Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic Mechanics

In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. The proposed formalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vector bundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system). From them, it is possible to formulate the Hamilton-Jacobi equation, obtaining as a particular case, the classical theory. The main application in this paper is to nonholonomic mechanical systems. For it, we first construct the linear almost Poisson structure on the dual space of the vector bundle of admissible directions, and then, apply the Hamilton-Jacobi theorem. Another important fact in our paper is the use of the orbit theorem to symplify the Hamilton-Jacobi equation, the introduction of the notion of morphisms preserving the Hamiltonian system; indeed, this concept will be very useful to treat with reduction procedures for systems with symmetries. Several detailed examples are given to illustrate the utility of these new developments.


Introduction
The standard Hamilton-Jacobi equation is the first-order, non-linear partial differential equation, for a function S(t, q A ) (called the principal function) and where H is the Hamiltonian function of the system. Taking S(t, q A ) = W (q A ) − tE, where E is a constant, we rewrite the previous equations as where W is called the characteristic function. Equations (1.1) and (1.2) are indistinctly referred as the Hamilton-Jacobi equation (see [1,11]; see also [6] for a recent geometrical approach). The motivation of the present paper is to extend this theory for the case of nonholonomic mechanical systems, that is, those mechanical systems subject to linear constraints on the velocities. In Remark 5.11 of our paper, we carefully summarize previous approaches to this subject. These tried to adapt the standard Hamilton-Jacobi equations for systems without constraints to the nonholonomic setting. But for nonholonomic mechanics is necessary to take into account that the dynamics is obtained from an almost Poisson bracket, that is, a bracket not satisfying the Jacobi identity. In this direction, in a recent paper [20], the authors have developed a new approach which permits to extend the Hamilton-Jacobi equation to nonholonomic mechanical systems. However, the expression of the corresponding Hamilton-Jacobi equation is far from the standard Hamilton-Jacobi equation for unconstrained systems. This fact has motivated the present discussion since it was necessary to understand the underlying geometric structure in the proposed Hamilton-Jacobi equation for nonholonomic systems.
To go further in this direction, we need a new framework, which captures the non-Hamiltonian essence of a nonholonomic problem. Thus, we have considered a more general minimal "Hamiltonian" scenario. The starting point is a vector bundle τ D : D −→ Q such that its dual vector bundle τ D * : D * −→ Q is equipped with a linear almost Poisson bracket {·, ·} D * , that is, a linear bracket satisfying all the usual properties of a Poisson bracket except the Jacobi identity. The existence of such bracket is equivalent to the existence of an skew-symmetric algebroid structure ([[·, ·]] D , ρ D ) on τ D : D −→ Q (i.e. a Lie algebroid structure eliminating the integrability property), or even, the existence of an almost differential d D on τ D : D −→ Q, that is, an operator d D which acts on the "forms" on D and it satisfies all the properties of an standard differential except that (d D ) 2 is not, in general, zero. We remark that skew-symmetric algebroid structures are almost Lie structures in the terminology of [34] (see also [35]) and that the one-to-one correspondence between skew-symmetric algebroids and almost differentials was obtained in [34]. We also note that an skew-symmetric algebroid also is called a pre-Lie algebroid in the terminology introduced in some papers (see, for instance, [15,16,23]) and the relation between linear almost Poisson brackets and skew-symmetric algebroid structures was discussed in these papers (see also [12,13] for some applications to Classical Mechanics).
In this framework, a Hamiltonian system is given by a Hamiltonian function h : D * −→ R. ] D is the usual Lie bracket of vector fields which is related with the canonical Poisson bracket on T * Q, so that d D is just, in this case, the usual exterior differential. Another important fact is the introduction of the notion of morphisms preserving the Hamiltonian system; indeed, this concept will be very useful to treat with reduction procedures for nonholonomic systems with symmetries. We remark that this type of procedures were intensively discussed in the seminal paper [3] by Bloch et al.
In the above framework we can prove the main result of our paper: Theorem 4.1. In this theorem, we obtain the Hamilton-Jacobi equation whose expression seems a natural extension of the classical Hamilton-Jacobi equation for unconstrained systems, as appears, for instance, in [1]. Moreover, our construction is preserved under the natural morphims of the theory. This fact is proved in Theorem 4.12.
Furthermore, using the orbit theorem (see [2]), we will show that the classical form of the Hamilton-Jacobi equation: H • α = constant, with α : Q → D * satisfying d D α = 0, remains valid for a special class of nonholonomic mechanical systems: those satisfying the condition of being completely nonholonomic. See Section 3 for more details and also the paper by Ohsawa and Bloch [31] for the particular case when D is a distribution on Q.
The above theorems are applied to the theory of mechanical systems subjected to linear nonholonomic constraints on a Lie algebroid A. The ingredients of this theory are a Lie algebroid τ A : A → Q over a manifold Q, a Lagrangian function L : A → R of mechanical type, and a vector subbundle τ D : D → Q of A. The total space D of this vector subbundle is the constraint submanifold (see [7]). Then, using the corresponding linear Poisson structure on A * , one may introduce a linear almost Poisson bracket on D * , the so-called nonholonomic bracket. A linear almost Poisson bracket on D which is isomorphic to the nonholonomic bracket was considered in [7]; however, it should be remarked that our formalism simplifies very much the procedure to obtain it. Using all these ingredients one can apply the general procedure (Theorems 4.1 and 4.12) to obtain new and interesting results. We also remark that the main part of the relevant information for developing the Hamilton-Jacobi equation for the nonholonomic system (L, D) is contained in the vector subbundle D or, equivalently, in its dual D * (see Theorems 4.1 and 4.12). Then, the computational cost is lower than in previous approximations to the theory.
In the particular case when A is the standard Lie algebroid τ T Q : T Q → Q then the constraint subbundle is a distribution D on Q. The linear almost Poisson bracket on D * is provided by the classical nonholonomic bracket (which is usually induced from the canonical Poisson bracket on T * Q), clarifying previous constructions [5,19,39]. In addition, as a consequence, we recover some of the results obtained in [20] about the Hamilton-Jacobi equation for nonholonomic mechanical systems (see Corollary 5.9). Moreover, we apply our results to an explicit example: the two-wheeled carriage. On the other hand, if our Lagrangian system on an arbitrary Lie algebroid A is unconstrained (that is, the constraint subbundle D = A) then, using our general theory, we recover some results on the Hamilton-Jacobi equation for Lie algebroids (see Corollary 5.1) which were proved in [25]. Furthermore, if A is the standard Lie algebroid τ T Q : T Q → Q then we directly deduce some well-known facts about the classical Hamilton-Jacobi equation (see Corollary 5.2).
Another interesting application is discussed; the particular case when the Lie algebroid is the Atiyah algebroid τĀ :Ā = T Q/G →Q = Q/G associated with a principal G-bundle F : Q → Q = Q/G. In such a case, we have a Lagrangian functionL :Ā → R of mechanical type and a constraint subbundle τD :D →Q of τĀ :Ā = T Q/G →Q. This nonholonomic system is precisely the reduction, in the sense of Theorem 4.12, of a constrained system (L, D) on the standard Lie algebroid τ A : A = T Q → Q. In fact, using Theorem 4.12, we deduce that the solutions of the Hamilton-Jacobi equations for both systems are related in a natural way by projection. We also characterize the nonholonomic bracket onD * . All these results are applied to a very interesting example: the snakeboard. In this example, an explicit expression of the reduced nonholonomic bracket is found; moreover, the Hamilton-Jacobi equations are proposed and it is shown the utility of our framework to integrate the equations of motion.
We expect that the results of this paper will be useful for analytical integration of many difficult systems (see, as an example, the detailed study of the snakeboard in this paper and the examples in [31]) and the key for the construction of geometric integrators based on the Hamilton-Jacobi equation (see, for instance, Chapter VI in [18] and references therein for the particular case of standard nonholonomic mechanical systems).
The structure of the paper is as follows. In Section 2, the relation between linear almost Poisson structures on a vector bundle, skew-symmetric algebroids and almost differentials is obtained. In Section 3, we introduce the notion of a completely nonholonomic skew-symmetric algebroid and we prove that on an algebroid D of this kind with connected base Q the space We also prove that on an arbitrary skew-symmetric algebroid D the condition d D f = 0 implies that f is constant on the leaves of a certain generalized foliation (see Theorem 3.4). For this purpose, we will use the orbit theorem. In Section 4, we consider Hamiltonian systems associated with a linear almost Poisson structure on the dual bundle D * to a vector bundle and a Hamiltonian function on D * . Then, the Hamilton-Jacobi equation is proposed in this setting. Moreover, using the results of Section 3, we obtain an interesting expression of this equation. In Section 5, we apply the previous results to nonholonomic mechanical systems and, in particular, to some explicit examples. Moreover, we review in this section some previous approaches to the topic. We conclude our paper with the future lines of work and an appendix with the proof of some technical results.

Linear almost Poisson structures, skew-symmetric algebroids and almost differentials
Most of the results contained in this section are well-known in the literature (see [14,15,16,34,35]). However, to make the paper more self-contained, we will include their proofs.
Let τ D : D → Q be a vector bundle of rank n over a manifold Q of dimension m. Denote by D * the dual vector bundle to D and by τ D * : D * → Q the corresponding vector bundle projection.
(iii) {·, ·} D * is linear, that is, if ϕ and ψ are linear functions on D * then {ϕ, ψ} D * is also a linear function.
If, in addition, the bracket satisfies the Jacobi identity then we have that {·, ·} D * is a linear Poisson structure on D * . Properties (i) and (ii) in Definition 2.1 imply that there exists a 2-vector Λ D * on D * such that Λ D * is called the linear almost Poisson 2-vector associated with the linear almost Poisson structure {·, ·} D * .
Note that there exists a one-to-one correspondence between the space Γ(τ D ) of sections of the vector bundle τ D : D → Q and the space of linear functions on D * . In fact, if X ∈ Γ(τ D ) then the corresponding linear functionX on D * is given bŷ Proof. Let Y be an arbitrary section of τ D : D → Q.
Using Definition 2.1, we have that is a linear function on D * . Thus, since (f • τ D * ){X,Ŷ } D * also is a linear function, it follows that {X, f • τ D * } D * is a basic function with respect to τ D * . This proves (i).
On the other hand, using (i) and Definition 2.1, we deduce that If (q i ) are local coordinates on an open subset U of Q and {X α } is a basis of sections of the vector bundle τ −1 D (U ) → U then we have the corresponding local coordinates (q i , p α ) on D * . Moreover, from Proposition 2.2, it follows that αβ and ρ j α real C ∞ -functions on U . Consequently, the linear almost Poisson 2-vector Λ D * has the following local expression 3. An skew-symmetric algebroid structure on the vector bundle τ D : and a vector bundle morphism ρ D : D → T Q, the anchor map, such that: (ii) If we also denote by ρ D : Γ(τ D ) → X(Q) the morphism of C ∞ (Q)-modules induced by the anchor map then If the bracket [[·, ·]] D satisfies the Jacobi identity, we have that the pair ([[·, ·]] D , ρ D ) is a Lie algebroid structure on the vector bundle τ D : D → Q.
is a Lie algebroid over Q, we may consider the generalized distributioñ D whose characteristic space at a point q ∈ Q is given byD(q) = ρ D (D q ), where D q is the fibre of D over q. The distributionD is finitely generated and involutive. Thus,D defines a generalized foliation on Q in the sense of Sussmann [38].D is the Lie algebroid foliation on Q associated with D.
Now, we will denote by LAP(D * ) (respectively, LP(D * )) the set of linear almost Poisson structures (respectively, linear Poisson structures) on D * . Denote also by SSA(D) (respectively, LA(D)) the set of skew-symmetric algebroid (respectively, Lie algebroid) structures on the vector bundle τ D : D → Q. Then, we will see in the next theorem that there exists a one-to-one correspondence between LAP(D * ) (respectively, LP(D * )) and the set of skew-symmetric algebroid (respectively, Lie algebroid) structures on τ D : D → Q.
On the other hand, from (2.2), we obtain that Therefore, αβ and ρ j α are called the local structure functions of the skew-symmetric algebroid structure ([[·, ·]] D , ρ D ) with respect to the local coordinates (q i ) and the basis {X α }.
Next, we will see that there exists a one-to-one correspondence between SSA(D) and the set of almost differentials on the vector bundle τ D : D → M . Definition 2.6. An almost differential on the vector bundle τ D : D → Q is a R-linear map If (d D ) 2 = 0 then d D is said to be a differential on the vector bundle τ D : D → Q.
Denote by AD(D) (respectively, D(D)) the set of almost differentials (respectively, differentials) on the vector bundle τ D : D → Q.
is an skew-symmetric algebroid structure on the vector bundle τ D : D → Q then the corresponding almost differential d D is defined by for α ∈ Γ(Λ k τ D * ) and X 0 , . . . , X k ∈ Γ(τ D ).
Let ([[·, ·]] D , ρ D ) be a skew-symmetric algebroid structure on the vector bundle τ D : D → Q and d D be the corresponding almost differential. If (q i ) are local coordinates on an open subset U of Q and {X α } is a basis of sections of the vector bundle τ −1 D (U ) → U such that C γ αβ and ρ j α are the local structure functions of the skew-symmetric algebroid structure, then for all i and γ. From Theorems 2.5 and 2.7, we conclude the following result

Skew-symmetric algebroids and the orbit theorem
Let (D, [[·, ·]] D , ρ D ) be a skew-symmetric algebroid over Q and d D be the corresponding almost differential.
We can consider the vector space over R Note that if D is a Lie algebroid we have that H 0 (d D ) is the 0 Lie algebroid cohomology group associated with D.
On the other hand, it is clear that if Q is connected and D is a transitive skew-symmetric algebroid, that is, (3.1) Condition (3.1) will play an important role in Section 4.1.
Next, we will see that (3.1) holds if the skew-symmetric algebroid is completely nonholonomic with connected base space.
LetD be the generalized distribution on Q whose characteristic space at the point q ∈ Q is It is clear thatD is finitely generated. Note that the C ∞ -module Γ(D) is finitely generated (see [17]). Now, denote by Lie ∞ (D) the smallest Lie subalgebra of X(Q) containingD. Then Lie ∞ (D) is comprised of finite R-linear combinations of vector fields of the form with k ∈ N, k = 0, andX 1 , . . . ,X k ∈ X(Q) satisfying X l (q) ∈D q , for all q ∈ Q (see [2]).
For each q ∈ Q, we will consider the vector subspace Lie ∞ q (D) of T q Q given by The leaf L of this foliation over the point q 0 ∈ Q is the orbit ofD over the point q 0 , that is, is the flow of the vector fieldX l at the timet l (for more details, see [2]).
Thus, if Q is a connected manifold, it follows that D is completely nonholonomic if and only if the orbit ofD over any point q 0 ∈ Q is Q.
(i) Definition 3.1 may be extended for anchored vector bundles. A vector bundle τ D : D → Q over Q is said to be anchored if it admits an anchor map, that is, a vector bundle morphism ρ D : D → T Q. In such a case, the vector bundle is said to be completely  [44]. In this sense it is formulated in the literature the classical Rashevsky-Chow theorem: If Lie ∞ q (D) = T q Q, for all q ∈ Q, then each orbit is equal to the whole manifold Q.

Now, we deduce the following result
Thus, first we have that Secondly, since D is completely nonholonomic then Lie ∞ q 0 (D) = T q 0 Q. Therefore, there exists a finite sequence of vector fields on Q,X 1 , . . . ,X k such thatX i (q) ∈D q , for all i ∈ {1, . . . , k} and From both considerations, we deduce the result.
However, the condition H 0 (d D ) R does not imply, in general, that the skew-symmetric algebroid D is completely nonholonomic.
In fact, let D be the tangent bundle to R 2 If (x, y) are the standard coordinates on R 2 , it follows that . So, we can consider the skew-symmetric algebroid structure ([[·, ·]] T R 2 , ρ T R 2 ) on T R 2 which is characterized by the following conditions Then, the generalized distributionD = T R 2 on R 2 is generated by the vector fields Thus, the Lie subalgebra Lie ∞ (D) of T R 2 is generated by the vector fields This implies that Consequently, using that f ∈ C ∞ (R 2 ), we obtain that f is constant.
Next, we will discuss the case when the generalized foliation Lie ∞ (D) = T Q. In fact, we will prove the following result.
Suppose that L is an orbit ofD and that D L is the vector bundle over L given by for all q ∈ L. Thus, we have a vector bundle morphism On the other hand, we may define a R-bilinear skew-symmetric bracket with U an open subset of Q and q ∈ U . Note that the condition with V an open subset of Q and In addition, it is clear that This proves (i).
On the other hand, it follows that the condition d D f = 0 implies that and, since H 0 (d D L ) R (as a consequence of the first part of the theorem), we conclude that f is constant on L.
It will be also interesting to characterize under what conditions there exist functions f ∈ C ∞ (Q) such that (d D ) 2 f = 0. Using Equations (2.4) we easily deduce that: for all X, Y ∈ Γ(τ D ). Now, consider the generalized distributionD on Q whose characteristic spacē D q at the point q ∈ Q is: It is also clear thatD is finitely generated. Denote by Lie ∞ (D) the smallest Lie subalgebra of X(Q) containingD. Observe that Lie ∞ (D) ⊆ Lie ∞ (D). We deduce that (d D ) 2 f = 0 if and only if f is constant on any orbitL ofD. Of course ifD is completely nonholonomic then the unique functions The triplet (D, {·, ·} D * , h) is said to be a Hamiltonian system.
We will denote by Λ D * the linear almost Poisson 2-vector associated with {·, ·} D * . Then, we may introduce the vector field H Λ D * h on D * given by Therefore, the Hamilton equations are
Then, the aim of this section is to prove the following result. (i) If c : I → Q is an integral curve of the vector field H Λ D * h,α , that is,  In order to prove Theorem 4.1, we will need some previous results: Thus, if we apply Proposition 4.3 we obtain that α is a closed 1-form if and only if α(Q) is a Lagrangian submanifold of T * Q. This is a well-known result in the literature (see, for instance, [1]). on Q and D * , respectively, are α-related, that is, Therefore, we must prove that Then, using (4.2), (4.3) and (4.4), we deduce that Consequently, from Proposition 4.3, we obtain that Now, using Proposition 4.5 and the fact that η α(q) ∈ (L α,D (q)) 0 , we conclude that which implies that (see (4.3) and (4.5)) Therefore, it follows that and, from Proposition 4.3, we obtain that there exists This implies that Let (F , F ) be a vector bundle morphism between the vector bundles τ D * : D * → Q and τD * : D * →Q. IfX is a section of τD :D →Q then we may define the section (F , F ) * X of τ D : D → Q characterized by the following condition α q (((F , F ) * X )(q)) =F (α q )(X(F (q))), (4.7) for all q ∈ Q and α q ∈ D * q .    Thus, from (2.2) and (4.6), we obtain that which implies that (4.8) holds.
Then, from (4.9), it follows that Thus, using condition (C), we deduce that This implies that (4.8) holds.
Remark 4.10. Let (F , F ) be a linear almost Poisson morphism between the vector bundles τ D * : D * → Q and τD * :D * →Q. Moreover, suppose that F is surjective and that (F , F ) is a fiberwise injective vector bundle morphism.
(i) From Theorem 4.9, we deduce that the condition H 0 (d D ) R implies that H 0 (dD) R. In general, the converse does not hold. However, if H 0 (dD) R and f ∈ C ∞ (Q) is a F -basic function such that d D f = 0 then f is constant. (ii) If F is a surjective submersion with connected fibers, V q F ⊆D q = ρ D (D q ), for all q ∈ Q, and d D f = 0 then f is a F -basic function. Here, V F is the vertical bundle to F . Now, we will introduce the following definition. In addition, from Theorem 4.9, we deduce the following result If G : A × Q A → R is a bundle metric on A then the Levi-Civita connection is determined by the formula for X, Y, Z ∈ Γ(A). Using the covariant derivative induced by ∇ G , one may introduce the notion of a geodesic of ∇ G as follows. A curve σ : An admissible curve σ : I → A is said to be a geodesic if ∇ G σ(t) σ(t) = 0, for all t ∈ I. The geodesics are the integral curves of a vector field ξ G on A, the geodesic flow of A, which is locally given by (for more details, see [7,9]). The Lagrangian function L : A → R of an (unconstrained) mechanical system on A is defined by , for a ∈ A, V : Q → R being a real C ∞ -function on Q. In other words, L is the kinetic energy induced by G minus the potential energy induced by V .
Note that if ∆ is the Liouville vector field of A then the Lagrangian energy E L = ∆(L) − L is the real C ∞ -function on A given by , for a ∈ A. On the other hand, we may consider the section grad G V of τ A : A → Q characterized by the following condition . Then, the solutions of the Euler-Lagrange equations for L are the integral curves of the vector field ξ L on A defined by is the standard vertical lift of the section grad G V . The local expression of the Euler-Lagrange equations iṡ for all i and E (see [7,9]). Now, we will denote by G : A → A * the vector bundle isomorphism induced by G and by # G : A * → A the inverse morphism. If α : Q → A * is a section of the vector bundle τ A * : A * → Q we also consider the vector field ξ L,α on Q defined by ξ L,α (q) = (T # G (α(q)) τ A )(ξ L (# G (α(q)))), for q ∈ Q.
Corollary 5.1. Let α : Q → A * be a 1-cocycle of the Lie algebroid A, that is, d A α = 0. Then, the following conditions are equivalent: is a solution of the Euler-Lagrange equations for L. (ii) α satisfies the Hamilton-Jacobi equation Proof. The Legendre transformation associated with the Lagrangian function L is the vector bundle isomorphism G : A → A * between A and A * induced by the bundle metric G (for the definition of the Legendre transformation associated with a Lagrangian function on a Lie algebroid, see [25]). Thus, if we denote by G * the bundle metric on A * then, the Hamiltonian function H L = E L • # G induced by the hyperregular Lagrangian function L is given by Therefore, if Λ A * is the corresponding linear Poisson 2-vector on A * and H Λ A * H L is the Hamiltonian vector field of H L with respect to Λ A * , we have that the solutions of the Hamilton equations are the integral curves of the vector field H Λ A * H L . In fact, the vector fields ξ L and H Λ A * H L are G -related, that is, Consequently, if σ : I → A is a solution of the Euler-Lagrange equations for L then G • σ : I → A * is a solution of the Hamilton equations for H L and, conversely, if γ : I → A * is a solution of the Hamilton equations for H L then # G • γ : I → A is a solution of the Euler-Lagrange equations for L (for more details, see [25]).
Next, we will apply Corollary 5.1 to the particular case when A is the standard Lie algebroid T Q and α is a 1-coboundary, that is, α = dS with S : Q → R a real C ∞ -function on Q. Note that, in this case, the bundle metric G on T Q is a Riemannian metric g on Q and that # G • α = # g • dS is just the gradient vector field of S, grad g S, with respect to g.
, where G * (respectively,Ḡ * ) is the bundle metric on A * (respectively,Ā * ). Note that the first condition implies thatF q is injective and an isometry. A particular example of the above general construction is the following one. Let F : Q →Q = Q/G be a principal G-bundle. Denote by φ : G × Q → Q the free action of G on Q and by T φ : G × T Q → T Q the tangent lift of φ. T φ is a free action of G on T Q. Then, we may consider the quotient vector bundle τĀ = τ T Q/G :Ā = T Q/G →Q = Q/G. The sections of this vector bundle may be identified with the vector fields on Q which are G-invariant. Thus, using that a G-invariant vector field is F -projectable and that the standard Lie bracket of two G-invariant vector fields is also a G-invariant vector field, we can define a Lie algebroid structure ([[·, ·]]Ā, ρĀ) on the quotient vector bundle τĀ = τ T Q/G :Ā = T Q/G →Q = Q/G. The resultant Lie algebroid is called the Atiyah (gauge) algebroid associated with the principal bundle F : Q →Q = Q/G (see [25,27]).
On the other hand, denote by T * φ : G × T * Q → T * Q the cotangent lift of the action φ. Then, the space of orbits of T * φ, T * Q/G, may be identified with the dual bundleĀ * toĀ. Under this identification, the linear Poisson structure onĀ * is characterized by the following condition: the canonical projectionF : A * = T * Q → T * Q/G Ā * is a Poisson morphism, when on A * = T * Q we consider the linear Poisson structure induced by the standard Lie algebroid τ A = τ T Q : A = T Q → Q, that is, the Poisson structure induced by the canonical symplectic structure of T * Q (an explicit description of the linear Poisson structure onĀ * T * Q/G may be found in [32]).
Thus, (F , F ) is a linear Poisson morphism between A * = T * Q andĀ * T * Q/G and, in addition, F is a fiberwise bijective vector bundle morphism. Now, suppose that G = g is a G-invariant Riemannian metric on Q and that V : Q → R is a Ginvariant function on Q. Then, we may consider the corresponding mechanical Lagrangian function L : A = T Q → R. Moreover, it is clear that g and V induce a bundle metricḠ onĀ = T Q/G and a real functionV :Q → R and, therefore, a mechanical Lagrangian functionL : On the other hand, we have that for each q ∈ Q the mapF q : is a linear isometry. Consequently, using Corollary 5.4, we deduce the following result An explicit example: The Elroy's Beanie. This system is probably the most simple example of a dynamical system with a non-Abelian Lie group of symmetries. It consists in two planar rigid bodies attached at their centers of mass, moving freely in the plane (see [29]). So, the configuration space is Q = SE(2) × S 1 with coordinates q = (x, y, θ, ψ), where the three first coordinates describe the position and orientation of the center of mass of the first body and the last one the relative orientation between both bodies. The Lagrangian L : T Q → R is where m denotes the mass of the system and I 1 and I 2 are the inertias of the first and the second body, respectively; additionally, we also consider a potential function of the form V (ψ). The kinetic energy is associated with the Riemannian metric G on Q given by G = m(dx 2 + dy 2 ) + (I 1 + I 2 )dθ 2 + I 2 dθ ⊗ dψ + I 2 dψ ⊗ dθ + I 2 dψ 2 .

Mechanical systems subjected to linear nonholonomic constraints on a Lie algebroid.
Let τ A : A → Q be a Lie algebroid over a manifold Q and denote by ([[·, ·]] A , ρ A ) the Lie algebroid structure on A. This kind of systems were considered in [7,9,14]. We will denote by i D : D → A the canonical inclusion. We also consider the orthogonal decomposition A = D ⊕ D ⊥ and the associated orthogonal projectors P : A → D and Q : A → D ⊥ . Then, the solutions of the dynamical equations for the nonholonomic (constrained) system (L, D) are just the integral curves of the vector field ξ (L,D) on D defined by where ξ L is the solution of the free dynamics (see Section 5.1) and T P : T A → T D is the tangent map to the projector P .
In Moreover, if ρ i B and C E BC are the local structure functions of A, we have that the local expression of the vector field ξ (L,D) is Thus, the dynamical equations for the constrained system (L, D) arė On the other hand, the constrained connection∇ : Γ(τ A ) × Γ(τ A ) → Γ(τ A ) associated with the system (L, D) is given by∇ . Therefore, ifΓ E BC are the coefficients of∇, we have thať ,Γ a γν = 0. Consequently, Eqs. (5.4) are just the Lagrange-D'Alembert equations for the system (L, D) considered in [7] (see also [9,14]).
Next, we will introduce a linear almost Poisson structure {·, ·} D * on D * . Denote by {·, ·} A * the linear Poisson bracket on A * induced by the Lie algebroid structure on A. It is easy to prove that {·, ·} D * is a linear almost Poisson bracket on D * . Moreover, if (q i , p B ) = (q i , p γ , p b ) are the dual coordinates of (q i , v B ) = (q i , v γ , v b ) on A * then it is clear that (q i , p γ ) are local coordinates on D * and, in addition, the local expressions of i * D and P * are i * D (q i , p γ , p b ) = (q i , p γ ), P * (q i , p γ ) = (q i , p γ , 0). (5.6) Thus, from (2.1), (5.5) and (5.6), we have that for ϕ, ψ ∈ C ∞ (D * ). On the other hand, one may introduce a linear Poisson bracket {·, ·} A on A in such a way that the vector bundle map G : A → A * is a Poisson isomorphism, when on A * we consider the linear Poisson structure {·, ·} A * . Since the local expression of G is we deduce that the local expression of the linear Poisson bracket {·, ·} A is Using the bracket {·, ·} A , one may define a linear almost Poisson bracket {·, ·} nh on D as follows. Ifφ andψ are real C ∞ -functions on D then We have that Thus, a direct computation proves that {·, ·} nh is just the nonholonomic bracket introduced in [7]. Note that, using (5.3) and (5.8), we obtain that ξ (L,D) is the Hamiltonian vector field of the function (E L ) |D with respect to the nonholonomic bracket {·, ·} nh , i.e., forφ ∈ C ∞ (D) (see also [7]). Moreover, if G D is the restriction of the bundle metric G to D and G D : D → D * is the corresponding vector bundle isomorphism then, from (5.7) and (5.8), we deduce that For this reason, {·, ·} D * will also be called the nonholonomic bracket associated with the constrained system (L, D).
We will denote by ([[·, ·]] D , ρ D ) (respectively, d D ) the corresponding skew-symmetric algebroid structure (respectively, almost differential) on the vector bundle τ D : D → Q and by # G D : D * → D the inverse morphism of G D : D → D * .
Then, from (2.2) and (5.5), it follows that for X, Y ∈ Γ(τ D ). Therefore, using (2.5), we have that On the other hand, if α : Q → D * is a section of the vector bundle τ D * : D * → Q one may consider the vector field ξ (L,D),α on Q given by (α(q)))), for q ∈ Q.
where g is a Riemannian metric on Q and V : Q → R is a real C ∞ -function on Q. Suppose also that D is a distribution on Q. Then, the pair (L, D) is a mechanical system subjected to linear nonholonomic constraints on the standard Lie algebroid τ T Q : T Q → Q. Note that, in this case, the linear Poisson structure on A * = T * Q is induced by the canonical symplectic structure on T * Q. Moreover, the corresponding nonholonomic bracket {·, ·} D * on D * was considered by several authors or, alternatively, other almost Poisson structures (on D or on g (D) ⊆ A * = T * Q) which are isomorphic to {·, ·} D * also were obtained by several authors (see [5,19,22,39]). Now, denote by # g : T * Q → T Q (respectively, # g D : D * → D) the inverse morphism of the musical isomorphism g : T Q → T * Q (respectively, g D : D → D * ) induced by the Riemannian metric g (respectively, by the restriction g D of g to D), by d the standard exterior differential on Q (that is, d = d T Q is the differential of the Lie algebroid τ T Q : T Q → Q), by ξ (L,D) ∈ X(D) the solution of the nonholonomic dynamics and by ξ (L,D)α ∈ X(Q) its projection on Q, α being a section of the vector bundle τ D * : D * → Q (see (5.11)). Using this notation, Corollary 5.6 and Remark 5.7, we deduce the following result Corollary 5.9. Let α : Q → D * be a section of the vector bundle τ D * : D * → Q such that d(P * •α) ∈ I(D 0 ). Then, the following conditions are equivalent: (i) If c : I → Q is an integral curve of the vector field ξ (L,D)α on Q we have that # g D •α•c : I → D is a solution of the Lagrange-D'Alembert equations for the constrained system (L, D).
Remark 5.10. As we know, the Legendre transformation associated with the Lagrangian function L : T Q → R is the musical isomorphism g : T Q → T * Q. Moreover, it is clear that X(Q) ⊆ D, where X is the vector field on Q given by X = # g D • α. Thus, Corollary 5.9 is a consequence of some results which were proved in [20] (see Theorem 4.3 in [20]). On the other hand, if H 0 (d D ) R (or if Q is connected and the distribution D is completely nonholonomic in the sense of Vershik and Gershkovich [44]) then (i) and (ii) in Corollary 5.9 are equivalent to the condition A Hamiltonian version of this last result was proved by Ohsawa and Bloch [31] (see Theorem 3.1 in [31]).
Remark 5.11. Previous approaches. There exists some different attempts in the literature of extending the classical Hamilton-Jacobi equation for the case of nonholonomic constraints [10,33,36,40,41,42,43]). These attempts were non-effective or very restrictive (and even erroneous), because, in many of them, they try to adapt the standard proof of the Hamilton-Jacobi equations for systems without constraints, using Hamilton's principle. See [37] for a detailed discussion on the topic.
To fix ideas, consider a lagrangian system L : for v q ∈ T q Q, and nonholonomic constraints determined by a distribution D of Q, whose annihilator is D 0 = span{µ b i dq i }. The idea of many of these previous approaches consist in looking for a function S : Q −→ R called the characteristic function which permits characterize the solutions of the nonholonomic problem. For it, define first the generalized momenta which satisfy the constraint equations G ij p i µ a j = 0. These last conditions univocally determine λ b as functions of q and ∂S/∂q and therefore we find the momenta as functions (5.14) By inserting these expressions for the generalized momenta in the Hamiltonian of the system, we obtain a version of the Hamilton-Jacobi equation (in its time-independent version): However, if we start with a curve c : I → Q satisfying the differential equationṡ in general, it is not true that the curve γ(t) = (c i (t), p i (t)) is a solution of the nonholonomic equations. This is trivially checked since from Equation (5.15) we deduce that: but, on the other hand,ṗ Substituting Equation (5.17) in Equation (5.18), a curve γ(t) = (c i (t), p i (t)) satisfying (5.16) is solution of the nonholonomic equations (that is, if it verifies the following condition: It is well-known (see [36,37]) that condition (5.19) takes place when the solutions of the nonholonomic problem are also of variational type. However, nonholonomic dynamics is not, in general, of variational kind (see [8,24,26]). Indeed, a relevant difference with the unconstrained mechanical systems is that a nonholonomic system is not Hamiltonian in the standard sense since the dynamics is obtained from an almost Poisson bracket, that is, a bracket not satisfying the Jacobi identity (see [5,19,22,39]).
An explicit example: The two-wheeled carriage (see [30]). The system has configuration space Q = SE(2)×T 2 , where SE(2) represents the rigid motions in the plane and T 2 the angles of rotation of the left and right wheels. We use standard coordinates (x, y, θ, ψ 1 , ψ 2 ) ∈ SE(2)×T 2 . Imposing the constraints of no lateral sliding and no sliding on both wheels, one gets the following nonholonomic constraints:ẋ sin θ −ẏ cos θ = 0, x cos θ +ẏ sin θ + rθ + aψ 1 = 0, x cos θ +ẏ sin θ − rθ + aψ 2 = 0, where a is the radius of the wheels and r is the half the length of the axle.
Assuming, for simplicity, that the center of mass of the carriage is situated on the center of the axle the Lagrangian is given by: where m is the mass of the system, J the moment of inertia when it rotates as a whole about the vertical axis passing through the point (x, y) and C the axial moment of inertia. Note that L is the kinetic energy associated with the Riemannian metric g on Q given by g = m(dx 2 + dy 2 ) + Jdθ 2 + Cdψ 2 1 + Cdψ 2 2 .
The constraints induce the distribution D locally spanned by the following g-orthonormal vector fields where Λ 1 = 4Cr 2 + a 2 J + am 1 r 2 Λ 2 = (a 2 J + 2Cr 2 )(2C + m 1 a 2 )(a 2 J + 4Cr 2 + a 2 r 2 m 1 ) We will denote by (x, y, θ, ψ 1 , ψ 2 , v 1 , v 2 ) the local coordinates on D induced by the basis {X 1 , X 2 }. In these coordinates, the restriction, L |D : D −→ R, of L to D is: The distribution D ⊥ orthogonal to D is generated by Moreover, since the standard Lie bracket [X 1 , X 2 ] of the vector fields X 1 and X 2 is orthogonal to D, it follows that (see (5.9)) In particular, taking α = K 1 X 1 + K 2 X 2 , with K 1 , K 2 ∈ R trivially is satisfied the 1-cocycle condition.
In addition, since E L = L, we deduce that which implies that (E L ) |D • # g D • α = K 2 1 + K 2 2 = constant. Thus, using Corollary 5.6, we conclude that to integrate the nonholonomic mechanical system (L, D) is equivalent to find the integral curves of the vector field on Q = S 1 × S 1 × R 2 given by which are easily obtained.
It is also interesting to observe that, in this particular example, and, thus, D is not completely nonholonomic. From Theorem 3.4 it is necessary to restrict the initial nonholonomic system to the orbits of D, that in this case are to obtain a completely nonholonomic skew-symmetric algebroid structure on the vector bundle τ D L k : D L k → L k . Note that on L k we can use, for instance, coordinates (x, y, ψ 1 , ψ 2 ).

5.2.2.
The particular caseĀ = T Q/G. Let F : Q →Q = Q/G be a principal G-bundle and τĀ = τ T Q/G :Ā = T Q/G →Q = Q/G be the Atiyah algebroid associated with the principal bundle (see Section 5.1). Suppose that g is a G-invariant Riemannian metric on Q, that V : Q → R is a G-invariant real C ∞ -function and that D is a G-invariant distribution on Q. Then, we may consider the corresponding nonholonomic mechanical system (L, D) on the standard Lie algebroid τ A = τ T Q : A = T Q → Q.
Denote by ξ (L,D) ∈ X(D) the nonholonomic dynamics for the system (L, D) and by {·, ·} D * the nonholonomic bracket on D * .
The Riemannian metric g and the function V : Q → R induce a bundle metricḠ on the Atiyah algebroid τĀ = τ T Q/G :Ā = T Q/G →Q = Q/G and a real C ∞ -functionV :Q → R onQ such that V • F = V , where F : Q →Q = Q/G is the canonical projection. Moreover, the space of orbitsD of the action of G on D is a vector subbundle of the Atiyah algebroid τĀ = τ T Q/G :Ā = T Q/G → Q = Q/G. Thus, we may consider the corresponding nonholonomic mechanical system (L,D) on A = T Q/G.
LetF : A = T Q →Ā = T Q/G be the canonical projection. Then, (F , F ) is a fiberwise bijective morphism of Lie algebroids andF (D) =D. Therefore, using some results in [7] (see Theorem 4.6 in [7]) we deduce that the vector field ξ (L,D) isF D -projectable on the nonholonomic dynamics ξ (L,D) ∈ X(D) of the system (L,D). Here,F D : D →D = D/G is the canonical projection.
On the other hand, if P : A = T Q → D andP :Ā = T Q/G →D = D/G are the orthogonal projectors then it is clear thatF whereF : A * = T * Q →Ā * T * Q/G andF D : D * →D * D * /G are the canonical projections. Moreover, if onĀ * we consider the linear Poisson structure induced by the Atiyah algebroid τĀ = τ T Q/G :Ā = T Q/G →Q = Q/G then, as we know,F : A * = T * Q →Ā * T * Q/G is a Poisson morphism. Thus, using this fact, (5.5) and (5.21), we deduce the following result Note that Proposition 5.12 characterizes the nonholonomic bracket {·, ·}D * . We also note that the linear map ( is a linear isometry, for all q ∈ Q. Therefore, from Remark 5.7 and Corollary 5.8, it follows An explicit example: The snakeboard. The snakeboard is a modified version of the traditional skateboard, where the rider uses his own momentum, coupled with the constraints, to generate forward motion. The configuration manifold is Q = SE(2) × T 2 with coordinates (x, y, θ, ψ, φ) (see [4,21]).
The system is described by a Lagrangian L(q,q) = 1 2 m(ẋ 2 +ẏ 2 ) + 1 2 where m is the total mass of the board, J > 0 is the moment of inertia of the board, J 0 > 0 is the moment of inertia of the rotor of the snakeboard mounted on the body's center of mass and J 1 > 0 is the moment of inertia of each wheel axles. The distance between the center of the board and the wheels is denoted by r. For simplicity, as in [21], we assume that J + J 0 + 2J 1 = mr 2 . The inertia matrix representing the kinetic energy of the metric g on Q defined by the snakeboard is g = mdx 2 + mdy 2 + mr 2 dθ 2 + J 0 dθ ⊗ dψ + J 0 dψ ⊗ dθ + J 0 dψ 2 + 2J 1 dφ 2 .
Next, we will denote by {X 1 , X 2 , X 3 } the g-orthonormal basis of D given by where f (φ) = J 0 − J 2 0 sin 2 φ mr 2 . The vector fields {X 1 , X 2 } describe changes in the internal angles φ and ψ, while X 3 represents the instantaneous rotation when the internal angles are fixed.
Using the left translations on SE(2), we have that the tangent bundle of SE(2) may be identified with the product manifold SE(2) × se(2) and therefore the Atiyah algebroid is identified with the vector bundleτ T 2 = τĀ :Ā = T T 2 × se(2) −→ T 2 . The canonical basis of τĀ : T T 2 × se(2) −→ T 2 is ∂ ∂ψ , ∂ ∂φ , ξ 1 , ξ 2 , ξ 3 . The anchor map and the linear bracket of the Lie algebroid τĀ : T T 2 ×se(2) −→ being equal to zero the rest of the fundamental Lie brackets. We select the orthonormal basis of sections, {X 1 , X 2 , X 3 , X 4 , X 5 }, where and {X 4 , X 5 } is an orthonormal basis of sections of the orthogonal complement toD,D ⊥ , with respect to the induced bundle metric GĀ.

Conclusions and Future Work
In this paper we have elucidated the geometrical framework for the Hamilton-Jacobi equation. Our formalism is valid for nonholonomic mechanical systems. The basic geometric ingredients are a vector bundle, a linear almost Poisson bracket and a Hamiltonian function both on the dual bundle. We also have discussed the behavior of the theory under Hamiltonian morphisms and its applicability to reduction theory. Some examples are studied in detail and, as a consequence, it is shown the utility of our framework to integrate the dynamical equations. However, in this direction more work must be done.
In particular, as a future research, we will study new particular examples, testing candidates for solutions of the nonholonomic Hamilton-Jacobi equation of the form α = d D f , for some f ∈ C ∞ (Q) (if there exists) and moreover we will study the complete solutions for the Hamilton-Jacobi equation using the groupoid theory. In this line, we will study the construction of numerical integrators via Hamilton-Jacobi theory [18]. We will also discuss the extension of our formalism to time-dependent Lagrangian systems subjected to affine constraints in the velocities. It would be interesting to describe the Hamilton-Jacobi theory for variational constrained problems, giving a geometric interpretation of the Hamilton-Jacobi-Bellman equation for optimal control systems. Finally, extensions to classical field theories in the present context could be developed.