-Papers OnLine , 51(3), 1 – 6, 2018.37] A. van der Schaft, B. Maschke, Geometry of thermodynamic processes,
Entropy , 20(12), 925–947, 2018.[38] A.J. van der Schaft, B. Maschke, About some system-theoretic propertiesof port-thermodynamic systems, pp. 228–238 in