A HAMILTON-JACOBI THEORY ON POISSON MANIFOLDS

. In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.


1.
Introduction. The standard formulation of the Hamilton-Jacobi problem is to find a function S(t, q i ) (called the principal function) such that ∂S ∂t where h = h(q i , p i ) is the hamiltonian function of the system. If we put S(t, q i ) = W (q i ) − tE, where E is a constant, then W satisfies W is called the characteristic function.
The Hamilton-Jacobi equation helps to solve the Hamilton equations for h Indeed, if we find a solution W of the Hamilton-Jacobi equation (2) then any solution (q i (t)) of the first set of equations (3) gives a solution of the Hamilton equations by taking p i (t) = ∂W ∂q i . This result can be founded in [1]. Moreover, one can rephrase the above result by stating that if W is a solution of the Hamilton-Jacobi equation, then dW (a 1-form on Q) transforms the integral curves of the vector field X dW h = T π Q • X h • dW into the integral curves of X h ; here, X h is the Hamiltonian vector field defined by the hamiltonian h and π Q : T * Q −→ Q is the canonical projection. Of course we can think in a more general situation where we look for general 1-forms on Q that play a similar role to dW .
This geometrical procedure has been succesfully applied to many other different contexts, including nonholonomic mechanics (see [7,8,11,13]), singular lagrangian systems [15,16], and even classical field theories [12,17,14]. Notice that in these frameworks, we don't have a symplectic framework; for instance, in nonholonomic mechanics the natural geometric framework is provided by a (2,0)-tensor field (an almost Poisson tensor) on the constraint submanifold that it is not integrable (that is, it is not satisfies the Jacobi identity). The almost-Poisson bracket in nonholonomic mechanics has been firstly introduced by A. van der Shaft and B.M. Mashke ( [24]). All these scenarios are just the motivation for the investigation developed in this paper.
Our goal is to develop a Hamilton-Jacobi theory in a unifying and more general setting, say hamiltonian systems on an almost-Poisson manifold, that is, a manifold equipped with a skew-symmetric (2, 0)-tensor field which does not necesarily satisfies the Jacobi identity. We also assume that the almost-Poisson manifold has a fibered structure over another manifold. The Hamilton-Jacobi problem now is to find a section of the fibered manifold such that its image is a lagrangian submanifold and the differential of the given hamiltonian vanishes on the tangent vectors to the section and belonging to the characteristic distribution.
The theory includes the case of classical hamiltonian systems on the cotangent bundle of the configuration manifold as well as the case of nonholonomic mechanical systems. We also apply the theory to time-dependent hamiltonian systems and systems with external forces.
We also discuss the existence of complete solutions and prove that if a complete solution exists then we obtain first integral in involution, which is a remarkable fact since our framework is just almost-Poisson.
Along the paper, all the manifolds are real, second countable and C ∞ . The maps are assumed to be also C ∞ . Sum over all repeated indices is understood 2. Hamilton-Jacobi theory on almost-Poisson manifolds.

2.1.
Hamiltonian systems on almost-Poisson manifolds. Assume that (E, Λ) is an almost-Poisson manifold, that is, E is a manifold equipped with an almost-Poisson structure Λ, which means that Λ is a skew-symmetric (2, 0)-tensor field on E. Notice that Λ does not necessarily satisfy the Jacobi identy; in that case, we will have a Poisson tensor, and E will be a Poisson manifold. For the moment, one only needs to ask (E, Λ) be an almost-Poisson manifold.
Therefore, Λ defines a vector bundle morphism for all α, β ∈ T * E. Of course, we shall also denote by the induced morphism of C ∞ -modules between the spaces of 1-forms and vector fields on E. Notice that we will use the notation Λ if there is danger of confusion.
We denote by C the characteristic distribution defined by Λ, that is The rank of the almost-Poisson structure at p is the dimension of the space C p . Notice that C is a generalized distribution and, moreover, is not (in general) integrable since Λ is not Poisson in principle.
The following lemma will be useful Lemma 2.1. Let (E, Λ) be an almost-Poisson manifold, then we have where C • denotes the annihilator of C.
Proof. Observe that and thus, the result holds.
We also have the following definition 19,23]) A submanifold N of E is said to be a lagrangian submanifold if the following equality holds To have dynamics we need to introduce a hamiltonian function h : E −→ R, and thus we obtain the corresponding hamiltonian vector field X h = (dh).

2.2.
Hamilton-Jacobi theory on almost-Poisson manifolds. Assume now that the almost-Poisson manifold E with almost-Poisson tensor Λ fibres over a manifold M , say π : E −→ M is a surjective submersion (in other words, a fibration).
Assume that γ is a section of π : E −→ M , i.e. π • γ = id M . Define the vector field X γ h on M by X γ h = T π • X h • γ. The following diagram summarizes the above construction: The following result relates the integral curves of X h and X γ h . Theorem 2.3. Assume that Im(γ) is a lagrangian submanifold of (E, Λ). Then the following assertions are equivalent: Assume that X h and X γ h are γ-related. Then X h = T γ(X γ h ) and since X h ∈ C, we have X h ∈ T Im(γ) ∩ C. But Im(γ) is a lagrangian submanifold, so there exists β ∈ (T Im(γ)) • such that X h = (β), along the image of γ.
Then, along Im(γ) we have Since Im(γ) is a lagrangian submanifold, we have and then X h = X α1 ∈ T Im(γ) ∩ C.
Therefore we deduce that X h and X γ h are γ-related since both are tangent to the submanifold Im γ.
Assume that (E, Λ) is a transitive Poisson manifold (a symplectic manifold), that is, C = T E. Then, we have that a submanifold N of a transitive Poisson manifold (E, Λ) is a lagrangian submanifold if and only if (T N • ) = T N. Therefore, the above Theorem 2.3 takes the following classical form.
Theorem 2.4. Assume that Im(γ) is a lagrangian submanifold of a transitive Poisson manifold (E, Λ). Then the following assertions are equivalent: 3. Computations in local coordinates. Assume that (x i , y a ) are local coordinates adapted to the fibration π : E −→ M , that is, π(x i , y a ) = (x i ), where (x i ) are local coordinates in M .
The above local expressions implies that Using (4) we deduce that a hamiltonian vector field X h for a hamiltonian function h ∈ C ∞ (E) is locally expressed by Now, let γ : we obtain Proof. First of all, let us observe that T Im(γ) is locally generated by the local vector fields annihilates T Im(γ) (recall that this is only valid along the points of Im(γ)) we deduce the following conditions on the coefficients: Now, a simple computation shows that Assume now that Im(γ) is a lagrangian submanifold. Then, (α) ∈ T Im(γ), with α ∈ T γ(M ) • , and using (10) we deduce that and Substituting the values of λ j given by (11) in equation (12) we obtain 4. Complete solutions. The essential idea in the standard Hamilton-Jacobi theory consists in finding a complete family of solutions to the problem (not only one particular solution). Therefore, in this section we shall discuss the notion of complete solutions for the Hamilton-Jacobi equation in this general framework of almost-Poisson manifolds. As a consequence we recover in a really simple and geometric way the results about complete solutions proved in [7]. First of all, we shall introduce the notion of complete solution. Assume that we have a hamiltonian system given by a hamiltonian function h : E −→ R on an almost Poisson manifold (E, Λ) fibered over a base manifold M . We assume that dim M = n and dim E = n + k .
is a solution of the Hamilton-Jacobi problem for h.
For the sake of simplicity we will assume that Φ is a global diffeomorphism. Consider f a : E → R given the composition of Φ −1 with the projection over the a-th component of R k . We obtain Proof. Given p ∈ E we will show that {f a , f b }(p) = 0. Suppose that f a (p) = λ a , for each a = 1, . . . , k, and observe that p ∈ Im(Φ λ ) = ∩ a f −1 a (λ a ). Therefore, we deduce that df a (p) |T Im(Φ λ ) = 0 since f a • Φ λ is a constant function. By the hypothesis, Im(Φ λ ) is a lagrangian submanifold in the sense previously explained, so

Applications.
5.1. Classical hamiltonian systems. (see [1,2,18]) A classical hamiltonian system is given by a hamiltonian function h defined on the cotangent bundle T * Q of the configuration manifold Q.
If it is the case, then E = T * Q and Λ is the canonical Poisson structure Λ Q on T * Q provided by the canonical symplectic form ω Q on T * Q. Recall that now we can take canonical bundle coordinates ( Since in bundle coordinates The notion of lagrangian submanifold defined in Section 2 in the almost-Poisson setting reduces to the well-known in the symplectic setting, that is, it is isotropic and coisotropic with respect to the symplectic form ω Q . If we compute the condition (9) for the current case we obtain ∂γ i ∂q j = ∂γ j ∂q i which just means that γ is a closed form, i.e., dγ = 0. So we recover the classical result (see [1,2]). Proposition 3. Given a 1-form γ on Q, we have that Im(γ) is a lagrangian submanifold of (T * Q, Λ Q ) if and only if γ is closed.
As a consequence, we deduce the classical result directly from Theorem 2.4: Theorem 5.1. Let γ be a closed 1-form on Q. Then the following assertions are equivalent: 1. X h and X γ h are γ-related; 5.2. Nonholonomic mechanical systems. In this section we will recover the results obtained in two previous papers [11,13] (see also [8,10,21]). A nonholonomic mechanical system is given by a lagrangian function L : T Q −→ R subject to contraints determined by a linear distribution D on the configuration manifold Q. We will denote by D the total space of the corresponding vector sub-bundle (τ Q ) |D : D −→ Q defined by D, where (τ Q ) |D is the restriction of the canonical projection τ Q : T Q −→ Q.
We will assume that the lagrangian L is defined by a Riemannian metric g on Q and a potential energy V ∈ C ∞ (Q), so that If {µ a }, 1 ≤ a ≤ k is a local basis of the annihilator D o of D, then the constraints are locally expressed as µ a i (q)q i = 0, where µ a = µ a i (q) dq i . The nonholonomic equations can be written as µ a i (q)q i = 0, for some Lagrange multipliers λ i to be determined.
Let S (respectively, ∆) be the canonical vertical endomorphism (respectively, the Liouville vector field) on T Q. In local coordinates, we have Therefore, we can construct the Poincaré-Cartan 2-form ω L = −dS * (dL) and the energy function function E L = ∆(L) − L, such that the equation has a unique solution, ξ L , which is a SODE on T Q (that is, S(ξ L ) = ∆). Furthermore, its solutions coincide with the solutions of the Euler-Lagrange equations for L: d dt ∂L ∂q i − ∂L ∂q i = 0. If we modify (13) as follows: X ∈ T D the unique solution X nh is again a SODE whose solutions are just the ones of the nonholonomic equations. Let F L : T Q −→ T * Q be the Legendre transformation given by F L is a global diffeomorphism which permits to reinterpret the nonholonomic mechanical system in the hamiltonian side. Indeed, we denote by h = E L • F L −1 the hamiltonian function and by M = F L(D) the constraint submanifold of T * Q.
The nonholonomic equations are then given by whereλ i are new Lagrange multipliers to be determined. As above, the symplectic equation which gives the hamiltonian vector field X h should be modified as follows to take into account the nonholonomic constraints: where F is a distribution along M whose annihilator F o is obtained from S * ((T D) o ) through F L. Equations (16) and (17) have a unique solution, the nonholonomic vector field X nh . An alternative way to obtain X nh is to consider the Whitney sum decomposition where the complement is taken with respect to ω Q . If is the canonical projection onto the first factor, one easily proves that Moreover, one can introduce an almost-Poisson tensor Λ nh on M by Λ nh (α, β) = Λ Q (P * α, P * β); its associated bracket is called the nonholonomic bracket. Let us recall that this bracket has been firstly introduced and studied by A. van der Shaft and B.M. Mashke ( [24]). Obviously, we have X nh = nh (dh), where nh stands for Λ nh .
An alternative way to define the nonholonomic bracket is as follows. Consider the distribution T M ∩ F along M . A direct computation shows that the subspace T p M ∩ F p is symplectic within the symplectic vector space (T p (T * Q), ω Q (p)), for every p ∈ M (see [4,6]).

Thus we have a second Whitney sum decomposition
where the complement is taken with respect to ω Q . IfP : is the canonical projection onto the first factor, one easily proves that Moreover, it is possible to write Λ nh in terms of the projectionP as follows the the nonholonomic almost-Poisson tensorΛ nh on M is now rewritten as Λ nh (α, β) = Λ Q (P * α,P * β) = ω Q (P (X α ),P (X β )) (see [6] for a proof).
Consider now the fibration (M, Λ nh ) π Q |M Q and the hamiltonian h |M (also denoted by h for sake of simplicity). We can easily prove that Indeed,we have nh (α), β = −ω Q (P X α , X β ) = ω Q (X β ,P X α ) = (i X β ω Q )(P X α ) = β,P X α which implies nh (α) =P (X α ). Furthermore, the symplectic structure Ω p on C p at any point p ∈ M is given by the restriction of the canonical symplectic structure ω Q on T * Q to C p . Proof. We notice that F = {v ∈ T (T * Q) such that T π Q (v) ∈ D}, and an easy computation in local coordinates shows that dim(F ∩ T M ) = 2 dim(D). Thus, we have T Im(γ) ∩ C = T γ(D).
On the other hand, it is clear that our definition of lagrangian submanifold is equivalent to T Im(γ) ∩ C be lagrangian with respect to the simplectic structure Ω on the vector space C. Since Ω is the restriction of ω Q , given X, Y ∈ D we have Ω(T γ(X), T γ(Y )) = Ω Q (T γ(X), T γ(Y )) = dγ(X, Y ).
So, after a careful counting of dimensions, we deduce that Im(γ) is lagrangian with respect to Λ nh if and only if dγ(X, Y ) = 0 for all X, Y ∈ D.
Using this proposition we can recover the Nonholonomic Hamilton-Jacobi Theorem as a consequence of Theorem 2.3 (see [11,13,21]). We will get a suitable expression for the nonholonomic almost-Poisson tensor Λ nh defined on the constraint submanifold M of T * Q (see [24,6]). This local representation can be also used to prove Proposition 4.
Let us recall that the constraints were defined through a distribution D on Q. Let D a complementary distribution of D in T Q and assume that {X α }, 1 ≤ α ≤ n − k is a local basis of D and that {Y a }, 1 ≤ a ≤ k is a local basis of D . Notice that where {µ a }, 1 ≤ a ≤ k is a local basis of the annihilator D o of D.
Next we introduce new coordinates in T * Q as follows: In these new coordinates we deduce that the constraints becomẽ p n−k+a = 0. Therefore, we can take local coordinates (q i ,p α ) on M .
A direct computation shows now that the nonholonomic almost-Poisson tensor Λ nh on M is given by [6] Λ nh (dq i , dq j ) = 0 , Λ nh (dq i , dp α ) = X i α Λ nh (dp α , dp β ) = X i β p j ∂X j β ∂q i − X i α p j ∂X j β ∂q i . Summarizing the above discussion we can apply the results obtained in Section 2 to the hamiltonian system given by h on the almost-Poisson manifold (M, Λ nh ).
Assume that γ : Q −→ M is a section of π Q |M : M −→ Q. Then, we have Since γ is a 1-form on Q taking values on M we deduce and since it takes values in M we get

MANUEL DE LEÓN, DAVID MARTÍN DE DIEGO AND MIGUEL VAQUERO
which can be equivalently written as Therefore, γ(Q) is a lagrangian submanifold of (M, Λ nh ) if and only if dγ ∈ I(D o ), where I(D o ) denotes the ideal of forms generated by D o . Indeed, notice that (18) holds if and only if dγ = a ξ a ∧ µ a , for some 1-forms ξ a . 5.3. Time dependent systems. In this section we will follow [18] for the decription of time dependent mechanical systems. Now we are going to develop a timedependent version of the previous construction. If we have the fibration E → M such that E is equipped with an almost-Poisson structure Λ, we can construct the following fibration in the obvious way where now R × E is equiped with the almost-Poisson structureΛ obtained in a natural way extending Λ in the trivial manner, that is, if f is a function on R × E, f (0, p), and where we are using the natural identifications for the tangent and cotangent vector spaces of product manifolds. Here, = Λ and˜ = Λ .
We can consider the "extended" version of this diagram, that is, consider T * R×E, equipped with the almost-Poisson structure Λ ext given by the addition of the canonical Poisson structure on T * R and Λ. Notice that if we consider global coordinates (t, e) on T * R ∼ = R × R, then and the canonical projection is According to the above notation, diagram (19) becomes Given a time dependent hamiltonian h : R × E → R, the dynamics are given by the evolution vector field ∂ ∂t + X h ∈ X(R × E). We can introduce the extended hamiltonian h ext : T * R × E → R given by h ext = µ * h + e and the respective hamiltonian vector field X hext = Λext (dh ext ). Notice that µ * (X hext ) = ∂ ∂t + X h . We will denote by C ext the characteristic distribution of Λ ext . Notice that C ext (t, e, p) = ∂ ∂t , ∂ ∂e + C p , under the obvious identifications.
If γ is a section ofπ, we can consider the section of π R given by µ • γ and define the vector field ( ∂ ∂t + X h ) γ on R × M as follows: Now, we can state the time-dependent version of Theorem 2.3.
Theorem 5.3. If Im(γ) is a lagrangian manifold in (T * R × E, Λ ext ), then the following assertions are equivalent.
Assume that ( ∂ ∂t + X h ) and ( ∂ ∂t + X h ) γ are µ • γ-related, which means that given m ∈ M , then since any tangent vector in T γ(m) (T * R × E) which projects by µ onto ∂ ∂t + X h is of the form Using the same argument that we used in Theorem 2.3 we can conclude that dh ext (γ(m)) + Bdt ∈ (T γ(m) Im(γ) ∩ C ext (γ(m))) • and so dh ext ∈ (T Im(γ) ∩ C ext ) • + dt .

External forces.
In this section we will apply the above general scheme to time-dependent systems and systems with external forces (see [9,5]). A force is represented by a semi-basic 1-form F (t, v q ) = α i (t, q,q) dq i , wich is equivalent to give a bundle mappping [9] for details). Assuming that our dynamical system is described by an hyperregular lagrangian L : T Q −→ R and the force F , then using the Legendre transformation F L : T Q → R we can transport F to the hamiltonian side and definẽ We have Given a hamiltonian h : R × T * Q → R, then the evolution of the system with external forceF is now given by ∂ ∂t where VF is the vector field determined by VF (t, α q ) = Λ Q (π * Q (F (t, α Q ))), Λ Q being the canonical Poisson structure on T * Q.
In bundle coordinates ∂ ∂t + X h + VF provides the differential equatioṅ We can equip T * (R × Q) with the almost-Poisson structureΛ given byΛ = Λ R×Q + V F ∧ ∂ ∂e (recall the definition of e in the previous section). Here Λ R×Q denotes the canonical Poisson tensor on T * (R × Q).
In local coordinates It is easy to see that the characteristic distribution ofΛ is the whole space. Indeed, using the local expression ofΛ we have We can define h ext = µ * h + e, where µ is defined in the same way that in 20, and construct the hamiltonian vector field X hext = Λ (dh ext ). Due to the definition of Λ it is easy to see that µ * (X hext ) = ∂ ∂t + X h + VF . The following diagram summarizes our construction If γ is a section of π R×Q (a 1-form on R × Q) we can consider the section of π given by µ • γ and define the vector field ( ∂ ∂t and we can state the following. Theorem 5.4. If Im(γ) is a lagrangian manifold in (T * (R × Q),Λ , then the following assertions are equivalent.
Proof. The proof is analogous to that in Theorem 5.3.
Next, we shall characterize when a section γ is lagrangian.
Proposition 5. Let γ be a 1-form on R × Q; then the image of γ is a lagrangian submanifold with respect toΛ if and only if Proof. Using (21), it is easy to see that Λ is an isomorphism, and so we can define the corresponding almost-symplectic structureΩ, that is ( Λ ) −1 = Ω , and thus Here Ω denotes the induced mapping from tangent vector to 1-forms defined by the 2-formΩ. So we can conclude that The image of the 1-form γ will be lagrangian forΩ if and only if and the result follows.

Remark 1.
Our result generalizes the Hamilton-Jacobi theorem derived in [3] for the case of linear forces and time-dependent systems [20].
6.1. Hamilton-Jacobi equation for the nonholonomic particle. Let a particle of unit mass be moving in space Q = R 3 , with lagrangian and subject to the constraint Passing to the hamiltonian point of view we define the hamiltonian h(x, y, z, p x , p y , p z ) = 1 2 (p 2 x + p 2 y + p 2 z ) + V (x, y, z) , and the submanifold M of T * Q given by p z − yp x = 0. We have the following decomposition along M T (T * Q) Therefore, the projector P : T (T * Q) |M −→ T M is given by P = id T (T * Q) |M − 1 1 + y 2 ∂ ∂p z − y ∂ ∂p x ⊗ (dp z − y dp x − p x dy) and the solution of the nonholonomic dynamics is the following vector field Taking noncanonical coordinates (x, y, z,p 1 ,p 2 ,p 3 ) on T * Q wherep 1 = p x + yp z , p 2 = p y andp 3 = p z − yp x then (x, y, z,p 1 ,p 2 ) are coordinates for M . Therefore, the nonholonomic bracket defined on M is given by {x,p 1 } nh = 1, {y,p 2 } nh = 1, {z,p 1 } nh = y, {p 1 ,p 2 } nh = 1 1 + y 2 , and the remaining brackets are zero.
Thus, the map Φ : is a complete solution of the nonholonomic problem. Therefore it is only necessary to integrate the vector fieldẋ = k 1 1 + y 2 y = k 2 z = yk 1 1 + y 2 to obtain all the solutions of the nonholonomic problem.