A new perspective on nonholonomic brackets and Hamilton–Jacobi theory

The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket on the cotangent bundle of the configuration manifold. This bracket was defined in [6, 21] although there was already some particular and less direct definition. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids [13, 20]. Recently, reviewing two older papers by R. J. Eden [17, 18], we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton–Jacobi theory and the quantization of nonholonomic systems.


Introduction
One of the most important objects in mechanics is the Poisson bracket, which allows us to obtain the evolution of an observable by bracketing it with the Hamiltonian function, or to obtain new conserved quantities of two given ones, using the Jacobi identity satisfied by the bracket.Moreover, the Poisson bracket is fundamental to proceed with the quantization of the system using what Dirac called the analogy principle, also known as the correspondence principle, according to which the Poisson bracket becomes the commutator of the operators associated to the quantized observables.
For a long time, no similar concept existed in the case of nonholonomic mechanical systems, until van der Schaft and Maschke [26] introduced a bracket similar to the canonical Poisson bracket, but without the benefit of integrability (see also [23]).Later, in [5,6] (see also [21]), we have developed a geometric and very simple way to define nonholonomic brackets, in the time-dependent as well time-independent cases.Indeed, it is possible to decompose the tangent bundle and the cotangent bundle along the constraint submanifold in two different ways.Both result in that the nonholonomic dynamics can be obtained by projecting the free dynamics.Furthermore, if we evaluate the projections of the Hamiltonian vector fields of two functions on the configuration manifold (after arbitrary extensions to the whole cotangent) by the canonical symplectic form, two non-integrable brackets are obtained.The first decomposition is due to de León and Martín de Diego [14] and the second one to Bates and Sniatycki [4].The advantage of this second decomposition is that it turns out to be symplectic, and it is the one we will use in the present paper.In any case, we proved that both brackets coincide on the submanifold of constraints [6].Refer to Section 4 in [6] for a detailed analysis explaining how this bracket generalizes the one defined by van der Schaft and Maschke in [26].
On the other hand, by studying the Hamilton-Jacobi equation, we develop a description of nonholonomic mechanics in the setting of skew-symmetric (or almost Lie) algebroids.Note that the "almost" is due to the lack of integrability of the distribution determining the constraints, showing the consistency of the description.In [13,20] we defined a new almost Poisson bracket that we also called nonholonomic.So far, although both nonholonomic brackets have been used in these two different contexts as coinciding, no such proof has ever been published.This paper provides this evidence for the first time.
But the issue does not end there.In 1951, R. J. Eden wrote his doctoral thesis on nonholonomic mechanics under the direction of P.A.M. Dirac (S-Matrix; Nonholonomic Systems, University of Cambridge, 1951), and the results were collected in two publications [17,18].In the first paper, Eden introduced an intriguing γ operator that mapped free states to constrained states.With that operator (a kind of tensor of type (1,1) that has the properties of a projector) Eden obtained the equations of motion, could calculate brackets of all observables, obtained a simple Hamilton-Jacobi equation, and even used it to construct a quantization of the nonholonomic system.These two papers by Eden have had little impact despite their relevance.Firstly, because they were written in terms of coordinates that made their understanding difficult, and secondly, because it was not intil the 1980s when the study of nonholonomic systems became part of the mainstream of geometric mechanics.
Recently, we have carefully studied these two papers by Eden, and realized that the operator γ is nothing else a projection defined by the orthogonal decomposition of the cotangent bundle provided by the Riemannian metric given by the kinetic energy.Consequently, we have defined a new bracket that we call Eden bracket, and proved that coincides with the previous nonholonomic brackets.We are sure that this new approach to the dynamics of nonholonomic mechanical systems opens new and relevant lines of research.Furthermore, this paper may be used as a reference for the reader interested in the different bracket formulations of nonholonomic mechanics.
The paper is structured as follows.In Section 2, we review some elementary notions on Lagrangian and Hamiltonian mechanics within a geometric framework.In Section 3, we recall the main aspects of nonholonomic mechanics and present the corresponding dynamics in both Lagrangian and Hamiltonian settings.We also briefly discuss the skew-symmetric algebroid approach.In Section 4, we introduce the nonholonomic bracket as defined using the cotangent bundle approach and the symplectic decomposition of its tangent bundle along the constraint submanifold.Additionally, we define the nonholonomic bracket in the skew-symmetric algebroid context.In Section 5 we introduce the notion of Eden bracket.The main results of the paper are presented in Section 6, where we prove that these three almost Poisson brackets coincide.In Section 7, we show how the Eden approach is very useful to discuss Hamilton-Jacobi theory for nonholonomic mechanical systems.The above results are illustrated with two examples in Section 8: the nonholonomic particle and the rolling ball.Finally, in Section 9, we point out some interesting future lines of research opened up by the results of this paper.We remark that some of the results in this paper (Theorems 1 and 5) were presented preliminarily, without proofs, in the form of a conference paper [11].
2 Lagrangian and Hamiltonian mechanics: a brief survey

Lagrangian mechanics
Let L : T Q → R be a Lagrangian function, where Q is a configuration n-dimensional manifold.Then, L = L(q i , qi ), where (q i ) are coordinates in Q and (q i , qi ) are the induced bundle coordinates in T Q.We denote by τ Q : T Q → Q the canonical projection such that τ Q (q i , qi ) = (q i ).
We will assume that L is regular, that is, the Hessian matrix and the 2-form Then, ω L is symplectic if and only if L is regular.Consider now the vector bundle isomorphism and the Hamiltonian vector field where E L = ∆(L) − L is the energy, and ∆ is the Liouville vector field generating the dilations on the fibers of T Q.In bundle coordinates, ∆ = qi ∂/∂ qi .The vector field ξ L , called the Euler-Lagrange vector field, is locally given by where ) is an integral curve of ξ L , then it satisfies the usual Euler-Lagrange equations

Legendre transformation
Let us recall that the Legendre transformation Indeed, F L is the fiber derivative of L.
In local coordinates, the Legendre transformation is given by Hence, L is regular if and only if F L is a local diffeomorphism.
Along this paper we will assume that F L is in fact a global diffeomorphism (in other words, L is hyperregular), which is the case when L : T Q → R is a Lagrangian of mechanical type, namely

Hamiltonian description
The Hamiltonian counterpart is developed on the cotangent bundle T * Q of Q. Denote by ω Q = dq i ∧ dp i the canonical symplectic form, where (q i , p i ) are the canonical coordinates on T * Q.
The Hamiltonian function is just H = E L • F L −1 and the Hamiltonian vector field is the solution of the symplectic equation Since F L * ω Q = ω L , we deduce that ξ L and X H are F L-related, and consequently F L transforms the solutions of the Euler-Lagrange equations (1) into the solutions of the Hamilton equations ( 2).
On the other hand, we can define a bracket of functions, called the canonical Poisson bracket, The local expression of the Poisson bracket is Remark 1.If (q i ) are local coordinates on Q, {e i } = {e i = e j i ∂/∂q j } is a local basis of vector fields on Q, and {µ i = µ i j dq j } is the dual basis of 1-forms (that is, µ i j e j k = δ i k ), then we van consider the corresponding local coordinates (q i , π i ) on T * Q.In fact, if α q = π i µ i (q) ∈ T * q Q, then (q i , π i ) will be the local coordinates for α q .Under these considerations, we have that where [e i , e j ] = C k ij (q)e k .Here [ , ] denotes the Lie bracket of vector fields (see [12]).♢ The bracket { , } can is a Poisson bracket, that is, { , } can is R-bilinear and: • It is skew-symmetric: {G, F } can = −{F, G} can ; • It satisfies the Leibniz rule: Moreover, the Poisson bracket { , } can may be used to give the evolution of an observable Remark 2. Given a 1-form α on a manifold N , we can define a vertical vector field α V on the cotangent bundle using the formula where ω N is the canonical symplectic form on T * N and π N : The vector field α V is called the vertical lift of α to T * N (see [16,27]).♢ 3 Nonholonomic mechanical systems

The Lagrangian description
A nonholonomic mechanical system is a quadruple (Q, g, V, D) where • Q is the configuration manifold of dimension n; • g is a Riemannian metric on Q; • D is a non-integrable distribution of rank k < n on Q.
As in Subsection 2.2, the metric g and the potential energy V define a Lagrangian function L : T Q → R of mechanical type by for v q ∈ T q Q, q ∈ Q.In bundle coordinates (q i , qi ) we have The nonholonomic dynamics is provided by the Lagrangian L subject to the nonholonomic constraints given by D, which means that the permitted velocities should belong to D. The nonholonomic problem is to solve the equations of motion where {µ A } is a local basis of D • (the annihilator of D) such that µ A = µ A i dq i .Here λ A are Lagrange multipliers to be determined.
A geometric description of the equations above can be obtained using the symplectic form ω L and the vector bundle of 1-forms, F , defined by F = τ * Q (D • ).More specifically, equations (5) are equivalent to These equations have a unique solution, ξ nh , which is called the nonholonomic vector field.
The Riemannian metric g induces a linear isomorphism and also a vector bundle isomorphism over Q The corresponding inverses of the three morphisms ♭ g will be denoted by ♯ g .
We can define the orthogonal complement, D ⊥g , of D with respect to g, as follows: D ⊥g q = {v q ∈ T q Q | g(v q , w q ) = 0, ∀ w q ∈ D} .The set D ⊥g is again a distribution on Q, or, if we prefer, a vector sub-bundle of T Q such that we have the Whitney sum

The Hamiltonian description
We can obtain the Hamiltonian description of the nonholonomic system (Q, g, V, D) using the Legendre transformation which in our case coincides with the isomorphism ♭ g associated to the metric g.Indeed, We can thus define the corresponding: Therefore, we obtain a new orthogonal decomposition (or Whitney sum) This decomposition is orthogonal with respect to the induced metric on tangent covectors, and it is the translation of the decomposition (6) to the Hamiltonian side.Similarly to the Lagrangian framework, M and D • are vector sub-bundles of π Q : and the orthogonal projection γ : The equations of motion for the nonholonomic system on T * Q can now be written as together with the constraint equations Notice that here the λA 's are Lagrange multipliers to be determined.Now the vector bundle of constrained forces generated by the 1-forms τ * Q (µ A ), can be translated to the cotangent side and we obtain the vector bundle generated by the 1-forms Therefore, the nonholonomic Hamilton equations ( 8) can be rewritten in intrinsic form as These equations have a unique solution, X nh , which is called the nonholonomic vector field.The vector fields X nh and ξ nh are related by the Legendre transformation restricted to D, namely,

The skew-symmetric algebroid approach
In [13] (see also [20]) we have developed an approach to nonholonomic mechanics based on the skew-symmetric algebroid setting.We denote by i D : D → T Q the canonical inclusion.The canonical projection given by the decomposition Then, the vector bundle The anchor map is just the canonical inclusion i D : D → T Q, and the skew-symmetric bracket ∥ , ∥ on the space of sections Γ(D) is given by Here, [ , ] is the standard Lie bracket of vector fields.
We also have the vector bundle morphisms provided by the adjoint operators: where D * is the dual vector bundle of D.
We define now an almost Poisson bracket on M as follows (see [13]): Remark 3. Suppose that (q i ) are local coordinates on Q, and that {e i } = {e a , e A } is a local basis of vector fields on Q such that {e a } (resp.{e A }) is a local basis of Γ(D) (resp.Γ(D ⊥g )) with Then, we can consider the dual basis {µ i } = {µ a , µ A } of 1-forms on Q and the corresponding local coordinates )) are local coordinates on D (resp.on D * ).In addition, we have the following simple expressions of i D : D → T Q, P : T Q → D and their dual morphisms Hence, using equations ( 3) and ( 10) we deduce that where [e a , e b ] = C c ab (q)e c + C A ab (q)e A .♢ The bracket { , } D * has the same properties as a Poisson bracket, although it may not satisfy the Jacobi identity, that is, { , • It satisfies the Leibniz rule in each argument: However, one may prove that { , } D * is a Poisson bracket if and only if the distribution D is integrable (see [19]).
Moreover, if is the nonholonomic Legendre transformation given by and Y nh is the nonholonomic dynamics in D * , then the bracket { , } D * may be used to give the evolution of an observable ϕ ∈ C ∞ (D * ).In fact, if h : D * → R is the constrained Hamiltonian function defined by

The nonholonomic bracket
Consider the vector sub-bundle T D M over M defined by As we know [4,6], T D M is a symplectic vector sub-bundle of the symplectic vector bundle where the restriction of ω Q to any fiber of T M (T * Q) is also denoted by ω Q .Thus, we have the following symplectic decomposition where (T D M ) ⊥ω Q denotes the symplectic orthogonal complement of T D M .Therefore, we have associated projections One of the most relevant applications of the above decomposition is that along M , where X nh denotes the nonholonomic vector field defined by ( 9).In addition, the above decomposition allows us to define the so-called nonholonomic bracket as follows.Given f, g ∈ C ∞ (M ), we set where i M : M → T * Q is the canonical inclusion, and f , g are arbitrary extensions to T * Q of f and g, respectively (see [6,21]).Since the decomposition ( 13) is symplectic, one can equivalently write Remark 4. Notice that f • γ and g • γ are natural extensions of f and g to T * Q.Hence, we can also define the above nonholonomic bracket as follows ♢ The bracket { , } nh is an almost Poisson bracket on M .In fact, { , } nh satisfies the Jacobi identity if and only if the distribution D is integrable (see [21,26]).In addition, if then, using the nonholonomic bracket, we can obtain the evolution of an observable Remark 5.If x ∈ M and f ∈ C ∞ (M ) then, using equations ( 16) and ( 17), we deduce that Section 4] for a thorough comparison between this bracket and the one defined by van der Schaft and Maschke in [26].

Eden bracket
Using the orthogonal projector γ : T * Q → M as defined in (7), we can define another almost Poisson bracket on M as follows: This bracket will be called Eden bracket.
Remark 6.Let (q i , π a , π A ) be local coordinates on T * Q as in Remark 3.Then, we have that the constrained submanifold M = (D ⊥g ) • is locally described by Thus, (q i , π a ) are local coordinates on M and the expression of the inclusion i Hence, using equations ( 3) and ( 18), we deduce that Eden bracket is locally characterized by

♢
As the bracket { , } D * , the Eden bracket satisfies all the properties of a Poisson bracket, with the possible exception of the Jacobi identity.Proof.Using that M = (D ⊥g ) • , it is easy to deduce that i M,D * is an isomorphism of vector bundles over the identity of Q.Thus, it remains to be seen that

Comparison of brackets
A direct proof comes from the commutativity of the following diagram: where γ is the orthogonal projector induced by the Riemannian metric, as defined in (7).In fact, given ϕ, ψ ∈ C ∞ (D * ), using equations ( 10) and ( 18), and the following facts we have Remark 7.An alternative proof can be given if we consider adapted bases on D and D ⊥g .Indeed, the local basis {e i } = {e a , e A } of vector fields on Q such that {e a } is a local basis of D and {e A } is a local basis of D ⊥g , defines coordinates (q i , v i ) = (q i , v a , v A ) on T Q as in Remark 3. Therefore, (q i , v a ) and (q i , v A ) define coordinates on D and D ⊥g , respectively.
Analogously, we can consider the dual local basis {µ i } = {µ a , µ A }, and the induced coordinates on D * and T * Q, say (q i , y a ) and (q i , π a , π A ), respectively, as in Remark 3.
In addition, (q i , π a ) are local coordinates on M in such a way that the canonical inclusion i M : M → T * Q is given by i M (q i , π a ) = (q i , π a , 0) .
Therefore, using equations (11) and (20), we obtain that the local expression of i M,D * : M → D * is just the identity i M,D * : (q i , π A ) → (q i , π A ) , and Theorem 1 immediately follows from equations ( 12) and (19).♢ Next, we will prove that the Eden bracket is just the nonholonomic bracket defined in [6,21].
Proposition 2. Given the projection P defined in (14), we have that where γ is the projection defined in (7).
which, using that T γ takes values in T M , implies that T γ(Z) ∈ T D M .
Next, we will prove that q , where (ϵ q ) V βq ∈ T βq (T * Q) is just the vertical lift of ϵ q to T βq (T * Q) defined by (see Remark 2).Indeed, ) is a vertical tangent vector, and hence Z − T γ(Z) = (ϵ q ) V βq , for some 1-form ϵ q ∈ T * q Q, with q ∈ Q.Let X : Q → D be a section of the vector sub-bundle D, and denote by X its associated fiberwise linear function: Then, we have ) , since we are assuming that we are taking tangent vectors at a point (q i , p i ) ∈ M , which implies that or, equivalently, (ϵ q ) V βq ( X) = ϵ q (X(q)) = 0 , for all X ∈ Γ(D).This proves that ϵ q ∈ D • q .Now, we will see that Indeed, if W ∈ T D βq M , then, using standard properties of the canonical symplectic structure ω Q ) (see equation ( 4)), and the fact that (T βq π Q )(W ) ∈ D and ϵ q ∈ D • , we deduce that This proves equation (22) and, thus, Since T γ(Z) ∈ T D M , we have that P(T γ(Z)) = T γ(Z)), which implies that Proposition 3.For any function f ∈ C ∞ (M ) and x ∈ M , we have , where γ is the projection defined in (7).In consequence, for any x ∈ M .
Proof.Let ϵ be a section of the vector bundle D • → Q.Then, we have Using Remark 5 and Propositions 2 and 3, we conclude that Corollary 4. For every x ∈ M , we have This result shows that T γ does not project the Hamiltonian dynamics X H onto the nonholonomic dynamics X nh .However, we can achieve this by modifying the Hamiltonian function using the projector γ, i.e. considering considering X H•γ instead of X H . Proof.Indeed, if x ∈ M then, using Propositions 2 and 3, we have Remark 8.In his paper, Eden writes the dynamics in terms of the constrained variables that he denotes by (q i * , p * i ) = (q i • γ, p i • γ).Then, he computes the Poison brackets of the observables substituting the canonical variables (q i , p i ) by the constrained variables (q i * , p * i ).Indeed, this coincides with computing the Eden brackets of the original observables.This can be seen explicitly in equation (3.4) in [18], where Eden computes the commutation relations of the constrained variables.Indeed, if those are taken as structure constants, they define the Eden bracket.♢ 7 Application to the Hamilton-Jacobi theory

Hamilton-Jacobi theory for standard Hamiltonian systems
Given a Hamiltonian H = H(q i , p i ), the standard formulation of the Hamilton-Jacobi problem is to find a function S(t, q i ), called the principal function, such that If we put S(t, q i ) = W (q i ) − tE, where E is a constant, then W satisfies The function W is called the characteristic function.Equations ( 23) and ( 24) are indistinctly referred as the Hamilton-Jacobi equation.See [1,2] for more details.
Let Q be the configuration manifold, and T * Q its cotangent bundle equipped with the canonical symplectic form ω Q .Let H : T * Q → R be a Hamiltonian function and X H the corresponding Hamiltonian vector field (see Subsection 2.3).
Let λ be a closed 1-form on Q, i.e. dλ = 0 (then, locally λ = dW ).We have that Theorem 6.The following conditions are equivalent: (i) If σ : I → Q satisfies the equation If λ is a closed 1-form on Q, one may define a vector field on Q: The following conditions are equivalent: (i) If σ : I → Q satisfies the equation Moreover, Theorem 6 may be reformulated as follows.
Theorem 7. Let λ be a closed 1-form on Q.Then the following conditions are equivalent: (i) X λ H and X H are λ-related; In that case, λ is called a solution of the Hamilton-Jacobi problem for H.
♢ One may find in the literature (see Theorem 2 in [7]) an extension of Theorem 7 for the more general case in which the 1-form λ is not necessarily closed.Theorem 8. Let λ be a 1-form on Q.Then, the following conditions are equivalent: In Subsection 7.4 (see Theorem 10), we will prove a nonholonomic version of Theorem 8, which will be useful for our interests.

Hamilton-Jacobi theory for nonholonomic mechanical systems
Let H : T * Q → R be a mechanical Hamiltonian function subject to nonholonomic constraints given by a distribution D on Q, as in the previous sections.We will continue using the same notations.Hence, we have the decomposition The vector field X nh ∈ X(M ) will denote the corresponding nonholonomic dynamics in the Hamiltonian side.
Thus, we conclude that X λ nh (q) = ♯ g (λ(q)) , as in the free case (see Remark 9).In particular, since λ(q) ∈ M q = (D ⊥g q ) • , we have that for all q ∈ Q. ♢ Moreover, in [22] the authors proved the following result.(i) X λ nh and X nh are λ-related; In consequence, the Hamilton-Jacobi equation for the nonholonomic system is Notice that dλ ∈ I(D • ) if and only if for all v 1 , v 2 ∈ D (see [13,25]).We can improve the results in the above theorem when the distribution D is completely nonholonomic (or bracket-generating), that is, if D along with all of its iterated Lie brackets [D, D], [D, [D, D]], . . .spans the tangent bundle T Q.
Indeed, using Chow's theorem, one can prove that if Q is a connected differentiable manifold and D is completely nonholonomic, then there is no non-zero exact one-form in the annihilator [13,25]).
On the other hand, we can give a different proof of Theorem 9 using the properties of the the Eden bracket and some general results in [13].A sketch of this proof is the following one.
Using Theorem 1 and the fact that the almost Poisson bracket { , } D * is linear on the vector bundle D * (see [13]), we directly deduce that the Eden bracket { , } E is also linear on the vector subbundle M = (D ⊥g ) • ⊆ T * Q.So, { , } E induces an skew-symmetric algebroid structure on the dual bundle M * = ((D ⊥g ) • ) * (see Theorem 2.3 in [13]).Note that M * may be identified with the vector subbundle D. Indeed, the dual isomorphism just an skew-symmetric algebroid isomorphism when on D we consider the skew-symmetric algebroid structure (∥ , ∥, i D ) induced by the linear almost Poisson bracket { , } D * .This structure (∥ , ∥, i D ) was described at the beginning of Subsection 3.3.Now, using this description, and the general Theorem 4.1 in [13], we directly deduce Theorem 9.

A new formulation of the Hamilton-Jacobi theory for nonholonomic mechanical systems
It is really interesting to express the projection γ in bundle coordinates.We can consider a basis {e a } of Γ(D) and As in the original papers by R. Eden [17,18], we can consider the regular matrix with components C ab = g(e a , e b ) and define E kj = e k a C ab e j b , where C ab are the components of the inverse matrix of (C ab ).Then a direct computation shows that the projector γ defined in (7) can be written as γ(q i , p i ) = (q i , γ j i p j ) , where Notice that γ maps free state phases into constrained state phases, i.e. points in T * Q into points in M .However, γ does not map the free dynamics into the nonholonomic dynamics, i.e. it does not map integral curves of X H into integral curves of X nh .Nevertheless, γ maps the free dynamics of a modified Hamiltonian into the nonholonomic dynamics (see Corollary 4).With the above notations, one can see that equation ( 29) can be locally written as On the other hand, the condition λ(Q) ⊆ M can be locally written as Therefore, the solutions of the Hamilton-Jacobi equation for the nonholonomic system are 1-forms λ ∈ Ω 1 (Q) satisfying the following conditions: or, in bundle coordinates, Observing the above equations, we can notice that if λ is a solution for the unconstrained Hamilton-Jacobi problem (and λ is assumed to be closed), then λ would be a solution for the nonholonomic Hamilton-Jacobi problem if and only if λ takes values in M .

Generalized nonholonomic Hamilton-Jacobi equation
In this section, we will proof a nonholonomic version of Theorem 8. Assume that (Q, g, V, D) is a nonholonomic mechanical system, and let As above, we denote by X nh ∈ X(M ) the nonholonomic dynamics in the Hamiltonian side, and by X λ nh on Q given by so that the following diagram commutes: As we know, for every q ∈ Q (see Remark 10).
Then, the vector fields X λ nh and X nh are λ-related if and only if Equation (33) will be called the generalized nonholonomic Hamilton-Jacobi equation.
Thus, since X λ nh (q) ∈ D q for every q ∈ Q, we deduce that the generalized nonholonomic Hamilton-Jacobi equation (33) may be equivalently written as where d D is the pseudo-differential of the skew-symmetric algebroid D.
We also remark the following facts, on results related with Theorem 10, that one may find in the literature: • In [8], the authors obtain a similar result but in the Lagrangian formulation.
• In [3], the authors discuss the Hamilton-Jacobi equation for nonholonomic mechanical systems subjected to affine nonholonomic constraints but in the skew-symmetric algebroid settting.The appearance of the Hamilton-Jacobi equation in [3] is similar to equation ( 34), but the relevant space in [3] is the affine dual of the constraint affine subbundle (which is different from our constraint vector subbundle M ).

♢
In order to prove Theorem 10, we will need the following lemmas.
Lemma 12.For every q ∈ Q, we have In addition, Proof.It is easy to see that In addition, we have Furthermore, it is easy to see that
Proof.Indeed, using equations ( 4) and (31), we have We can now prove the theorem above.
Taking into account that X λ nh (q) ∈ D q (see Remark 10), it follows that On the other hand, from equation ( 9) and since u q ∈ D q , we obtain that Finally, using Remark 11 and equations (36), ( 37) and (38) we deduce the result.
Following the same notation as in Subsection 7.3, proceeding as in that subsection and using the fact that X λ nh = ♯ g (λ), we deduce that a 1-form λ = λ i dq i ∈ Ω 1 (Q) satisfies the condition λ(Q) ⊆ M and the generalized nonholonomic Hamilton-Jacobi equation (33) if and only if λ i = γ j i λ j , for all i and 8 Examples

The nonholonomic particle
Consider a particle of unit mass be moving in space and subject to the constraint Φ = ż − y ẋ = 0 .
Following the previous notations, this means that the distribution D is generated by the global vector fields Moreover, we have and Passing to the Hamiltonian side, we obtain the Hamiltonian function with constraints given by the function We have an orthogonal decomposition where a simple computation shows that M = ⟨dy, dx + ydz⟩ .
Thus, we can take global coordinates (x, y, z, π 1 , π 2 ) on M , and using equation ( 19) we obtain the following equations for the Eden bracket: the rest of the brackets between the coordinates being zero.
Next, a straightforward calculation shows that and γ ≡ E. Hence, if p x dx + p y dy + p z dz is a 1-form on R 3 then px dx + py dy + pz dz = γ(p x dx + p y dy + p z dz) ∈ Γ(M ), and we have that Let λ ∈ Ω 1 (Q) be a solution of the Hamilton-Jacobi equation (30).The condition γ • λ = λ implies that λ is of the form λ = λ x dx + λ y dy + yλ x dz .
On the other hand, the condition dλ| D×D = 0 holds if and only if dλ(e 1 , e 2 ) = 0 , or, equivalently, These equations coincide with those obtained in Example 6.1 from [15].In particular, if the Hamiltonian is purely kinetical (i.e.V = 0), then a solution for the Hamilton-Jacobi equation is given by for some constants E and µ (see Example 5.3.1 in [9]).

The rolling ball
Consider a sphere of radius r and mass 1 which rolls without sliding on a horizontal plane.The configuration space is where ω = (ω 1 , ω 2 , ω 3 ) denotes the angular velocity of the ball, and I the moment of inertia of the sphere with respect to its center of mass.Assume that the sphere is homogeneous.Then, I = diag(I, I, I).
The ball rotates without sliding, i.e. its subject to the nonholonomic constraints denote the standard basis of right-invariant vector fields on SO(3).Let ρ 1 , ρ 2 , ρ 3 be the right Maurer-Cartan 1-forms, which form a basis of T Then, the constraint 1-forms are Their brackets are given by From the orthogonal basis {e a , e b , e c , e α , e β } and using the Euler angles (θ, φ, ψ) as coordinates of SO(3), we have local coordinates (x, y, θ, φ, ψ, v a , v b , v c , v α , v β ) in T Q, where In these new coordinates, the Lagrangian function is given by Since it is purely kinetical, the Lagrangian energy coincides with the Lagrangian function, namely, E L = L.The constraint submanifold D ⊆ T Q is given by The constrained Lagrangian function is The Legendre transformation and its inverse are given by y, θ, φ, ψ, p a , p b , p c ) → (x, y, θ, φ, ψ, p a , p b , p c , 0, 0) .
Thus, the constrained Hamiltonian function is Let {µ a , µ b , µ c , µ α , µ β } denote the dual basis of {e a , e b , e c , e α , e β } .Then, and its inverse is (39) On the other hand, we have that X λ nh (q) = ♯ g (λ(q)) , for each q ∈ Q, where ♯ g : T * q Q → T q Q denotes the isomorphism defined by the Riemannian metric g.Hence, It is worth remarking that from this solution one can obtain 3 independent first integrals of the nonholonomic dynamics.Indeed, the map ψ : Q × R 3 → M given by ψ : (q, c a , c b , c c ) → c a µ a (q) + c b µ b (q) + c c µ c (q) is a global trivialization of M .Its inverse is given by ψ −1 : M ∋ (q, p a , p b , p c ) → (q, p a , p b , p c ) ∈ Q × R 3 .
Define the functions f a , f b , f c : M → R given by f a = p a , f b = p b , f c = p c .
Using equations ( 3) and (4), we have and thus f a , f b and f c are first integrals of the nonholonomic dynamics.Translating these first integrals to the Lagrangian formalism, we obtain that the functions are first integrals for the nonholonomic Lagrangian dynamics.Hence, v a , v b and v c are also first integrals for the nonholonomic Lagrangian dynamics.Therefore, ω 1 , ω 2 and ω 3 are first integrals as well.As a matter of fact, they coincide with the first integrals obtained in [24, pp. 194-198].

Conclusions and future work
We have presented the concept of the Eden bracket and contrasted it with other nonholonomic brackets.It is noteworthy that there exist almost Poisson isomorphisms among the three nonholonomic mechanics formulations.Hence, one can make use of the formulation that is more convenient for each problem, and translate it to the other formulations via these isomorphisms.The use of this new description of the nonholonomic bracket following Eden's ideas opens up many possibilities to simplify some developments in nonholonomic mechanics, including the following: • We are going to study the quantization of nonholonomic systems [10].More specifically, following the original ideas by Eden [18], we plan to study what is the quantum counterpart of a mechanical system with nonholonomic constraints.
• We would also like to discuss the connection between complete solution of the generalized nonholonomic Hamilton-Jacobi equation, complete systems of first integrals of the nonholonomic system and symmetries of the system.In addition, the Eden bracket could be used to study of the reduction by symmetries and define a new version of the nonholonomic momentum map.
• Moreover, we plan to construct a discrete version of the operator γ in order to develop geometric integrators for nonholonomic mechanical systems.
comments on first integrals of the rolling ball.All the authors would like to express their appreciation to the referee for their valuable feedback and constructive comments, which greatly improved the clarity of this paper.

First
of all, we shall prove that the almost Poisson brackets defined on D * and M are isomorphic.Theorem 1.The vector bundle isomorphism i M,D * : M → D * over the identity of Q, given by the composition i M,D * = i * D • i M , is an almost Poisson isomorphism between the almost Poisson manifolds (M, { , } E ) and (D * , { , } D * ) .

Theorem 9 .
Let λ be a 1-form on Q taking values into M and satisfying dλ ∈ I(D • ), where I(D • ) denotes the ideal defined by D • .Then the following conditions are equivalent:

r 2 I
+ mr 2 p b , p c I .Let us now look for a solution λ ∈ Ω 1 (Q) for the generalized nonholonomic Hamilton-Jacobi equation (34).The condition λ(Q) ⊆ M implies that λ is of the formλ = λ a µ a + λ b µ b + λ c µ c ,for some functions λ a , λ b , λ c : Q → R.Then, λ is a solution of the generalized nonholonomic Hamilton-Jacobi equation if and only ifd D (H • λ) + i X λ nh d D λ = 0, where d D denotes the pseudo-differential of the skew-symmetric algebroid D. For simplicity's sake, assume that d D (H • λ) = 0 and i X λ nh d D λ = 0. We have thatd D (H • λ) = r 2 λ a I + mr 2 dλ a + r 2 λ b I + mr 2 dλ b + λ c I dλ c , which vanishes if λ a = ca , λ b = c b and λ c = c c for some constants c a , c b , c c ∈ R.Then, we have that dλ (e a , e b ) = −λ [e a , e b ] = c c r 2 , dλ (e a , e c ) = −λ [e a , e c ] = − Ic b I + mr 2 , dλ (e b , e c ) = −λ [e b , e c ] = Ic a I + mr 2 , and thus d D λ = c c r 2 µ a ∧ µ b − Ic b I + mr 2 µ a ∧ µ c + Ic a I + mr 2 µ b ∧ µ c .

X λ nh = c a r 2 I
+ mr 2 e a + c b r 2 I + mr 2 e b + c c I e c .(40)Making use of the local expressions (39) and (40), we obtain that i X λ nh d D λ = 0 , and conclude that λ is a solution of the generalized nonholonomic Hamilton-Jacobi equation.