The ubiquity of the symplectic hamiltonian equations in mechanics

In this paper, we derive a"hamiltonian formalism"for a wide class of mechanical systems, including classical hamiltonian systems, nonholonomic systems, some classes of servomechanism... This construction strongly relies in the geometry characterizing the different systems. In particular, we obtain that the class of the so-called algebroids covers a great variety of mechanical systems. Finally, as the main result, a hamiltonian symplectic realization of systems defined on algebroids is obtained.


Introduction
One of the most important equations in Mathematics and Physics are certainly the Hamilton where (q i , p i ) are canonical coordinates. Symplectic geometry allows us to write these equations in an intrinsic way (see [1,24]), (1.1) In equation (1.1), H : T * Q → R represents a Hamiltonian function defined on the cotangent bundle T * Q of a configuration manifold Q and ω Q is the canonical symplectic form of the cotangent bundle (in canonical coordinates, ω Q = dq i ∧ dp i ). The skew-symmetry of the canonical symplectic form leads to conservative properties for the Hamiltonian vector field X H (preservation of the energy). On the contrary, in other type of systems this conservative behavior is not required. For instance, from the symmetry of a riemmanian metric it follows dissipative properties for the gradient vector field (see [11]).
Our approximation adopts a new point of view. First, it is necessary to understand the underlying geometry of Equation (1.1) which will permit us to conclude that Hamilton's equations have an ubiquity property: many different mechanical systems can be described by a symplectic equation precisely, ω Q = −dλ Q (see [1,24] for details). In symplectic geometry terms we say that (T * Q, ω Q ) is an exact symplectic manifold. This structure induces a linear Poisson tensor field Π T * Q on T * Q defined by where X F and X G are the hamiltonian vector fields corresponding to the functions F : T * Q → R and G : T * Q → R, respectively. In canonical coordinates A trivial, but interesting, comment is that the classical bracket of vector fields (the standard Lie bracket) is induced by the linear Poisson tensor Π T * Q (and viceversa). In fact, there exists a one-toone correspondence between the space of vector fields on Q and the space of linear functions on T * Q.
Indeed, for each vector field X ∈ X(Q) the corresponding linear function X : T * Q → R is given by where , is the natural pairing between vectors and covectors, and κ q ∈ T * q Q. Therefore, for X, Y ∈ X(Q), the bracket of the two vector fields X and Y is characterized as the unique vector field associated to the linear function −Π T * Q (dX, dŶ ). Observe that in coordinates In a schematic way we have linear Poisson tensor Π T * Q ←→ standard Lie bracket on Q.
That is, we have that there exists a one-to-one correspondence between linear Poisson tensors on T * Q and Lie algebra structures of vector fields on Q (see [12]).
As a preliminary conclusion, classical hamiltonian formulations strongly relies on the standard Lie bracket of vector fields (or equivalently, the linear Poisson tensor on T * Q). Modifications of this bracket (or the associated linear tensor) will presumably change the properties of the dynamics. For example, if we do not impose the skew-symmetry of the bracket we will have a dissipative behavior, since, in the cotangent bundle, we will obtain a 2-contravariant linear tensor field which is not necessarily skew-symmetric [11]. Another property that it is possible to drop is the Jacobi identity, which is related with the preservation of the symplectic form by the flow of the hamiltonian vector field. In many interesting cases, as for instance nonholonomic mechanics (see [9,10,14]), it is well known that the Jacobi identity is equivalent to the integrability of the constraints, that is, to holonomic mechanics. Since our objective is to obtain a geometric framework including all these cases, it is necessary to work without imposing Jacobi identity, from the beginning, to our tensor field or associated bracket.
Moreover, the role of the tangent bundle is not essential, and we may change it for an arbitrary vector bundle, and then the linear contravariant tensor field will be now defined on its dual bundle E * (see [38]).
We will show that the category of algebroids is general enough to cover all the cases that we want to analyze. An algebroid (see [15,18,19,21,31]) is, roughly speaking, a vector bundle τ E : E → Q, equipped with a bilinear bracket of sections B E : Γ(τ E ) × Γ(τ E ) → Γ(τ E ) and two vector bundle morphisms ρ l E : E → T Q and ρ r E : E → T Q satisfying a Leibniz-type property (see (2.1)). Observe that properties like skew-symmetry or Jacobi identity are not considered in this category. This general structure is equivalent to give a linear 2-contravariant tensor field Π E * on its dual bundle E * . In conclusion, we have that linear contravariant tensor field Π E * ←→ algebroid structure on E .
The main objective of this paper is to show that the general construction of the hamiltonian symplectic formalism in classical mechanics remains essentially unchanged starting from the more

Algebroids and Hamiltonian Mechanics
Let τ E : E → Q be a real vector bundle over a manifold Q and Γ(τ E ) be the space of sections of Definition 2.
1. An algebroid structure on E is a R-bilinear bracket together with two vector bundles morphisms ρ l E , ρ r E : E → T Q (left and right anchors) such that for f, f ′ ∈ C ∞ (Q) and σ, σ ′ ∈ Γ(τ E ). The algebroid structure on τ E : E → Q was defined in [15,18,19] and it is called a Leibniz algebroid in [21,31].
If the R-bilinear bracket B E is skew-symmetric we have a skew-symmetric algebroid structure [15] (an almost Lie algebroid structure in the terminology of [23] or an almost-Lie structure in the terminology of [32]). In such a case, the left anchor coincides with the right anchor: ρ l E = ρ r E . In the sequel, we will denote the bracket of sections in this skew-symmetric case by [ is a Lie algebroid structure on the vector bundle τ E : E → Q (see [28]).
Another interesting case is when the R-bilinear bracket B E is symmetric, then we have a symmetric algebroid structure. In such a case, ρ l E = −ρ r E . Now, note that there exists a one-to-one correspondence between the space Γ(τ E ) of sections of the vector bundle τ E : E → Q and the space of linear functions on E * . In fact, if σ ∈ Γ(τ E ) then the corresponding linear function σ on E * is given by where τ E * : E * → Q is the vector bundle projection.
In the particular case when E is a skew-symmetric algebroid it follows that Π E * is a linear 2-vector on E * (or an almost Poisson structure on E * in the terminology of [23]). If E is a Lie algebroid, the bracket {·, ·} Π E * satisfies the Jacobi identity and Π E * is a Poisson structure on E * (see [12,23,27,38] The integral curves of the vector field H Π E * H are the solutions of the Hamilton equations for H.
Next, we will obtain some local expressions.
Suppose that (q i ) are local coordinates on Q and that {σ α } is a local basis of the space Γ The local functions (B E ) γ αβ , (ρ l E ) i α and (ρ r E ) j β are called the local structure functions of algebroid τ E : E → Q.
Denote by (q i , p α ) the induced local coordinates on E * . Then using (2.2) and (2.3), it follows that Therefore, the Hamiltonian vector field of H is given by which implies that the local expression of the Hamilton equations is Remark 2.2. Working with not in general skew-symmetric or symmetric tensors it is possible to distinguish two different dynamics (one on the left and one on the right). Therefore, we can obtain also a different Hamiltonian vector field H Π E * H of H with respect to Π E * : In coordinates, In the sequel we only consider the vector field H Π E * H since the analysis for H Π E * H is similar.
2.1. First example. A symmetric case: Gradient extension of a dynamical system. Let Q be an n-dimensional manifold. Let G be a riemannian metric on Q, i.e, a positive-definite symmetric (0, 2)-tensor on Q. Associated to G we have the associated musical isomorphisms where X, Y ∈ X(Q) and µ ∈ Λ 1 (Q). In coordinates (q i ) on Q, the metric is expressed as G = Fixed a function f ∈ C ∞ (M ), it is defined the gradient vector field associated to f as grad G (f ) = ♯ G (df ). In coordinates, Associated with the metric G there is an affine connection ∇ G , called the Levi-Civita connection Consider now the symmetric product: Locally, It is well known that the symmetric product is an element crucial in the study of various aspects of mechanical control systems such us controllability, motion planning, (see for example [3]) and also characterize when a distribution is geodesically invariant [26]. Now define the left and right anchors by, This structure induces a linear tensor Π T * Q of type (2,0) on T * Q. In local coordinates (q i , p i ) on T * Q, the bracket relations induced by this tensor field are: Given an arbitrary vector field X ∈ X(Q) one may define the function H X : The equations for its integral curves are: These equations are the gradient extension of the nonlinear equationq i = X i (q) (see [11] with G : E × Q E → R a bundle metric on E and D the total space of a vector subbundle τ D : D → Q of E such that rankD = m. The vector subbundle D is said to be the constraint subbundle.
Denote by i D : D → E the canonical inclusion and consider the orthogonal decomposition E = D ⊕ D ⊥ and the associated orthogonal projectors P : defined in a similar way than in (2.5) (see [10]). It is determined by the formula: for σ, σ ′ , σ ′′ ∈ Γ(τ E ). The solutions of the nonholonomic problem are the ρ E -admissible curves γ : I −→ D such that (see [9,10]) Now, we will derive the equations of motion (2.8) using the general procedure introduced in Section 2. First, we define on the vector bundle τ D : D −→ Q the following skew-symmetric algebroid structure: This skew-symmetric algebroid induces a linear almost-Poisson tensor field Π D * on the dual bundle D * . In [23], it is shown that this structure is also induced from the linear Poisson bracket {·, ·} Π E * on E * :  If we denote by (q i , p a ) the induced local coordinates on D * , then { , } Π D * is determined by the following bracket relations: Next, we consider the constrained Hamiltonian function H D * on D * given by Therefore, In the induced local coordinates: and the Hamilton equations are In other words, Thus, it is easy to prove that On the other hand, from (2.8), it follows that a curve is a solution of the nonholonomic problem if and only if Therefore, we deduce that a curve γ : In this section we will discuss Lagrangian systems on a Lie algebroid τ E : E → Q subjected to generalized nonholonomic constraints (see [2,6]).
As in the classical nonholonomic case, the kinematic constraints are described by a vector subbundle Therefore, to determine the dynamics it is only necessary to fix a bundle of reaction forces which vanishes on a vector subbundleD of E. We have that, in general, D =D.
The case D =D, classical nonholonomic mechanics, was studied in Subsection 2.2. The case D =D appears in many interesting problems, for instance when the restriction is realized by the action of a servo mechanism [29] or for rolling tyres as in [6].
We will call τD :D → Q the variational subbundle. It is important to note thatD can not be deduced from the kinematic constraints, as the virtual displacements are in classical nonholonomic mechanics. Now, let G : E × Q E → R be a bundle metric on the vector bundle τ E : E → Q and V : Q → R ∈ C ∞ (Q). If rankD = rankD and L : E → R is the Lagrangian function of mechanical type given by then the triple (L, D,D) will be called a mechanical system subject to generalized linear nonholonomic constraints on E.
We will assume thatD satisfies the compatibility condition withD ⊥ q the orthogonal complement ofD q in E q with respect to scalar product G q . It is obvious that, in particular, this property holds in the classical non-holonomic caseD = D. In the general case, we have that the equations of motion of such a system are given by δL γ(t) ∈D 0 τD(γ(t)) , for a ρ E -admissible curve γ : I → D, or equivalently, In other words, we have a local basis of sections adapted to the decomposition E = D ⊕D ⊥ . We will (2. 16) where the functions C c ab , (ρ l D ) i b and (ρ r D ) i b are properly deduced in Appendix B. Based on the compatibility condition, it seems natural to consider some decompositions of the original vector bundle E. In particular, we will use Next, denote by ([[·, ·]] E , ρ E ) the Lie algebroid structure on the vector bundle τ E : E → Q. Then the bracket on D given by and the anchors maps On the other hand, if we denote by (q i , p a ) the corresponding local coordinates on D * then the linear bracket { , } Π D * on D * is determined by the following relations: Now, we take the Hamiltonian function H E * : E * → R (the Hamiltonian energy) defined by Then, the constrained Hamiltonian function H D * : D * → R is given by Thus, which implies that the local expression of the Hamilton equations is In the induced local coordinates: and the Hamilton equations are

2.3.2.
Lagrangian mechanics for modifications of the standard Lie bracket. Now, we analyze another example that is of our interest since it has a non skew-symmetric bracket.
Consider the case of the standard tangent bundle τ T Q : T Q → Q, a Lagrangian L of mechanical type and an arbitrary (1, 2)-tensor field T : It is easy to show that if we modify the standard Lie bracket [·, ·] on X(Q) by is an algebroid.
If we take local coordinates (q i ) and Thus, the linear bracket {·, ·} Π T * Q on T * Q is characterized by the following relations In this case, the Hamiltonian function H T * Q : T * Q → R is given by and then the associated Hamilton equations are Note that {H T * Q , H T * Q } Π T * Q = −T k ij G il G jm p k p l p m and the dynamics has in general a dissipative behavior. An interesting case, is when the tensor field T is skew-symmetric (T (X, Y ) = −T (Y, X) for all X, Y ∈ X(Q)) then the hamiltonian H T * Q is preserved, dH T * Q /dt = 0 along the flow. An important example, when this condition is fulfilled, is the following one. Consider, as above, a riemannian manifold (Q, G) and an arbitrary affine connection ∇. Take the (1, 2) tensor field S which encodes the difference between it and the Levi-Civita connection corresponding to the riemannian metric, that is, This tensor field is called the contorsion tensor field (see [34]; and also [4,8]). Now, consider as T (X, Y ) = S(X, Y ) − S(Y, X), and the bracket of vector fields: We obtain (2.23) but now the flow preserves the Hamiltonian function. Equations (2.23) are important in the modellization of generalized Chaplygin systems ( [4,8] and references therein), where now the connection ∇ is a metric connection with torsion.

Exact symplectic algebroids and Hamiltonian vector fields
In this section we will introduce the notion of an exact symplectic algebroid and we will prove that for such an algebroid τĒ :Ē →Q, if F is a real C ∞ -function on the base manifoldQ, then F induces a Hamiltonian vector field onQ.
First, we will give the definitions of the two differentials of a real C ∞ -function F on the base manifoldQ of an arbitrary vector bundle τĒ :Ē →Q with an algebroid structure (BĒ, ρ lĒ , ρ rĒ ). We will also give the definition of the differential of a section of the vector bundle τĒ * :Ē * →Q.
In fact, the left differential d lĒ F of F is given by for σ ∈ Γ(τĒ).

Remark 4.4.
In the particular case when E is an skew-symmetric algebroid (that is, the bracket B E is skew-symmetric), the exact symplectic structure Ω T E l E * was considered by Popescu et al [33] in order to develop a symplectic description of the Hamiltonian mechanics on skew-symmetric algebroids.
In this case, the exact symplectic structure Ω T E l E * does not depend on the chosen connection.
To do this, consider an arbitrary bundle metric G : E × Q E → R on E and denote by ∇ G : On the other hand the curvature of the connection ∇ G is the tensor field of type (1,3) on τ E : Using (A.2) and the fact that ρ E is a Lie algebra morphism, it is easy to prove that By a similar argument, using (A.2) and (A.5), we have that Therefore, if we replace in (4.5) the tensor R by the curvature R ∇ G of the connection ∇ G then we obtain that Consequently, if σ (respectively, σ ′ ) is a section of τ E : E → Q and X (respectively, X ′ ) is a vector field on E * which is τ E * -projectable on ρ E (σ) (respectively, ρ E (σ ′ )) then (σ, X) and (σ ′ , X ′ ) are sections This implies that B T E E * is the canonical Lie bracket on Γ(τ T E E * ) (see [25]).
In Section 5 we will use the following properties of the curvature of the connection ∇ G : and Note that (4.12) follows using (4.9) and the fact that [[·, ·]] E satisfies the Jacobi identity.

Remark 4.5.
A situation which will be useful in the examples is the case when we start with a vector bundle τ E : E → Q with a skew-symmetric algebroid structure (B E , ρ E ) such that the anchor map ρ E : E → T Q is a skew-symmetric algebroid morphism, that is, Observe that this condition does not imply that E → Q is a Lie algebroid as in the previous remark.
Under this weaker condition it is still possible to choose the tensor R in such a way the bracket defined in equation (4.5) is again the usual bracket defined in (4.10).

Proof. [Theorem 4.3]
Suppose that (q i ) are local coordinates on Q, {σ α } is a local basis of Γ(τ E ) and that (B E ) γ αβ , (ρ l E ) i α and (ρ r E ) i α are the local structure functions of E with respect to the local coordinates (q i ) and to the basis {σ α }. Then, from Proposition 4.1, it is clear that Moreover, if (q i , p α ) are the corresponding local coordinates on E * , we have that (see (A.4) and Now, let H : E * −→ R be a hamiltonian function. From (3.2), (4.4) and (4.13), it follows that Therefore, from (3.4) and (4.8), we obtain that the right Hamiltonian section H Using (4.4) and (4.13), the right Hamiltonian section yields the left-right Hamiltonian vector field of H which is (4.14) Consequently, from (2.4) and (4.14), we deduce that is a skew-symmetric tensor of type (0,3) (respectively, a symmetric tensor of type (0,3)). Now, we may extend the definition of the differential to any (0,2)-tensor in Γ(τ ⊗ 0 2Ē * ). If T is a section of the vector bundle τ ⊗ 0 2Ē * : ⊗ 0 2Ē * →Q then the differential of T is the section d AS E T of the vector bundle τ ⊗ 0 σ)) and T S (σ,σ) = 1 2 (T (σ,σ) + T (σ, σ)), the canonical inclusion, equipped with an algebroid structure (B D , ρ l D , ρ r D ) and an arbitrary vector bundle morphism F : E → D. Then, we may construct the (1, 3)-tensor field: for all σ,σ,σ ∈ Γ(D). It follows that R D satisfies both conditions (5.2) and (5.3). Observe, for instance, that it is precisely the case of nonholonomic mechanics discussed in Subsection 2.2, where now F is the orthogonal projector P .
Note that lĒ * ), with Ψ S and µ S symmetric tensors and ∨ being the symmetric product. On the other hand, if α, β ∈ Γ(τĒ * ) we have that and thus

Remark 5.3.
(1) The skew-symmetric differential was defined in [23] as the almost differential on an almost Lie algebroid. Note that (d Ā E ) 2 = 0 if and only if (B Ā E , ρ Ā E ) is a Lie algebroid structure on τĒ :Ē → Q.
(2) LetĒ be the tangent bundle of the manifold Q and ∇ be a linear connection on Q. Then, Moreover, the corresponding symmetric differential d S T Q was considered in [20]. In fact, in [20] using the symmetric differential and the symmetric Lie derivative, the derivations of the algebra of symmetric tensors are classified and the Frölicher-Nijenhuis bracket for vector valued In this case, we have a Riemannian manifold (Q, G) and the vector bundle τ T Q : T Q → Q endowed with the symmetric product The anchor maps are ρ l T Q = id T Q and ρ r T Q = −id T Q . Thus, where D l (respectively, D r ) is the ρ l T Q -connection (respectively, the ρ r T Q -connection) defined by Moreover, it is easy to prove that the T Q-tangent bundle to T * Q, T T Q l T * Q, may be identified with the vector bundle τ T (T * Q) : T (T * Q) → T * Q. Under this identification, we have that (see (A.4)) where (q i , p i ) are fibred coordinates on T * Q and Γ k ij are the Christoffel symbols of the Levi-Civita connection ∇ G . Therefore, using (4.8), we deduce that the exact symplectic structure Ω T (T * Q) is just the canonical symplectic structure of T * Q Ω T (T * Q) = dq i ∧ dp i (6.2) (note that Γ k ij = Γ k ji ). On the other hand, from (4.4) and (6.1), it follows that which implies that the right-Hamiltonian section of H is the vector field on T * Q given by Thus, if we apply the above construction to the Hamiltonian function H = H X =X, with X ∈ X(Q), we reobtain the Hamilton equations (2.6).
Consider (g, [·, ·] g ) a Lie algebra of finite dimension. In this case, the Lie bracket is the Lie algebra structure [·, ·] g and the anchor map is the null map.
Consider now a nonholonomic mechanical system on g, that is a vector subspace d ⊂ g of kinematic constraints (d is not, in general, a Lie subalgebra) and a lagrangian function L : g → R of mechanical type induced by a scalar product G on g. As we did in Example 2.2 we assert that (d, [·, ·] d , 0) is a skew-symmetric algebroid with the bracket given by [ξ, η] In what follows we are going to use the formalism in T d d * proposed in Section 4 to find an exact symplectic form and the corresponding Hamilton equations.
Let us consider a basis {ξ a } of d and {ξ a } the dual basis of d * .
In this case, we choose the 0-connection D = D l = D r to be D ξa ξ b = 1 2 [ξ a , ξ b ] d and thus Γ c ab = 1 2 c c ab where c c ab are the structure constants of the skew-symmetric algebroid (d, [·, ·] d , 0). Now, it is easy to prove that T d d * may be identified with d * × d × d * and, under this identification, the vector bundle projection is just the canonical projection on the first factor Since d satisfies the hypotheses of Remark 4.5, it is easy to see that a suitable structure of skewsymmetric algebroid on T d d * ≃ d * × d × d * → d * is determined by the following relations: A straightforward computation shows that Now, we consider the basis {E a ,Ẽ a } of Γ(τ T d d * ) defined as E a = (·, ξ a , 0) such that E a (κ) = (κ, ξ a , 0) E a = (·, 0, ξ a ) such thatẼ a (κ) = (κ, 0, ξ a ).
Then, we obtain and the anchor map is Thus we conclude that Moreover, Therefore, we have that the unique solution of Equation Now, using the anchor map we obtain that the corresponding Hamiltonian vector field on d * : ).
Its integral curves are precisely the nonholonomic Lie-Poisson equations (see [9] and references therein)ṗ that is, using a classical notation,κ for ξ, η ∈ d and κ ∈ d * . Note that if d = g then ad d * = ad * is the infinitesimal coadjoint representation. As in the previous example, consider a Lie algebra (g, [·, ·] g ) of finite dimension, a subspace d ⊂ g and a lagrangian L : g → R of mechanical type induced by a scalar product G on g. Since we are considering a generalized nonholonomic system, d is endowed with an algebroid structure (d, B d , 0, 0) given by (2.18), (2.19) and (2.20), (in this case, the anchors are zero but the bracket is not necessarily skew-symmetric). In fact, for σ, σ ′ ∈ d and P : g = d ⊕ d ⊥ → d ⊆ g, Π : g =d ⊕ d ⊥ →d ⊆ g the corresponding projectors.
As in the previous example, the space T d d * may be identifiaed with the product d * × d × d * and, under this identification, the vector bundle projection is the canonical projection on the first factor Then, we obtain that a suitable bracket on Γ(T d d * ) has the following form The anchor maps, in this case, are In this paper, we suppose that the constraints (kinematic or variational) are linear. It would be interesting to discuss the more general case when the constraints are not linear and, more precisely, the case of affine constraints.
Another goal we have proposed is to develop a Klein formalism for Lagrangian systems on algebroids.
Finally, a different aspect on which we intend to work is a Hamilton-Jacobi theory for Hamiltonian systems on algebroids.
Remark A.3. Every vector bundle τ E : E → Q admits a ρ E -connection. In fact, let ∇ : be an standard linear connection on τ E : E → Q. Then, if we define the map it is easy to prove that D is a ρ E -connection.
Let D be a ρ E -connection on the anchored vector bundle (τ E : E → Q, ρ E ). If (q i ) are local coordinates on Q and {σ α } is a local basis of Γ(τ E ) then D γ αβ are the Christoffel symbols of the connection D with respect to the local basis {σ α }. Now, suppose that σ q is an element of the fiber E q , with q ∈ Q. Then, we may introduce the R-linear map D σq : Γ(τ E ) → E q given by where σ ∈ Γ(τ E ) and σ(q) = σ q . Note that, using (A.1), one deduces that the map D σq is well defined.
Thus, if κ q ∈ E * q , we may consider the linear map Dh κq is the tangent vector to E * at κ q which is characterized by the following conditions for f ∈ C ∞ (Q) and γ ∈ Γ(τ E ). Here, γ : E * → R is the linear function on E * induced by the section γ.
In particular, if σ ∈ Γ(τ E ) we may define the D-horizontal lift to E * as the vector field σ Dh on E * given by It is clear that for σ, σ ′ ∈ Γ(τ E ) and f ∈ C ∞ (Q).
Moreover, if (q i ) are local coordinates on Q and {σ α } is a local basis of Γ(τ E ), then we have the corresponding local coordinates (q i , p α ) on E * and (for more details, see [5]).
On the other hand, if κ ′ q ∈ E * q we may consider the standard vertical lift as the linear map being the tangent vector to E * at κ ′ q which is characterized by the following conditions for f ∈ C ∞ (Q) and γ ∈ Γ(τ E ).
Thus, if κ ∈ Γ(τ E * ) is a section of the dual vector bundle τ E * : E * → Q then the vertical lift to E * is the vector field κ v on E * given by It is clear that for κ, κ ′ ∈ Γ(τ E * ) and f ∈ C ∞ (Q).
Moreover, if (q i ) are local coordinates on Q, {σ α } is a local basis of Γ(τ E ), {σ α } is the dual basis of Γ(τ E * ) and (q i , p α ) the corresponding local coordinates on E * then is not, in general, a local basis of vector fields on E * . Note that ρ E is not, in general, an epimorphism.
Remark A.5. The ρ E -connection D induces a ρ E -connection D * on the dual vector bundle τ E * : for σ, γ ∈ Γ(τ E ) and κ ∈ Γ(τ E * ). If {σ α } is a local basis of Γ(τ E ) and {σ α } is the dual basis of Γ(τ E * ) then D * σα σ γ = −D γ αβ σ β , where D γ αβ are the Christoffel symbols of the connection D. Therefore, if σ ∈ Γ(τ E ) it is possible to consider the corresponding D * -horizontal lift to E as a vector field σ D * h on E.
The above results are a generalization of some lifting operations previously defined in [35,36,37] for the case E = T Q and ρ E = ρ T Q = id T Q .

Appendix B: generalized nonholonomic systems
Let us consider a vector bundle τ E : E → Q with a Lie algebroid structure ([[·, ·]] E , ρ E ). A linear generalized nonholonomic system on E is a mechanical system determined by a regular lagrangian function L : E → R and two distributions, the kinematic constraints described by a vector subbundle τ D : D → Q and the variational constraints given by the vector subbundle τD :D → Q. As we have explained in section 2.3.1, the distributionD is the subspace where the constraint forces are doing null work. It is clear that in the (classical) nonholonomic systems D =D. Generalized nonholonomic systems were studied in [2,6,7,29].
We will assume that the lagrangian is of mechanical type, that is, we have a bundle metric G on E and a real C ∞ -function V : Q → R such that L(e) = 1 2 G(e, e) − V (τ E (e)), for e ∈ E.
Moreover, we will assume that the following compatibility condition holds E = D ⊕D ⊥ whereD ⊥ is the orthogonal complement of the variational distributionD with respect to the bundle metric G.
Suppose that (q i ) are local coordinates on an open subset U of Q and that {σ α } = {σ a , σ A } is a basis of sections of the vector bundle τ −1 E (U ) → U adapted to the decomposition E = D⊕D ⊥ . We will denote by (q i , v α ) = (q i , v a , v A ) the corresponding local coordinates on E. We will assume that the bundle metric G can be locally written as G = G αβ σ α ⊗ σ β . We will also assume that σ a (respectively, σ A ) is an orthonormal basis of Γ(τ D ) (respectively, Γ(τD⊥ )). Thus, we have that G ab = δ b a (respectively G AB = δ B A ) and it is easy to see thatD = span{σ d − G dA σ A }. Then the system (B.1) can be written A straightforward computation shows that the system (B.2) is equivalent to where Γ c ab are the Christoffel symbols of the Levi-Civita connection in τ E : E → Q and with G αβ the inverse matrix of G αβ (note that G eC = −G ef G f C and that G ef + G eC G Cf = δ e f ). Now we can write these symbols Γ c ab in terms of the local structure functions of the Lie algebroid τ E : E → Q using the expression where [α, β; γ] = ∂G αβ ∂q i (ρ E ) i γ + C µ αβ G µγ (see [9,10]). Then, since G cA = −G cd G dA , it is easy to prove that Thus, if we denote byC a bc the real function given bỹ Proof. In the local basis {σ a , σ A } adapted to the decomposition D ⊕D ⊥ we have that P (σ a ) = σ a and P (σ A ) = G cA σ c .
Then it is simple to prove that P • Π |D = id D and from this it is obtained that the bracket and the anchor maps given above define an algebroid structure on τ D : D → Q.
On the other hand, +G cd (C a bd + G aA C A bd − G dA C a bA − G dA G aB C B bA )σ a . Now, using that G cd G dA G aA = G ca + δ c a , it follows that ∂G ca ∂q i = ∂G cd ∂q i G dA G aA + G cd ∂G dA ∂q i G aA + G cd G dA ∂G aA ∂q i which implies that (see (B.5)) B D (σ b , σ c ) =C a bc σ a .