The Hamilton--Jacobi theory for contact Hamiltonian systems

The aim of this paper is to develop a Hamilton--Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton-Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.


Introduction
The Hamilton-Jacobi equation is an alternative formulation of classical mechanics, equivalent to other formulations such as Lagrangian and Hamiltonian mechanics. The Hamilton-Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.
The Hamilton-Jacobi equation has been extensively studied in the case of symplectic Hamiltonian systems, more specifically, for Hamiltonian functions H defined in the cotangent bundle T * Q of the configuration space Q. The Hamiltonian vector field is obtained by the equation where ω Q is the canonical symplectic form on T * Q. As we know, bundle coordinates (q i , p i ) are also Darboux coordinates so that X H has the local form The Hamilton-Jacobi problem consists in finding a function S : Q −→ R such that for some E ∈ R. The above equation (1) is called the Hamilton-Jacobi equation for H. Of course, one easily see that (1) can be written as follows which opens the possibility to consider general 1-forms on Q (considered as sections of the cotangent bundle π Q : T * Q −→ Q).
Recently, the observation that given such a section γ : Q −→ T * Q permits to relate X H with its projection X γ H via γ onto Q, in the sense that X γ H and X H are γ-related if and only if (2) holds, provided that γ be closed (or, equivalently, its image be a Lagrangian submanifold of (T * Q, ω Q )) has opened the possibility to discuss the Hamilton-Jacobi problem in many other scenarios: nonholonomic systems, multisymplectic field theories, time-dependent mechanics, among others.
In [13] we have started the extension of the Hamilton-Jacobi theory for contact Hamiltonian systems (see also [14]). Let us recall that a contact Hamilton system is defined by a Hamiltonian function on a contact manifold, in our case, the extended cotangent bundle T * Q × R equipped with the canonical contact form η Q = dz − θ Q , where z is a global coordinate in R and θ Q the Liouville form on T * Q, with the obvious identifications.
Contact Hamiltonian systems are widely used in many fields of Physics, like thermodynamics, dissipative systems, cosmology, and even in Biology (the so-called neurogeometry). The corresponding Hamilton equations were obtained in 1940 by G. Herglotz using a variational principle that extends the usual one of Hamilton, but they can be alternatively derived using contact geometry.
The goal of this paper is to continue the study of the Hamilton-Jacobi problem in the contact context, using the two vector fields associated to the Hamiltonian H: • the Hamiltonian vector field • the evolution vector field We notice that the Hamilton-Jacobi problem has been treated by other authors [7,21], who establish a relationship between the Herglotz variational principle and the Hamilton-Jacobi equation, although their interests are analytical rather than geometrical The content of the paper is as follows. Section 2 is devoted to introduce the main ingredients of contact manifolds and contact Hamiltonian systems as well as the interpretation of a contact manifold as a Jacobi structure. In Section 3 we discuss the different types of submanifolds of a contact manifold. Section 4 is the main part of the paper; there, we discuss the Hamilton-Jacobi problem for a contact Hamiltonian vector field as well as for the corresponding evolution vector field. The results are more involved than in the case of symplectic Hamiltonian systems due to the different possibilities that may occur. In Section 5 we study the relations of the Hamilton-Jacobi problem for a contact Hamiltonian systems and its symplectification. Finally, some examples are discussed in Section 6.

Contact manifolds
Consider a contact manifold [8,9,5] (M, η) with contact form η; this means that η ∧ dη n = 0 and M has odd dimension 2n + 1. Then, there exists a unique vector field R (called Reeb vector field) such that There is a Darboux theorem for contact manifolds (see [16,24]) so that around each point in M one can find local coordinates (called Darboux coordinates) (q i , p i , z) such that η = dz − p i dq i and we have R = ∂ ∂z The contact structure defines an isomorphism between tangent vectors and covectors. For each x ∈ M, Similarly, we obtain a vector bundle isomorphism We will also denote by♭ : X(M) → Ω 1 (M) the corresponding isomorphism of C ∞ (M)-modules between vector fields and 1-forms over M; ♯ will denote the inverse of♭.
Therefore, we have that♭ (R) = η, so that, in this sense, R is the dual object of η.
For a Hamiltonian function H on M we define the Hamiltonian vector field X H by♭ (X H ) = dH − (R(H) + H) η In Darboux coordinates we get this local expression Therefore, an integral curve (q i (t), p i (t), z(t)) of X H satisfies the contact Hamilton equations In addition to the Hamiltonian vector field X H associated to a Hamiltonian function H, there is another relevant vector field, called evolution vector field defined by E H = X H + HR so that it reads in local coordinates as follows Consequently, the integral curves of E H satisfy the differential equations Remark 1. The evolution vector field plays a relevant role in the geometric description of thermodynamics (see [26,27]).
Given a contact 2n + 1 dimensional manifold (M, η), we can consider the following distributions on M, that we will call vertical and horizontal distribution, respectively: We have a Withney sum decomposition and, at each point x ∈ M: We will denote by π H and π V the projections onto these subspaces. We notice that dim H = 2n and dim V = 1, and that (dη) | H is non-degenerate and V is generated by the Reeb vector field R. Definition 1.
1. A diffeomorphism between two contact manifolds F : 3. A vector field X ∈ X(M) is an infinitesimal contactomorphism (respectively infinitesimal conformal contactomorphism) if its flow φ t consists of contactomorphisms (resp. conformal contactomorphisms).
Therefore, we have Proposition 1.

A vector field X is an infinitesimal contactomorphism if and only if
L X η = 0.
2. X is an infinitesimal conformal contactomorphism if and only if there exists g ∈ C ∞ (M) such that In this case, we say that (g, X) is an infinitesimal conformal contactomorphism.

Contact manifolds as Jacobi structures
Definition 2. A Jacobi manifold [22,25,24] is a triple (M, Λ, E), where Λ is a bivector field (a skew-symmetric contravariant 2-tensor field) and E ∈ X(M) is a vector field, so that the following identities are satisfied: where [·, ·] is the Schouten-Nijenhuis bracket.
Given a Jacobi manifold (M, Λ, E), we define the Jacobi bracket: This bracket is bilinear, antisymmetric, and satisfies the Jacobi identity. Furthermore, it fulfills the weak Leibniz rule: That is, (C ∞ (M), {·, ·}) is a local Lie algebra in the sense of Kirillov.
Conversely, given a local Lie algebra (C ∞ (M), {·, ·}), we can find a Jacobi structure on M such that the Jacobi bracket coincides with the algebra bracket.
Remark 2. The weak Leibniz rule is equivalent to this identity: {f, gh} = g{f, h} + h{f, g} + ghE(f ) Given a contact manifold (M, η) we can define the associated Jacobi structure (M, Λ, E) by For an arbitrary function f on M we can prove that the Hamiltonian vector field X f with respect to the contact structure η coincides with the one defined by its associated Jacobi structure, say where ♯ Λ is the vector bundle morphism from tangent covectors to tangent vectors defined by Λ, i.e.

Submanifolds
As in the case of symplectic manifolds, we can consider several interesting types of submanifolds of a contact manifold (M, η). To define them, we will use the following notion of complement for contact structures [8]: Let (M, η) be a contact manifold and x ∈ M. Let ∆ x ⊂ T x M be a linear subspace. We define the contact complement of ∆ x is the annihilator. We extend this definition for distributions ∆ ⊆ T M by taking the complement pointwise in each tangent space.
Here, Λ is the associated 2-tensor according to the previous section.
Definition 3. Let N ⊆ M be a submanifold. We say that N is: The coisotropic condition can be written in local coordinates as follows.
Let N ⊆ M be a k-dimensional manifold given locally by the zero set of functions φ a : U → R, with a ∈ {1, . . . , k}.
We have that According to (11), we conclude that N is coisotropic if and only if for all a, b.
Using the above results, one can easily prove the following characterization of a Legendrian submanifold. Consider a function f : Then, one immediately checks that j 1 f (Q) is a Legendrian submanifold of (T * Q × R, η Q ). Moreover, we have Remark 3. The above result is the natural extension of the well-known fact that a section σ of the cotangent bundle π Q : T * Q −→ Q is a Lagrangian submanifold with respect to the canonical symplectic structure ω Q = −dθ Q on T * Q if and only if σ is a closed 1-form (and hence, locally exact).

The Hamilton-Jacobi equations for a Hamiltonian vector field
We consider the extended phase space T * Q × R, and a Hamiltonian function can be locally expressed as follows (T * Q × R, η) is a contact manifold with Reeb vector field R = ∂ ∂z . Consider the Hamiltonian vector field X H for a given Hamiltonian function, say In coordinates, it reads We also have♭ where ♭ is the isomorphism previously defined. We also have that Recall that (T * Q × R, Λ, R) is a Jacobi manifold with Λ given in the usual way (see Section 2.2). The proposed contact structure provides us with the contact Hamilton equations.
Consider γ a section of π : The following diagram summarizes the above construction Assume that in local coordinates we have We can compute T γ(X γ H ) and obtain (15) and (19), we have that Assume now that Notice that the above two conditions imply that γ(Q × R) is foliated by Lagrangian leaves γ z (Q), z ∈ R.
We will discuss the consequences of the above conditions. The submanifold γ(Q × R) is locally defined by the functions Therefore, the first condition is equivalent to If, in addition, γ z (Q) is Lagrangian submanifold for any fixed z ∈ R, then we obtain and, using again (21), we get Under the above conditions (using 22 and 23), 20 becomes We can write down eq (24) in a more friendly way. First of all, consider the following functions and 1-forms defined on Q × R: is a Lagrangian submanifold of (T * Q, ω Q ), for any z ∈ R. Then, the vector fields X H and X γ H are γ-related if and only if (24) holds (equivalently, (25) holds).
Equations (24) and (25) are indistinctly referred as a Hamilton-Jacobi equation with respect to a contact structure. A section γ fullfilling the assumptions of the theorem and the Hamilton-Jacobi equation will be called a solution of the Hamilton-Jacobi problem for H.
Remark 4. Notice that if γ is a solution of the Hamilton-Jacobi problem for H, then X H is tangent to the coisotropic submanifold γ(Q × R), but not necesarily to the Lagrangian submanifolds γ z (Q), z ∈ R. This occurs when In such a case, we call γ an strong solution of the Hamilton-Jacobi problem.
A characterization of conditions on the submanifolds γ(T Q × R), γ z (T Q) can be given as follows. Let σ : Q × R → Λ k (T * Q) be a z-dependent k-form on Q. Let d Q σ be the exterior derivative at fixed z, that is where σ z = σ(·, z). In local coordinates, we have where f : Q × R → R is a function and α = α i dq i : That is, there exists locally a function f : are Lagrangian if and only if d Q γ = 0. By the Poincaré Lemma, locally γ = d Q f , Now also assume that γ(Q × R) is coisotropic. Then, equation (23) can be written as γ ∧ L ∂/∂z γ = 0, or, equivalently, that γ and L ∂/∂z γ are proportional. Locally, we obtain that d Q ∂f ∂z = σd Q f .

Complete solutions
Next, we shall discuss the notion of complete solutions of the Hamilton-Jacobi problem for a Hamiltonian H.

Definition 4. A complete solution of the Hamilton-Jacobi equation for a
Hamiltonian H is a diffeomorphism Φ : Q × R × R n → T * Q × R such that for any set of parameters λ ∈ R n , λ = (λ 1 , . . . , λ n ), the mapping is a solution of the Hamilton-Jacobi equation. If, in addition, any Φ λ is strong, then the complete solution is called an strong complete solution.
We have the following diagram where we define functions f i such that for a point p ∈ T * Q × R, it is satisfied and α : Q × R × R n → R n is the canonical projection.
The first immediate result is that where λ = (λ 1 , · · · , λ n ). In other words, Therefore, since X H is tangent to any of the submanifolds Im Φ λ , we deduce that So, these functions are conserved quantities. Moreover, we can compute Theorem 3. There exist no linearly independent commuting set of firstintegrals in involution (44) for a complete strong solution of the Hamilton-Jacobi equation on a contact manifold.
Proof: If all the particular solutions are strong, then the Reeb vector field R will be transverse to the coisotropic submanifold Φ λ (Q × R). Indeed, if R is tangent to that submanifold, we would have . So, Φ λ does not depend on z, hence it cannot be a diffeomorphism. Therefore, if the brackets {f i , f j } vanish, then we woul obtain that the functions f i cannot be linearly independent. Indeed, we should have for all i, j. But this would imply that f i and f j are linearly dependent in the case λ = (0, . . . , 0).

An alternative approach
Instead of considering sections of π : T * Q × R −→ Q × R as above, we could consider a section of the canonical projection π : In local coordinates, we have We want γ to fulfill Now, notice thatγ = ρ • γ is a 1-form on Q. Then, we locally haveγ = γ i (q) dq i . Next, we assume that γ(Q) is a Legendrian submanifold of (T * Q×R, η Q ). This implies thatγ(Q) is a Lagrangian submanifold of (T * Q, ω Q ).

By Proposition 3, γ(Q) is a Legendrian submanifold if and only if it is
locally the 1-jet of a function, namely γ = j 1 γ z , where we consider γ z as a function from Q to R. In other words, we have: If we assume that the section γ fulfills the above condition, we can see that equations (33) become Definition 5. Assume that a section γ such that γ(Q) is a Legendrian submanifold of (T * Q × R, η Q ) andγ(Q) is a Lagrangian submanifold of (T * Q, ω Q ). Then γ is called a solution of the Hamilton-Jacobi problem for the contact Hamiltonian H if and if equation (36) holds.
We could discuss the existence of complete solutions in a similar manner to the case of the Hamiltonian vector field. We omit the details that are left to the reader.

A first approach
Assume that E H is the evolution vector field defined for a Hamiltonian function H : T * Q × R −→ R. Then, we have Assume that γ is a section of the canonical projection π : In local coordinates we have Therefore, we can define the projected evolution vector field Assume now that 1. γ(Q × R) is a coisotropic submanifold of (T * Q × R, η Q ); 2. γ z (Q) is a Legendrian submanifold of (T * Q × R, η Q ), for any z ∈ R, where γ z (q) = γ(q, z).
Then, a direct computation shows that (38) becomes where is a coisotropic submanifold of (T * Q × R, η Q ), and γ z (Q) is a Legendrian submanifold of (T * Q × R, η Q ), for any z ∈ R. Then, the vector fields E H and E γ H are γ-related if and only if (39) holds. Equation (39) is referred as a Hamilton-Jacobi equation for the evolution vector field. A section γ fullfilling the assumptions of the theorem and the Hamilton-Jacobi equation will be called a solution of the Hamilton-Jacobi problem for the evolution vector field of H.

An alternative approach
We will maintain the notations of the previous subsection, but now γ is a section of the canonical projection π : T * Q × R −→ Q, say γ : Q → T * Q × R.
In local coordinates we have As in the above sections, we define the projected evolution vector field If we assume that γ = j 1 f , for some function f : Q −→ R (or, equivalently, γ(Q) is a Legendrian submanifold of (T * Q × R, η Q )), then Remark 5. Notice that f and γ z define (locally) the same 1-jet. Therefore, we have the following.
Theorem 5. Assume that a section γ of the projection T * Q×R → Q is such that γ(Q) is a Legendrian submanifold of (T * Q × R, η Q ). Then, the vector fields E H and E γ H are γ-related if and only if (42) holds. Equation (42) is referred as a Hamilton-Jacobi equation for the evolution vector field. A section γ fullfilling the assumptions of the theorem and the Hamilton-Jacobi equation will be called a solution of the Hamilton-Jacobi problem for the evolution vector field of H.

Complete solutions
As in the case of the Hamiltonian vector field, we can consider complete solutions for the evolution vector field.

Definition 6.
A complete solution of the Hamilton-Jacobi equation for the evolution vector field E H of a Hamiltonian H on a contact manifold (M, η) is a diffeomorphism Φ : Q × R × R n → T * Q × R such that for any set of parameters λ = (λ 0 , λ 1 , . . . , λ n ) ∈ R × R n , the mapping is a solution of the Hamilton-Jacobi equation.
For simplicity, we will use the notation (λ α , α = 0, 1, . . . , n). As in the previous case, we define functions f α such that for a point p ∈ T * Q × R, it is satisfied where π α : Q × R × R n → R is the canonical projection onto the α factor. A direct computation shows that Therefore, since under our hypthesis, E H is tangent to any of the submanifolds Im Φ λ , we deduce that So, these functions are conserved quantities for the evolution vector field. Moreover, we can compute Theorem 6. There exist no linearly independent commuting set of firstintegrals in involution (44) for a complete solution of the Hamilton-Jacobi equation for the evolution vector field.
Proof: Since the images of the sections are Legendrian then they are integral submanifolds of ker η Q . So, the Reeb vector field R will be transverse to them, and consequently, there is at least some index α 0 such that Therefore, if all the brackets {f α , f β } vanish, then we woul obtain that the functions f α cannot be linearly independent.
5 Symplectification of the Hamilton-Jacobi equation

Homogeneous Hamiltonian systems and contact systems
There is a close relationship between homogeneous symplectic and contact systems, see for example [20,29]. Here we briefly recall some facts about the symplectification of cotangent bundles. For any manifold M a function F : T * M → R is said to be homogeneous if, for any p q ∈ T * p M, we have F (λp q ) = λF (p q ) for any λ ∈ R. In this situation the function F can be projected to the projective bundle P(T * M) over M obtained by projectivization of every cotangent space. We are interested in the case that M = Q × R, with natural coordinates (q i , z, P i , P z ) on T * (Q × R). We note that this definition can be generalized to any vector bundle.
LetH be an homogeneous Hamiltonian function on T * (Q × R). Locally, we have thatH(q i , z, λP i , λP z ) = λH(q i , z, P i , P z ), for all λ ∈ R. Equivalently, one can writẽ for P z = 0, where H : With the above changes, we have identified the manifold T * Q × R as the projective bundle P(T * (Q × R)) of the cotangent bundle T * (Q × R) taking out the points at infinity, that is the subset defined by {P z = 0}.
Following [29, Section 4.1], the map sends the Hamiltonian symplectic system (T * (Q × R) \ {P z = 0}, ω Q×R ,H) onto the Hamiltonian contact system (T * Q × R, η Q , H), where ω Q×R = dq i ∧ dP i + dz ∧ dP z and η Q = dz − p i dq i are the canonical symplectic and contact forms, respectively. Observe that the natural coordinates of T * Q×R, denoted by (q i , p i , z), correspond to the homogeneous coordinates in the projective bundle. In fact, the map Φ is projectivization up to a minus sign; i.e., the map that sends each point in the fibers of T * (Q × R) to the line that passes through it and the origin. The map Φ satisfiesH = −P z Φ * (H) and ω Q = −d(P z Φ * (η Q )) It can be shown that Φ provides a bijection between conformal contactomorphisms and homogeneous symplectomorphisms. Moreover, Φ maps homogeneous Lagrangian submanifolds L ⊆ T * (Q × R) onto Legendrian submanifolds L = Φ(L) ⊆ T * Q × R. Indeed, if L is homogeneous, then L is Legendrian if and only if L is Lagrangian. Moreover, the Hamilton equations forH are transformed into the Hamilton equations for H, i.e., Φ * XH = X H . See [29,28] for more details on this topics.
We also remark that this construction is symplectomorphic to the symplectification defined in [20], which is given by where t is the (global) coordinate of the second R factor with the "symplectified" HamiltonianH ′ = e t H setting and then project it to the original contact manifold. That is,H ′ such that where pr 1 : T * Q × R × R → T * Q × R is the projection onto the first two factors.
The following map provides the symplectomorphism that is, Ψ = (Φ, − log(−P z ). This map is a symplectomorphism that maps H ontoH ′ . Moreover it is a fiber bundle automorphism over T Q × R (see the diagram below):

Relations for the Hamilltonian vector field
Now we will establish a relationship between solutions to the Hamilton-Jacobi problem in both scenarios. Suppose that is a solution of the symplectic Hamilton-Jacobi equation, i.e.,γ(Q × R) is Lagrangian and d(H •γ) = 0, or equivalently where XγH = T p•XH•γ is the projected vector field and p : T * (Q×R) → Q×R the canonical projection. We want to use the solutionγ of the Hamilton-Jacobi problem in the symplectification (which we will often refer to as "symplectic solution") to obtain a section that is a solution in the contact setting ("contact solution", for simplicity). We assumeγ t (q i , z) = 0 and take γ = Φ •γ : Q × R → T * Q × R. In local coordinates Proof. Letγ = (γ Q ,γ t ) be such that its image is Lagrangian. That is, dγ = 0. Splitting the part in Q and in R, we see that this is equivalent to Now, γ = −γ Q /γ t . By Theorem 2, it is necessary that d Q γ = 0 and (L ∂/∂z γ) ∧ γ = 0. We compute hence the images of γ z 0 are Lagrangian and the image of γ is coisotropic if and only if L ∂/∂z (γ Q ) is proportional toγ Q . Conversely, assume that γ satisfies d Q γ = 0 and L ∂/∂z γ = σγ. We must findγ t so that (54) are satisfied. Sinceγ Q = −γ t γ, we have that (54) are equivalent to A solution forγ t on the first equation above clearly solves the second one. Since we look for nonvanishingγ t , we let g = log •|γ t | so that is just if we let this equation can be written as we note that this vector fields commute, indeed If this PDE has local solutions, operating with the equations above, one has, This condition is clearly neccesary, and it is also sufficient by [23,Thm. 19.27].
We have that Combining the last two results, we obtain a correspondence between symplectic and contact solutions to the Hamilton-Jacobi problem. Conversely, given a contact solution γ of the Hamilton-Jacobi equation, there exists a symplectic solutionsγ such thatγ = ± exp(g), where g is a solution to the PDE d Q g + γL ∂/∂z g = −L ∂/∂z γ.

The alternative approach
For each z, we have sections γ = pr 2 •γ z : Q → T * Q × R of the form (q i ) → (q i ,γ j (q i , z),γ t (q i , z)), being pr 2 : (q i , p i , z, t) → (q i , p i , t). We know that γ is a solution of the contact Hamilton-Jacobi problem if and only if γ(Q) is Legendrian and H • γ = 0.
The condition that γ(Q) is Legendrian is equivalent to where we write γ(q i ) = (q i , γ j (q i ), γ z (q i )), which by definition of γ and using that γ(Q × R) is Lagrangian reads and thereforeγ i = e z g(q i ), with g i functions depending only on the (q i ). This can be summarized as follows: is a solution of the symplectified Hamilton-Jacobi problem. Then is a solution of the contact Hamilton-Jacobi problem if and only if

Relations for the evolution vector field
We now consider the evolution field E H . First, note that so that we cannot simply expect to project the vector field as before. In fact, one can easily prove that under the assumption that the symplectified Hamiltonian is of the formH = F (t, H), then the associated vector field XH such that i XH ω = dH will never verify T pr 1 • XH = E H • pr 1 We will now see that, despite this apparent obstruction, one can still establish some relations. Letγ : Q × R → T * (Q × R) be a solution of the symplectified problem and define the section γ = pr 2 •γ z : Q → T * Q × R. This will be a solution of the associated Hamilton-Jacobi problem for the evolution field if and only if γ(Q) is Legendrian and The Legendrian condition is equivalent to or, using thatγ(Q × R) is Lagrangian like in the previous section, On the other hand, we know thatγ is a solution of the symplectic problem and therefore d(H •γ) = 0, which by definition means e −γt(q i ,z) H(q i ,γ j (q i , z), z) = C with C constant. Since γ(q i ) = (q i ,γ j (q i , z),γ t (q i , z)), using the previous equation we obtain: Then the condition d(H • γ) = 0 reads which occurs if and only if at every point (q i ) we have: The functional form found for H • γ tells us that it is either non-zero at every point or it vanishes everywhere. If it does not vanish (everywhere), we claim that the second equation must be true. Indeed, suppose the first two equations do not hold. Then the third equation must be true not just at a given point but in an open neighborhood and we would havẽ where h i are arbitrary functions. Using again thatγ(Q × R) is Lagrangian we could write ∂γ t ∂q i = ∂h ∂q i = e z g i (q i ) = ∂γ i ∂z which would imply that h depends also on z. Therefore, if H • γ = 0 then the second equation is true at every point. Using thatγ(Q × R) is Lagrangian we see this is equivalent toγ i = 0. Therefore we find: Theorem 9. Letγ : Q × R → T * (Q × R) be a solution of the symplectified problem withγ i = e z g i , where g i : Q → R, and consider the section Then γ is a solution of the contact problem for the evolution field if and only if one of the two following conditions is fulfilled: 2.γ i = 0.

Application to thermodynamic systems
We consider thermodynamic systems in the so called energy representation.
Hence the thermodynamic phase space, representing the extensive variables, is the manifold T * Q × R, equipped with its canonical contact form The local coordinates on the configuration manifold Q are (q i , U), where U is the internal energy and q i 's denote the rest of extensive variables. Other variables, such as the entropy, may be chosen instead of the internal energy, by means of a Legendre transformation. The state of a thermodynamic system always lies on the equilibrium submanifold L ⊆ T * Q × R, which is a Legendrian submanifold. The pair (T * Q × R, L) is a thermodynamic system. The equations (locally) defining L are called the state equations of the system. On a thermodynamic system (T * Q×R, L), one can consider the dynamics generated by a Hamiltonian vector field X H associated to a Hamiltonian H. If this dynamics represents quasistatic processes, meaning that at every time the system is in equilibrium, that is, its evolution states remain in the submanifold L, it is required for the contact Hamiltonian vector field X H to be tangent to L. This happens if and only if H vanishes on L.
Using Hamilton-Jacobi theory, one sees that a section γ satisfied H •γ = 0 if and only if X γ H and X H are γ-related.

The classical ideal gas
This example is fully described in [17]. The classical ideal gas is described by the following variables.
Thus, the thermodynamic phase space is T * R 3 × R and the contact 1-form is The Hamilltonian function is where R is the constant of ideal gases. The Reeb vector field is R = ∂ ∂U . The Hamilltonian and evolution vector fields are just The Hamilonian vector field here represents an isochoric and isothermal process on the ideal gas. Assume that γ : R 3 → T * R 3 × R is the section locally given by γ(S, V, N) = (S, V, N, γ T , γ P , γ µ γ U ).
we know that γ(R 3 ) is a Legendrian submanifold of (T * R 3 × R, η) if and only if, The Hamilton-Jacobi equation is for some k ∈ R. That is, This is a first order linear PDE, whose solution is given by with F : R 2 → R an arbitrary function. The case k = 0, which is the one relevant for the thermodynamic interpretation, is given by