Non-holonomic Lagrangian systems on Lie algebroids

This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. The proposed novel formalism provides new insights into the geometry of nonholonomic systems, and allows us to treat in a unified way a variety of situations, including systems with symmetry, morphisms and reduction, and nonlinearly constrained systems. Various examples illustrate the results.


Introduction
The category of Lie algebroids has proved useful to formulate problems in applied mathematics, algebraic topology, and differential geometry. In the context of Mechanics, an ambitious program was proposed in [68] in order to develop formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and discrete mechanics on Lie groupoids. In the last years, this program has been actively developed by many authors, and as a result, a powerful mathematical structure is emerging.
The main feature of the Lie algebroid framework is its inclusive nature. Under the same umbrella, one can consider such disparate situations as systems with symmetry, systems evolving on semidirect products, Lagrangian and Hamiltonian systems on Lie algebras, and field theory equations (see [19,43] for recent topical reviews illustrating this). The Lie algebroid approach to Mechanics builds on the particular structure of the tangent bundle to develop a geometric treatment of Lagrangian systems parallel to Klein's formalism [22,36]. At the same time, the attention devoted to Lie algebroids from a purely geometrical viewpoint has led to an spectacular development of the field, e.g., see [5,13,48,60] and references therein. The merging of both perspectives has already provided mutual benefit, and will undoubtedly lead to important developments in the future.
The other main theme of this paper are nonholonomic Lagrangian systems, i.e., systems subject to constraints involving the velocities. This topic is a classic subject in Mathematics and Mechanics, dating back to the early times of Lagrange; a comprehensive list of classical references can be found in [57]. At the beginning of the nineties, the work [37] sparked a renewed interest in the geometric study of nonholonomic mechanical systems, with a special emphasis on symmetry aspects. In the last years, several authors have extended the ideas and techniques of the geometrical treatment of unconstrained systems to the study of nonholonomic mechanical systems, see the recent monographs [3,17]. These include symplectic [12,40,41], Hamiltonian [65], and Lagrangian approaches [23,38], the study of almost Poisson brackets [10,33,39], and symmetry and reduction of the dynamics [2,4,7,8,9,18,47].
In this paper we develop a comprehensive treatment of nonholonomic systems on Lie algebroids. This class of systems was introduced in [20] when studying mechanical control systems (see also [56] for a recent approach to mechanical systems on Lie algebroids subject to linear constraints). Here, we build on the geometry of Lie algebroids to identify suitable regularity conditions guaranteeing that the nonholonomic system admits a unique solution. We develop a projection procedures to obtain the constrained dynamics as a modification of the unconstrained one, and define an almost-Poisson nonholonomic bracket. We show that many of the properties that standard nonholonomic systems enjoy have their counterpart in the proposed setup. As important examples, we highlight that the analysis here provides a natural interpretation for the use of pseudo-coordinates techniques and lends itself to the treatment of constrained systems with symmetry, following the ideas developed in [20,53]. We carefully examine the reduction procedure for this class of systems, paying special attention to the evolution of the momentum map.
From a methodological point of view, the approach taken in the paper has enormous advantages. This fact must mainly be attributed to the inclusive nature of Lie algebroids. Usually, the results on nonholonomic systems available in the literature are restricted to a particular class of nonholonomic systems, or to a specific context. However, as illustrated in Table 1, many different nonholonomic systems fit under the Lie algebroid framework, and this has the important consequence of making the results proved here widely applicable. With the aim of illustrating this breadth, we consider various examples throughout the paper, including the Suslov problem, the Chaplygin sleigh, the Veselova system, Chaplygin Gyro-type systems, the twowheeled planar mobile robot, and a ball rolling on a rotating table. We envision that future developments within the proposed framework will have a broad impact in nonholonomic mechanics. In the course of the preparation of this manuscript, the recent research efforts [14,55] were brought to our attention. These references, similar in spirit to the present work, deal with nonholonomic Lagrangian systems and focus on the reduction of Lie algebroid structures under symmetry. Chaplygin's gyro [49,64] Symmetry-invariant Atiyah algebroid Nonholonomic Lagrange-Poincaré Snakeboard [4] Table 1. The Lie algebroid framework embraces different classes of nonholonomic systems.
The paper is organized as follows. In Section 2 we collect some preliminary notions and geometric objects on Lie algebroids, including differential calculus, morphisms and prolongations. We also describe classical Lagrangian systems within the formalism of Lie algebroids. In Section 3, we introduce the class of nonholonomic Lagrangian systems subject to linear constraints, given by a regular Lagrangian L : E −→ R on the Lie algebroid τ : E −→ M and a constraint subbundle D of E. We show that the known results in Mechanics for these systems also hold in the context of Lie algebroids. In particular, drawing analogies with d'Alembert principle, we derive the Lagrange-d'Alembert equations of motion, prove the conservation of energy and state a Noether's theorem. We also derive local expressions for the dynamics of nonholonomic Lagrangian systems, which are further simplified by the choice of a convenient basis of D. As an illustration, we consider the class of nonholonomic mechanical systems. For such systems, the Lagrangian L is the polar form of a bundle metric on E minus a potential function on M . In Section 3.2, we perform the analysis of the existence and uniqueness of solutions of constrained systems on general Lie algebroids, and extend the results in [2,8,10,18,40] for constrained systems evolving on tangent bundles. We obtain several characterizations for the regularity of a nonholonomic system, and prove that a nonholonomic system of mechanical type is always regular. The constrained dynamics can be obtained by projecting the unconstrained dynamics in two different ways. Under the first projection, we develop a distributional approach analogous to that in [2], see also [56]. Using the second projection, we introduce the nonholonomic bracket. The evolution of any observable can be measured by computing its bracket with the energy of the system. Section 4 is devoted to studying the reduction of the dynamics under symmetry. Our approach follows the ideas developed in [15], who defined a minimal subcategory of the category of Lie algebroids which is stable under Lagrangian reduction. We study the behavior of the different geometric objects introduced under morphisms of Lie algebroids, and show that fiberwise surjective morphisms induce consistent reductions of the dynamics. This result covers, but does not reduce to, the usual case of reduction of the dynamics by a symmetry group. In accordance with the philosophy of the paper, we study first the unconstrained dynamics case, and obtain later the results for the constrained dynamics using projections. A (Poisson) reduction by stages procedure can also be developed within this formalism. It should be noticed that the reduction under the presence of a Lie group of symmetries G is performed in two steps: first we reduce by a normal subgroup N of G, and then by the residual group. In Section 5, we prove a general version of the momentum equation introduced in [4]. In Section 6, we show some interesting examples and in Section 7, we extend some of the results previously obtained for linear constraints to the case of nonlinear constraints. The paper ends with our conclusions and a description of future research directions.

Preliminaries
In this section we recall some well-known facts concerning the geometry of Lie algebroids. We refer the reader to [6,32,48] for details about Lie groupoids, Lie algebroids and their role in differential geometry.
2.1. Lie algebroids. Let M be an n-dimensional manifold and let τ : E → M be a vector bundle. A vector bundle map ρ : E → T M over the identity is called an anchor map. The vector bundle E together with an anchor map ρ is said to be an anchored vector bundle (see [59]). A structure of Lie algebroid on E is given by a Lie algebra structure on the C ∞ (M )-module of sections of the bundle, (Sec(E), [· , ·]), together with an anchor map, satisfying the compatibility condition Here f is a smooth function on M , σ, η are sections of E and ρ(σ) denotes the vector field on M given by ρ(σ)(m) = ρ(σ(m)). From the compatibility condition and the Jacobi identity, it follows that the map σ → ρ(σ) is a Lie algebra homomorphism from the set of sections of E, Sec(E), to the set of vector fields on M , X(M ).
In what concerns Mechanics, it is convenient to think of a Lie algebroid ρ : E → T M , and more generally an anchored vector bundle, as a substitute of the tangent bundle of M . In this way, one regards an element a of E as a generalized velocity, and the actual velocity v is obtained when applying the anchor to a, i.e., v = ρ(a). A curve a : [t 0 , t 1 ] → E is said to be admissible ifṁ(t) = ρ(a(t)), where m(t) = τ (a(t)) is the base curve. We will denote by Adm(E) the space of admissible curves on E.
Given local coordinates (x i ) in the base manifold M and a local basis {e α } of sections of E, we have local coordinates (x i , y α ) in E. If a ∈ E is an element in the fiber over m ∈ M , then we can write a = y α e α (m) and thus the coordinates of a are (m i , y α ), where m i are the coordinates of the point m. The anchor map is locally determined by the local functions ρ i α on M defined by ρ(e α ) = ρ i α (∂/∂x i ). In addition, for a Lie algebroid, the Lie bracket is determined by the functions C γ αβ defined by [e α , e β ] = C γ αβ e γ . The functions ρ i α and C γ αβ are called the structure functions of the Lie algebroid in this coordinate system. They satisfy the following relations 2.2. Exterior differential. The anchor ρ allows to define the differential of a function on the base manifold with respect to an element a ∈ E. It is given by df (a) = ρ(a)f.
It follows that the differential of f at the point m ∈ M is an element of E * m . Moreover, a structure of Lie algebroid on E allows to extend the differential to sections of the bundle p E, which will be called p-sections or just p-forms. If ω ∈ Sec( p E), then dω ∈ Sec( p+1 E) is defined by It follows that d is a cohomology operator, that is, d 2 = 0. Locally the exterior differential is determined by Throughout this paper, the symbol d will refer to the exterior differential on the Lie algebroid E and not to the ordinary exterior differential on a manifold. Of course, if E = T M , then both exterior differentials coincide. The usual Cartan calculus extends to the case of Lie algebroids (see [48,58]). For every section σ of E we have a derivation i σ (contraction) of degree −1 and a derivation d {e ′α } the corresponding dual basis. The bundle map Φ is determined by the relations Φ ⋆ x ′i = φ i (x) and Φ ⋆ e ′α = φ α β e β for certain local functions φ i and φ α β on M . Then, Φ is admissible if and only if The map Φ is a morphism of Lie algebroids if and only if, in addition to the admissibility condition above, one has In these expressions, ρ i α , C α βγ are the local structure functions on E and ρ ′i α , C ′α βγ are the local structure functions on E ′ .

2.4.
Prolongation of a fibered manifold with respect to a Lie algebroid. Let π : P → M be a fibered manifold with base manifold M . Thinking of E as a substitute of the tangent bundle of M , the tangent bundle of P is not the appropriate space to describe dynamical systems on P . This is clear if we note that the projection to M of a vector tangent to P is a vector tangent to M , and what one would like instead is an element of E, the 'new' tangent bundle of M . A space which takes into account this restriction is the E-tangent bundle of P , also called the prolongation of P with respect to E, which we denote by T E P (see [43,51,52,59]). It is defined as the vector bundle τ E P : T E P → P whose fiber at a point p ∈ P m is the vector space We will frequently use the redundant notation (p, b, v) to denote the element (b, v) ∈ T E p P . In this way, the map τ E P is just the projection onto the first factor. The anchor of T E P is the projection onto the third factor, that is, the map ρ 1 : T E P → T P given by ρ 1 (p, b, v) = v. The projection onto the second factor will be denoted by T π : T E P → E, and it is a vector bundle map over π. Explicitly T π(p, b, v) = b.
An element z ∈ T E P is said to be vertical if it projects to zero, that is T π(z) = 0. Therefore it is of the form (p, 0, v), with v a vertical vector tangent to P at p.
Given local coordinates (x i , u A ) on P and a local basis {e α } of sections of E, we can define a local basis {X α , V A } of sections of T E P by Vertical elements are linear combinations of {V A }. The anchor map ρ 1 applied to a section Z of T E P with local expression Z = Z α X α + V A V A is the vector field on P whose coordinate expression is If E carries a Lie algebroid structure, then so does T E P . The associated Lie bracket can be easily defined in terms of projectable sections, so that T π is a morphism of Lie algebroids. A section Z of T E P is said to be projectable if there exists a section σ of E such that T π • Z = σ • π. Equivalently, a section Z is projectable if and only if it is of the form Z(p) = (p, σ(π(p)), X(p)), for some section σ of E and some vector field X on E (which projects to ρ(σ)). The Lie bracket of two projectable sections Z 1 and Z 2 is then given by It is easy to see that [Z 1 , Z 2 ](p) is an element of T E p P for every p ∈ P . Since any section of T E P can be locally written as a linear combination of projectable sections, the definition of the Lie bracket for arbitrary sections of T E P follows.
The Lie brackets of the elements of the basis are and the exterior differential is determined by 2.5. Prolongation of a map. Let Ψ : P → P ′ be a fibered map from the fibered manifold π : P → M to the fibered manifold π ′ : It is clear from the definition that T Φ Ψ is a vector bundle map from τ E P : [54] it is proved the following result. Proposition 2.1. The map T Φ Ψ is an admissible map. Moreover, T Φ Ψ is a morphism of Lie algebroids if and only if Φ is a morphism of Lie algebroids.
Given local coordinate systems (x i ) on M and (x ′i ) on M ′ , local adapted coordinates (x i , u A ) on P and (x ′i , u ′A ) on P ′ and a local basis of sections {e α } of E and {e ′ α } of E ′ , the maps Φ and Ψ are determined by Φ ⋆ e ′α = Φ α β e β and Ψ(x, u) = (φ i (x), ψ A (x, u)). Then the action of T Φ Ψ is given by We finally mention that the composition of prolongation maps is the prolongation of the composition. Indeed, let Ψ ′ be another bundle map from π ′ : P ′ → M ′ to another bundle π ′′ : P ′′ → M ′′ and Φ ′ be another admissible map from τ ′ : E ′ → M ′ to τ ′′ : E ′′ → M ′′ both over the same base map. Since Φ and Φ ′ are admissible maps then so is Φ ′ • Φ, and thus we can define the prolongation of In the particular case when the bundles P and P ′ are just P = E and P ′ = E ′ , whenever we have an admissible map Φ : . From the result above, we have that T Φ Φ is a Lie algebroid morphism if and only if Φ is a Lie algebroid morphism. In coordinates we obtain where (x i , y γ ) are the corresponding fibred coordinates on E. From this expression it is clear that T Φ Φ is fiberwise surjective if and only if Φ is fiberwise surjective.
2.6. Lagrangian Mechanics. In [51] (see also [59]) a geometric formalism for Lagrangian Mechanics on Lie algebroids was defined. Such a formalism is similar to Klein's formalism [36] in standard Lagrangian mechanics and it is developed in the prolongation T E E of a Lie algebroid E over itself. The canonical geometrical structures defined on T E E are the following: is the vector tangent to the curve a + tb at t = 0, • The vertical endomorphism S : T E E → T E E defined as follows: • The Liouville section which is the vertical section corresponding to the Liouville dilation vector field: Given a Lagrangian function L ∈ C ∞ (E) we define the Cartan 1-form θ L and the Cartan 2-form ω L as the forms on T E E given by and The real function E L on E defined by E L = d ∆ L − L is the energy function of the Lagrangian system. By a solution of the Lagrangian system (a solution of the Euler-Lagrange equations) we mean a sode section Γ of T E E such that The local expressions for the vertical endomorphism, the Liouville section, the Cartan 2-form and the Lagrangian energy are Thus, a sode Γ is a section of the form The sode Γ is a solution of the Euler-Lagrange equations if and only if the functions f α satisfy the linear equations The Euler-Lagrange differential equations are the differential equations for the integral curves of the vector field ρ 1 (Γ), where the section Γ is the solution of the Euler-Lagrange equations. Thus, these equations may be written aṡ In other words, if δL : Adm(E) → E * is the Euler-Lagrange operator, which locally reads where {e α } is the dual basis of {e α }, then the Euler-Lagrange differential equations read δL = 0. The function L is said to be regular Lagrangian if ω L is regular at every point as a bilinear map. In such a case, there exists a unique section Γ L of T E E which satisfies the equation i ΓL ω L − dE L = 0. Note that from (2.3), (2.4), (2.5) and (2.6), it follows that for X ∈ Sec(T E E). Thus, using (2.8), we deduce that which implies that Γ L is a sode section. Therefore, for a regular Lagrangian function L we will say that the dynamical equations (2.2) are just the Euler-Lagrange equations.
On the other hand, the vertical distribution is isotropic with respect to ω L , see [43]. This fact implies that the contraction of ω L with a vertical vector is a semibasic form. This property allows us to define a symmetric 2-tensor G L along τ by whereb is any element in T E a E which projects to b, i.e., T τ (b) = b, and a ∈ E. In coordinates G L = W αβ e α ⊗ e β , where the matrix W αβ is given by (2.10) It is easy to see that the Lagrangian L is regular if and only the matrix W is regular at every point, that is, if the tensor G L is regular at every point. By the kernel of G L at a point a we mean the vector space In the case of a regular Lagrangian, the Cartan 2-section ω L is symplectic (nondegenerate and d-closed) and the vertical subbundle is Lagrangian. It follows that a 1-form is semi-basic if and only if it is the contraction of ω L with a vertical element.
Finally, we mention that the complete lift σ C of a section σ ∈ Sec(E) is the section of T E E characterized by the two following properties: (1) projects to σ, i.e., T τ (2.12)

Linearly constrained Lagrangian systems
Nonholonomic systems on Lie algebroids were introduced in [20]. This class of systems includes, as particular cases, standard nonholonomic systems defined on the tangent bundle of a manifold and systems obtained by the reduction of the action of a symmetry group. The situation is similar to the non-constrained case, where the general equation δL = 0 comprises as particular cases the standard Lagrangian Mechanics, Lagrangian Mechanics with symmetry, Lagrangian systems with holonomic constraints, systems on semi-direct products and systems evolving on Lie algebras, see e.g., [51].
We start with a free Lagrangian system on a Lie algebroid E. As mentioned above, these two objects can describe a wide class of systems. Now, we plug in some nonholonomic linear constraints described by a subbundle D of the bundle E of admissible directions. If we impose to the solution curves a(t) the condition to stay on the manifold D, we arrive at the equations δL a(t) = λ(t) and a(t) ∈ D, where the constraint force λ(t) ∈ E * τ (a(t)) is to be determined. In the tangent bundle geometry case (E = T M ), the d'Alembert principle establishes that the mechanical work done by the constraint forces vanishes, which implies that λ takes values in the annihilator of the constraint manifold D. Therefore, in the case of a general Lie algebroid, the natural equations one should pose are (see [20]) and a(t) ∈ D.
In more explicit terms, we look for curves a(t) on E such that -they are admissible, ρ(a(t)) =ṁ(t), where m = τ • a, -they stay in D, a(t) ∈ D m(t) , -there exists λ(t) ∈ D • m(t) such that δL a(t) = λ(t). If a(t) is one of such curves, then (a(t),ȧ(t)) is a curve in T E E. Moreover, since a(t) is in D, we haveȧ(t) is tangent to D, that is, (a(t),ȧ(t)) ∈ T D D. Under some regularity conditions (to be made precise later on), we may assume that the above curves are integral curves of a section Γ, which as a consequence will be a sode section taking values in T D D. Based on these arguments, we may reformulate geometrically our problem as the search for a sode Γ (defined at least on a neighborhood of D) τ (a) and Γ(a) ∈ T D a D, at every point a ∈ D. In the above expression D • is the pullback of D • to T E E, that is, α ∈ D • τ (a) if and only if there exists λ ∈ D • τ (a) such that α = λ • T a τ . Definition 3.1. A nonholonomically constrained Lagrangian system on a Lie algebroid E is a pair (L, D), where L is a smooth function on E, the Lagrangian, and i : D ֒→ E is a smooth subbundle of E, known as the constraint subbundle. By a solution of the nonholonomically constrained Lagrangian system (L, D) we mean a sode section Γ ∈ T E E which satisfies the Lagrange-d'Alembert equations With a slight abuse of language, we will interchangeably refer to a solution of the constrained Lagrangian system as a section or the collection of its corresponding integral curves. The restriction of the projection τ : E → M to D will be denoted by π, that is, π = τ | D : D → M . . We want to stress that a solution of the Lagrange-d'Alembert equations needs to be defined only over D, but for practical purposes we consider it extended to E (or just to a neighborhood of D in E). We will not make any notational distinction between a solution on D and any of its extensions. Solutions which coincide on D will be considered as equal. See [30,40] for a more in-depth discussion. In accordance with this convention, by a sode on D we mean a section of T D D which is the restriction to D of some sode defined in a neighborhood of D. Alternatively, a sode on D is a section Γ of T D D such that T τ (Γ(a)) = a for every a ∈ D.
The above facts prove that a sode section Γ of T E E is a solution of the holonomic Lagrangian system (L, D) on E if and only if Γ |D is a solution of the Euler-Lagrange equations for the (unconstrained) Lagrangian function L D on the Lie algebroid D.
In other words, the holonomic Lagrangian system (L, D) on E may be considered as an unconstrained (free) Lagrangian system on the Lie algebroid D.
• Next, suppose that (L, D) is a nonholonomically constrained Lagrangian system on the Lie algebroid E. Then, the different spaces we will consider are shown in the following commutative diagram As an intermediate space in our analysis of the regularity of the constrained systems, we will also consider T E D, the E-tangent to D. The main difference between T E D and T D D is that the former has a natural Lie algebroid structure while the later does not.
The following two results are immediate consequences of the above form of the Lagrange-d'Alembert equations. Proof. Indeed, for every a ∈ D, we have Γ(a) ∈ T D a D, so that T τ (Γ(a)) ∈ D. Therefore i Γ D • = 0 and contracting 0 = i Γ (i Γ ω L − dE L ) = −d Γ E L at every point in D.
Theorem 3.5 (Noether's theorem). Let (L, D) be a constrained Lagrangian system which admits a unique sode Γ solution of the dynamics. If σ is a section of D such that there exists a function f ∈ C ∞ (M ) satisfying for σ, η, ζ ∈ Sec(E). The coefficients of the connection ∇ G are given by where G αν are the coefficients of the metric G, (G αν ) is the inverse matrix of (G αν ) and [α, β; γ] = ∂G αβ ∂x i ρ i γ + C µ αβ G µγ . Using the covariant derivative induced by ∇ G , one may introduce the notion of a geodesic of ∇ G as follows. An admissible curve a : I → E is said to be a geodesic if ∇ G a(t) a(t) = 0, for all t ∈ I. In local coordinates, the conditions for being a geodesic read da γ dt + 1 2 (Γ γ αβ + Γ γ βα )a α a β = 0, for all γ. The geodesics are the integral curves of a sode section Γ ∇ G of T E E, which is locally given by is called the geodesic flow (for more details, see [20]).
The class of systems that were considered in detail in [20] is that of mechanical systems with nonholonomic constraints 1 . The Lagrangian function L is of mechanical type, i.e., it is of the form with V a function on M . The Euler-Lagrange section for the unconstrained system can be written as In this expression, by grad G V we mean the section of E such that dV (m), a = G(grad G V (m), a), for all m ∈ M and all a ∈ E m , and where we remind that d is 1 In fact, in [20], we considered controlled mechanical systems with nonholonomic constraints, that is, mechanical systems evolving on Lie algebroids and subject to some external control forces.
the differential in the Lie algebroid. The Euler-Lagrange differential equations can be written asẋ Alternatively, one can describe the dynamical behavior of the mechanical control system by means of an equation on E via the covariant derivative. An admissible curve a : t → a(t) with base curve t → m(t) is a solution of the system (3.2) if and only if Note that If this mechanical control system is subject to the constraints determined by a subbundle D of E, we can do the following. Consider the orthogonal decomposition E = D ⊕ D ⊥ , and the associated orthogonal projectors P : E → D, Q : E → D ⊥ . Using the fact that G(P ·, ·) = G(·, P ·), one can write the Lagrange-d'Alembert equations in the form A specially well-suited form of these equations makes use of the constrained con-nection∇ defined by∇ σ η = P (∇ G σ η) + ∇ G σ (Qη). In terms of∇, we can rewrite this equation as∇ a(t) a(t) + P (grad G V (m(t))) = 0, Q(a) = 0, where we have used the fact that the connection∇ restricts to the subbundle D.
Moreover, following the ideas in [45], we proved in [20] that the subbundle D is geodesically invariant for the connection∇, that is, any integral curve of the spray Γ∇ associated with∇ starting from a point in D is entirely contained in D.
Since the terms coming from the potential V also belongs to D, we have that the constrained equations of motion can be simply stated aš ∇ a(t) a(t) + P (grad G V (m(t))) = 0, a(0) ∈ D. (3.4) Note that one can write the constrained equations of the motion as followṡ and that the restriction to D of the vector field The coordinate expression of equations (3.4) is greatly simplified if we take a basis {e α } = {e a , e A } of E adapted to the orthogonal decomposition E = D ⊕ D ⊥ , i.e., D = span{e a }, D ⊥ = span{e A }. Denoting by (y α ) = (y a , y A ) the induced coordinates, the constraint equations Q(a) = 0 just read y A = 0. The differential equations of the motion are theṅ whereΓ α βγ are the connection coefficients of the constrained connection∇. • In the above example the dynamics exists and is completely determined whatever the (linear) constraints are. As we will see in Section 3.2, this property is lost in the general case.
3.1. Lagrange-d'Alembert equations in local coordinates. Let us analyze the form of the Lagrange-d'Alembert equations in local coordinates. Following the example above, let us choose a special coordinate system adapted to the structure of the problem as follows. We consider local coordinates (x i ) on an open set U of M and we take a basis {e a } of local sections of D and complete it to a basis {e a , e A } of local sections of E (both defined on the open U). In this way, we have coordinates (x i , y a , y A ) on E. In this set of coordinates, the constraints imposed by the submanifold D ⊂ E are simply y A = 0. If {e a , e A } is the dual basis of {e a , e A }, then a basis for the annihilator Let us find the local expression of the Lagrange-d'Alembert equations in these coordinates. We consider a section Γ such that Γ |D ∈ Sec(T D D), which is therefore of the form Γ = g a X a + f a V a . From the local expression (2.5) of the Cartan 2-form and the local expression (2.6) of the energy function, we get If we assume that the Lagrangian L is regular, when we evaluate at y A = 0, we have that g a = y a and thus Γ is a sode. Moreover, contracting with X a , after a few calculations we get so that (again after evaluation at y A = 0), the functions f a are solution of the linear equations where all the partial derivatives of the Lagrangian are to be evaluated on y A = 0. As a consequence, we get that there exists a unique solution of the Lagranged'Alembert equations if and only if the matrix is regular. Notice that C ab is a submatrix of W αβ , evaluated at y A = 0 and that, as we know, if L is of mechanical type then the Lagrange-d'Alembert equations have a unique solution. The differential equations for the integral curves of the vector field ρ 1 (Γ) are the Lagrange-d'Alembert differential equations, which reaḋ Finally, notice that the contraction with X A just gives the components Remark 3.7 (Equations in terms of the constrained Lagrangian). In some occasions, it is useful to write the equations in the forṁ where, on the left-hand side of the second equation, all the derivatives can be calculated from the value of the Lagrangian on the constraint submanifold D. In other words, we can substitute L by the constrained Lagrangian L c defined by In what follows, we will assume that the Lagrangian L is regular at least in a neighborhood of D.
Let us now perform a precise global analysis of the existence and uniqueness of the solution of Lagrange-d'Alembert equations. In order to characterize geometrically those nonholonomic systems which are regular, we define the tensor G L,D as the restriction of G L to D, that is, for every a ∈ D and every b, c ∈ D τ (a) . In coordinates adapted to D, we have that the local expression of G L,D is G L,D = C ab e a ⊗ e b where the matrix C ab is given by equation (3.6).
A second important geometric object is the subbundle Finally, we also consider the subbundle ( The relation among these three objects is described by the following result.
Lemma 3.11. The following properties are satisfied: In terms of G L and writing c = T τ (u), the above equation reads −G L a (b, c) = ζ, c . By taking u ∈ T τ −1 (D), then c is in D and therefore is an arbitrary element of D τ (a) , so that the above condition reads G L a (b, c) = 0, for all c ∈ D τ (a) , which is precisely the condition for ξ V (a, b) to be in F a .
Theorem 3.12. The following properties are equivalent: (1) The constrained Lagrangian system (L, D) is regular, In the case of a constrained mechanical system, the tensor G L is given by , so that it is positive definite at every point. Thus the restriction to any subbundle D is also positive definite and hence regular. Thus, nonholonomic mechanical systems are always regular.
Proposition 3.13. Conditions (3) and (4) in Theorem 3.12 are equivalent, respectively, to Proof. The equivalence between (4) and (4') is obvious, since we are assuming that the free Lagrangian is regular, i.e., ω L is symplectic. The equivalence of (3) and (3') follows by computing the dimension of the corresponding spaces. The ranks of Thus rank(T E E) = rank(T E D) + rank(F ), and the result follows.

Projectors.
We can express the constrained dynamical section in terms of the free dynamical section by projecting to the adequate space, either T E D or T D D, according to each of the above decompositions of T E E| D . Of course, both procedures give the same result.
Projection to T E D. Assuming that the constrained system is regular, we have a direct sum decomposition Let us denote by P and Q the complementary projectors defined by this decomposition, that is, for all a ∈ D. Then we have, Theorem 3.14. Let (L, D) be a regular constrained Lagrangian system and let Γ L be the solution of the free dynamics, i.e., i ΓL ω L = dE L . Then the solution of the constrained dynamics is the sode We consider adapted local coordinates (x i , y a , y A ) corresponding to the choice of an adapted basis of sections {e a , e A }, where {e a } generate D. The annihilator D • of D is generated by {e A }, and thus D • is generated by {X A }. A simple calculation shows that a basis {Z A } of local sections of F is given by where Q a A = W Ab C ab and C ab are the components of the inverse of the matrix C ab given by equation (3.6). The local expression of the projector over F is then If the expression of the free dynamical section Γ L in this local coordinates is (where f α are given by equation (2.7)), then the expression of the constrained dynamical section is where all the functions f α are evaluated at y A = 0.
Projection to T D D. We have seen that the regularity condition for the constrained system (L, D) can be equivalently expressed by requiring that the subbundle T D D is a symplectic subbundle of (T E E, ω L ). It follows that, for every a ∈ D, we have a direct sum decomposition Let us denote byP andQ the complementary projectors defined by this decomposition, that is, for all a ∈ D. Then, we have the following result: Proof. From Theorem 3.14 we have that the solution Γ (L,D) of the constrained dynamics is related to the free dynamics by In adapted coordinates, a local basis of sections of ( where the sections Z A are given by (3.9) and the sections Y A are 3.4. The distributional approach. The equations for the Lagrange-d'Alembert section Γ can be entirely written in terms of objects in the manifold T D D. Recall that T D D is not a Lie algebroid. In order to do this, define the 2-section ω L,D as the restriction of ω L to T D D. If (L, D) is regular, then T D D is a symplectic subbundle of (T E E, ω L ). From this, it follows that ω L,D is a symplectic section on that bundle. We also define ε L,D to be the restriction of dE L to T D D. Then, taking the restriction of Lagrange-d'Alembert equations to T D D, we get the following equation 10) which uniquely determines the section Γ. Indeed, the unique solution Γ of the above equations is the solution of Lagrange-d'Alembert equations: if we denote by λ the constraint force, we have for every u ∈ T D a D that where we have taken into account that T τ (u) ∈ D and λ(a) ∈ D • . This approach, the so called distributional approach, was initiated by Bocharov and Vinogradov (see [67]) and further developed byŚniatycki and coworkers [2,24,62]. Similar equations, within the framework of Lie algebroids, are the base of the theory proposed in [56].
Remark 3.16 (Alternative description with T E D). One can also consider the restriction to T E D, which is a Lie algebroid, but no further simplification is achieved by this. Ifω is the restriction of ω L to T E D andε is the restriction of dE L to T E D, then the Lagrange-d'Alembert equations can be written in the form i Γω −ε =λ, whereλ is the restriction of the constraint force to T E D, which, in general, does not vanish. Also notice that the 2-formω is closed but, in general, degenerated. • 3.5. The nonholonomic bracket. Let f, g be two smooth functions on D and take arbitrary extensions to E denoted by the same letters (if there is no possibility of confusion). Suppose that X f and X g are the Hamiltonian sections on T E E given respectively by i X f ω L = df and i Xg ω L = dg. We define the nonholonomic bracket of f and g as follows: {f, g} nh = ω L (P (X f ),P (X g )). (3.11) Note that if f ′ is another extension of f , then (X f − X f ′ ) |D is a section of (T D D) ⊥ and, thus, we deduce that (3.11) does not depend on the chosen extensions. The nonholonomic bracket is an almost-Poisson bracket, i.e., it is skew-symmetric, a derivation in each argument with respect to the usual product of functions and does not satisfy the Jacobi identity.
In addition, one can prove the following formulȧ Equation (3.12) implies once more the conservation of the energy (by the skewsymmetric character of the nonholonomic bracket). Alternatively, since T D D is an anchored vector bundle, one can take the function f ∈ C ∞ (D) and its differentialdf ∈ Sec((T D D) * ). Since ω L,D is regular, we have a unique sectionX f ∈ Sec(T D D) defined by iX f ω L,D =df . Then the nonholonomic bracket of two functions f and g is {f,

Morphisms and reduction
One important advantage of dealing with Lagrangian systems evolving on Lie algebroids is that the reduction procedure can be naturally handled by considering morphisms of Lie algebroids, as it was already observed by Weinstein [68]. We study in this section the transformation laws of the different geometric objects in our theory and we apply these results to the study of the reduction theory.
Let ξ V and ξ ′V , S and S ′ , and ∆ and ∆ ′ , be the vertical liftings, the vertical endomorphisms, and the Liouville sections on E and E ′ , respectively. Then, Proof. For the first property, we notice that both terms are vertical, so that we just have to show that their action on functions coincide. For every function For the second property, we have ∆(a) = ξ V (a, a) so that applying the first property it follows that Finally, for any z = (a, b, V ) ∈ T E E, we obtain that which concludes the proof.
Proof. Indeed, for every Z ∈ T E E we have where we have used the transformation rule for the vertical endomorphism. The second property follows from the fact that The third one follows similarly and the fourth is a consequence of the second property and the definitions of the tensors G L and G L ′ .
Let Γ be a sode and L ∈ C ∞ (E) be a Lagrangian. For convenience, we define for every section Z of T E E. We notice that Γ is the solution of the free dynamics if and only if δ Γ L = 0. On the other hand, notice that the 1-form δ Γ L is semibasic, because Γ is a sode.
which concludes the proof.
Thus, the dynamics of both systems is uniquely defined.
Theorem 4.4 (Reduction of the free dynamics). Suppose that the Lagrangian functions L and L ′ are Φ-related, that is, L = L ′ • Φ. If Φ is a fiberwise surjective morphism and L is a regular Lagrangian then L ′ is also a regular Lagrangian. Moreover, if Γ L and Γ L ′ are the solutions of the free dynamics defined by L and L ′ then Therefore, if a(t) is a solution of the free dynamics defined by L, then Φ(a(t)) is a solution of the free dynamics defined by L ′ .
Proof. If Γ L and Γ L ′ are the solutions of the dynamics, then δ ΓL L = 0 and δ Γ L ′ L ′ = 0 so that the left-hand side in equation (4.1) vanishes. Thus for every Z ∈ Sec(T E E). Therefore, using that L ′ is regular and the fact that T Φ Φ is a fiberwise surjective morphism, we conclude the result.
We will say that the unconstrained dynamics Γ L ′ is the reduction of the unconstrained dynamics Γ L by the morphism Φ.

4.2.
Reduction of the constrained dynamics. The above results about reduction of unconstrained Lagrangian systems can be easily generalized to nonholonomic constrained Lagrangian systems whenever the constraints of one system are mapped by the morphism to the constraints of the second system. Let us elaborate on this.
Let (L, D) be a constrained Lagrangian system on a Lie algebroid E and let (L ′ , D ′ ) be another constrained Lagrangian system on a second Lie algebroid E ′ . Along this section, we assume that there is a fiberwise surjective morphism of Lie algebroids Φ : E → E ′ such that L = L ′ • Φ and Φ(D) = D ′ . The latter condition implies that the base map is also surjective, so that we will assume that Φ is an epimorphism (i.e., in addition to being fiberwise surjective, the base map ϕ is a submersion).
As a first consequence, we have for every a ∈ D and every b, c ∈ D π(a) , and therefore, if (L, D) is regular, then so is (L ′ , D ′ ).
we have the following properties: If P, Q and P ′ , Q ′ are the projectors associated with (L, D) and (L ′ , D ′ ), respectively, then With respect to the decompositions we have the following properties: Thus, one may consider the vector bundle morphisms Moreover, using that Φ is fiberwise surjective and that ϕ is a submersion, we deduce that the rank of the above morphisms is maximum. This proves (1) and (4). The proof of (5) is as follows. For every a ′ ∈ D ′ , one can choose a ∈ D such that Φ(a) = a ′ , and one can write any element In a similar way, using that For the proof of (2) we have that Thus, using that (1), it follows that (2) holds. Finally, (3) is an immediate consequence of (1) and (2), and similarly, (6) is an immediate consequence of (4) and (5).
From the properties above, we get the following result.
is a solution of Lagranged'Alembert differential equations for L, then Φ(a(t)) is a solution of Lagranged'Alembert differential equations for L ′ .
Proof. For the free dynamics, we have that T Φ Φ • Γ L = Γ L ′ • Φ. Moreover, from property (3) in Lemma 4.5, for every a ∈ D, we have that which concludes the proof.
We will say that the constrained dynamics Γ (L ′ ,D ′ ) is the reduction of the constrained dynamics Γ (L,D) by the morphism Φ.
is the nonholonomic bracket for the constrained system (L, D) (respectively, (L ′ , D ′ )). In other words, Φ : D → D ′ is an almost-Poisson morphism.
Proof. Using (2) in Proposition 4.2 and the fact that Φ is a Lie algebroid morphism, we deduce that Thus, since T Φ Φ is fiberwise surjective, we obtain that Now, from (3.11) and Lemma 4.5, we conclude that One of the most important cases in the theory of reduction is the case of reduction by a symmetry group. In this respect, we have the following result.  43,48]). Let q Q G : Q → M be a principal G-bundle, let τ : E → Q be a Lie algebroid, and assume that we have an action of G on E such that the quotient vector bundle E/G is well-defined. If the set Sec(E) G of equivariant sections of E is a Lie subalgebra of Sec(E), then the quotient E ′ = E/G has a canonical Lie algebroid structure over M such that the canonical projection q E G : E → E/G, given by a → [a] G , is a (fiberwise bijective) Lie algebroid morphism over q Q G . As a concrete example of application of the above theorem, we have the wellknown case of the Atiyah or Gauge algebroid. In this case, the Lie algebroid E is the standard Lie algebroid T Q → Q, the action is by tangent maps gv ≡ T ψ g (v), the reduction is the Atiyah Lie algebroid T Q/G → Q/G and the quotient map q T Q G : T Q → T Q/G is a Lie algebroid epimorphism. It follows that if L is a Ginvariant regular Lagrangian on T Q then the unconstrained dynamics for L projects to the unconstrained dynamics for the reduced Lagrangian L ′ . Moreover, if the constraints D are also G-invariant, then the constrained dynamics for (L, D) reduces to the constrained dynamics for (L ′ , D/G).
On a final note, we mention that the pullback of the distributional equation 4.3. Reduction by stages. As a direct consequence of the results exposed above, one can obtain a theory of reduction by stages. In Poisson geometry, reduction by stages is a straightforward procedure. Given the fact that the Lagrangian counterpart of Poisson reduction is Lagrangian reduction, it is not strange that reduction by stages in our framework becomes also straightforward.
The Lagrangian theory of reduction by stages is a consequence of the following basic observation: Let Φ 1 : E 0 → E 1 and Φ 2 : E 1 → E 2 be a fiberwise surjective morphisms of Lie algebroids and let Φ : E 0 → E 2 be the composition Φ = Φ 2 • Φ 1 . The reduction of a Lagrangian system in E 0 by Φ can be obtained by first reducing by Φ 1 and then reducing the resulting Lagrangian system by Φ 2 . This result follows using that T Φ Φ = T Φ2 Φ 2 • T Φ1 Φ 1 . Based on this fact, one can analyze one of the most interesting cases of reduction: the reduction by the action of a symmetry group. We consider a group G acting on a manifold Q and a closed normal subgroup N of G. The process of reduction by stages is illustrated in the following diagram ·/G Total reduction In order to prove our results about reduction by stages, we have to prove that E 0 , E 1 and E 2 are Lie algebroids, that the quotient maps Φ 1 : E 0 → E 1 , Φ 2 : E 1 → E 2 and Φ : E 0 → E 2 are Lie algebroids morphisms and that the composition Φ 1 •Φ 2 equals to Φ. Our proof is based on the following well-known result (see [15]), which contains most of the ingredients in the theory of reduction by stages.
Building on the previous results, one can deduce the following theorem, which states that the reduction of a Lie algebroid can be done by stages. The following diagram illustrates the above situation: In particular, for the unconstrained case one has the following result.
Theorem 4.11 (Reduction by stages of the free dynamics). Let q Q G : Q → Q/G be a principal G-bundle, and N a closed normal subgroup of G. Let L be a Lagrangian function on Q which is G-invariant. Then the reduction by the symmetry group G can be performed in two stages: 1. reduce by the normal subgroup N , 2. reduce the resulting dynamics from 1. by the residual symmetry group G/N .
Since the dynamics of a constrained system is obtained by projection of the free dynamics, we also the following result. 1. reduce by the normal subgroup N , 2. reduce the resulting dynamics from 1. by the residual symmetry group G/N .

The momentum equation
In this section, we introduce the momentum map for a constrained system on a Lie algebroid, and examine its evolution along the dynamics. This gives rise to the so-called momentum equation.
5.1. Unconstrained case. Let us start by discussing the unconstrained case. Let τ E : E → M be a Lie algebroid over a manifold M and L : E → R be a regular Lagrangian function. Suppose that τ K : K → M is a vector bundle over M and that Ψ : K → E is a vector bundle morphism (over the identity of M ) between K and E. Then, we can define the unconstrained momentum map J (L,Ψ) : E → K * associated with L and Ψ as follows If σ : M → K is a section of τ K : K → M then, using the momentum map J (L,Ψ) , we may introduce the real function J σ (L,Ψ) : E → R given by

2)
where d T E E is the differential of Lie algebroid T E E → E. In particular, if < d T E E L, (Ψ • σ) c >= 0, then the real function J σ (L,Ψ) is a constant of the motion for the Lagrangian dynamics associated with the Lagrangian function L.
is the vertical lift of (Ψ•σ) ∈ Sec(E) then, using (5.1) and the fact that where θ L is the Cartan 1-form associated with L. Thus, from (5.3), we deduce that , ω L being the Cartan 2-form associated with L.
Therefore, if E L : E → R is the Lagrangian energy, we obtain that (5.4) Now, from (2.11) and since Γ L is a sode section, it follows that where ∆ ∈ Sec(T E E) is the Liouville section. Consequently, using (5.4) we deduce that (5.2) holds. where ξ M ∈ X(M ) is the infinitesimal generator of the action ψ associated with ξ ∈ g.
A direct computation proves that the (unconstrained) momentum map J L,Ψ) : E = T M → K * = M × g * associated with L and Ψ is given by where J : T M → g * is the standard momentum map associated with L and the action ψ defined by If x is point of M we consider the vector subspace K D x of K x given by x , for all x ∈ M , and (Ψ • σ) c ∈ Sec(T E E) is the complete lift of (Ψ • σ) ∈ Sec(E) then we have that Moreover, the pair (J , j) is a Lie algebroid monomorphism which implies that ). Thus, using that J σ (L,D,Ψ) = J σ (L,Ψ) and proceedings as in the proof of Theorem 5.1, we deduce that Consequently, Finally, using (5.7), (5.8) and (5.9), we conclude that (5.6) holds.
where g x is the vector subspace of g given by We also remark that the sets K D and (K D ) * may be identified with the sets Under this identification, the nonholonomic momentum map J (L,D,Ψ) : E → (K D ) * associated with the system (L, D) and the morphism Ψ is just the standard nonholonomic momentum map J nh : T M → (g D ) * associated with the system (L, D) and the action ψ (see [4,8,9]). Now, if ξ : M → g is an smooth map the ξ defines, in a natural way, a section ." The above equality is an intrinsic expression of the standard nonholonomic momentum equation. In addition, using again Theorem 5.3 we also deduce another well-known result (see [4,8,9]): "If the Lagrangian function L : T M → R is invariant under the tangent action T ψ of G on T M and ξ ∈ g is a horizontal symmetry (that is, ξ ∈ g x , for all x ∈ M ) then the real function (J nh e ξ ) |D is a constant of the motion for the constrained Lagrangian dynamics, where ξ : M → g is the constant map ξ(x) = ξ, for all x ∈ M." •

Examples
As in the unconstrained case, constrained Lagrangian systems on Lie algebroids appear frequently. We show some examples next.
6.1. Nonholonomic Lagrangian systems on Lie algebras. Let g be a real algebra of finite dimension. Then, it is clear that g is a Lie algebroid over a single point. Now, suppose that (l, d) is a nonholonomic Lagrangian system on g, that is, l : g → R is a Lagrangian function and d is a vector subspace of g. If w : I → g is a curve on g then dl(ω(t)) ∈ T * ω(t) g ∼ = g * , ∀t ∈ I, and thus, the map dl • ω may be considered as a curve on g * dl • ω : I → g * . Therefore, (dl • ω) ′ (t) ∈ T dl(ω(t)) g * ∼ = g * , ∀t ∈ I. Moreover, from (3.7), it follows that ω is a solution of the Lagrange-d'Alembert equations for the system (l, d) if and only if where ad * : g × g * → g * is the infinitesimal coadjoint action. The above equations are just the so-called Euler-Poincaré -Suslov equations for the system (l, d) (see [28]). We remark that in the particular case when the system is unconstrained, that is, d = g, then one recovers the the standard Euler-Poincaré equations for the Lagrangian function l : g → R.
If G is a Lie group with Lie algebra g then nonholonomic Lagrangian systems on g may be obtained (by reduction) from nonholonomic LL mechanical systems with configuration space the Lie group G.
In fact, let e be the identity element of G and I : g → g * be a symmetric positive definite inertia operator. Denote by g e : g×g → R the corresponding scalar product on g given by g e (ω, ω ′ ) =< I(ω), ω ′ >, for ω, ω ′ ∈ g ∼ = T e G.
g e induces a left-invariant Riemannian metric g on G. Thus, we way consider the Lagrangian function L : T G → R defined by In other words, L is the kinetic energy associated with the Riemannian metric g. Now, let D be a left-invariant distribution on G. Then, since L is a left-invariant function, the pair (L, D) is an standard nonholonomic LL system in the terminology of [28].
Therefore, if ω : I → g is a curve on g, we have that (dl • ω)(t) = I(ω(t)), for all t and, using (6.1), it follows that ω is a solution of the Lagrange-d'Alembert equations for the system (l, d) if and only iḟ ω − I −1 (ad * ω(t) I(ω(t))) ∈ d ⊥ , ω(t) ∈ d, for all t, where d ⊥ is the orthogonal complement of the subspace d, that is, Two simple examples of the above general situation are the following ones.
The Suslov system. The most natural example of LL system is the nonholonomic Suslov problem, which describes the motion of a rigid body about a fixed point under the action of the following nonholonomic constraint: the body angular velocity vector is orthogonal to a some fixed direction in the body frame.
The configuration space of the problem is the group G = SO(3). Thus, in this case, the Lie algebra g may be identified with R 3 and, under this identification, the Lie bracket on g is just the cross product × on R 3 .
The Chaplygin sleigh. The Chaplygin sleigh is a rigid body sliding on a horizontal plane. The body is supported at three points, two of which slide freely without friction while the third is a knife edge, a constraint that allows no motion orthogonal to this edge. This mechanical system was introduced and studied in 1911 by Chaplygin [16] (see also [57]). The configuration space of this system is the group SE(2) of Euclidean motions of the two-dimensional plane R 2 . As we know, we may choose local coordinates (θ, x, y) on SE(2). θ and (x, y) are the angular orientation of the blade and position of the contact point of the blade on the plane, respectively. Now, we introduce a coordinate system called the body frame by placing the origin at the contact point and choosing the first coordinate axis in the direction of the knife edge. Denote the angular velocity of the body by ω =θ and the components of the linear velocity of the contact point relative to the body frame by v 1 , v 2 . The set (ω, v 1 , v 2 ) is regarded as an element of the Lie algebra se (2). Note that v 1 =ẋ cos θ +ẏ sin θ, v 2 =ẏ cos θ −ẋ sin θ.
The position of the center of mass is specified by the coordinates (a, b) relative to the body frame. Let m and J denote the mass and moment of inertia of the sleigh relative to the contact point. Then, the corresponding symmetric positive definite inertia operator I : se(2) → se(2) * and the reduced nonholonomic Lagrangian system (l, d) on se(2) are given by (see [28]). Thus, the Lagrange-d'Alembert equations for the system (l, d) arė Multidimensional generalizations of the Chaplygin sleigh were discussed in [28] (see also [57] and [70]).

Nonholonomic LR systems and right action Lie algebroids.
Here, we show how the reduction of a nonholonomic LR system produces a nonholonomic Lagrangian system on a right action Lie algebroid.
Let us start by recalling the definition of a right action Lie algebroid (see [32]). Let (F, [·, ·] F , ρ F ) be a Lie algebroid over a manifold N and π : M → N be a smooth map. A right action of F on π : M → N is a R-linear map (T m π)(Ψ(X)(m)) = ρ F (X(π(m))), for f ∈ C ∞ (N ), X, Y ∈ Sec(F ) and m ∈ M . If Ψ : Sec(E) → X(M ) is a right action of F on π : M → N and τ F : F → N is the vector bundle projection then the pullback vector bundle of F over π, , for X, Y ∈ Sec(E) and m ∈ M . The triple (E, [·, ·] E , ρ E ) is called the right action Lie algebroid of F over π and it is denoted by π Ψ F (see [32]).
Note that if the Lie algebroid F is a real Lie algebra g of finite dimension and π : M → {a point} is the constant map then a right action of g on π is just a right infinitesimal action Ψ : g → X(M ) of g on the manifold M . In such a case, the corresponding right action Lie algebroid is the trivial vector bundle pr 1 : M × g → M .
Next we recall the definition of a nonholonomic LR system following [26,35]. Let G be a compact connected Lie group with Lie algebra g and < ·, · >: g × g → R be an Ad G -invariant scalar product on g. Now, suppose that I : g → g is a inertia operator which is symmetric and definite positive with respect to the scalar product < ·, · >. Denote by g the left-invariant Riemannian metric given by for h ∈ G and v h , v ′ h ∈ T h G. Then, the Lagrangian function L : T G → R of the system is

4)
V : G → R being the potential energy. The constraint distribution D is a rightinvariant distribution on G. Thus, if e is the identity element of G and d = D e , we have that The nonholonomic Lagrangian system (L, D) on T G is called a nonholonomic LR system in the terminology of [26,35]. Note that, since L is a Lagrangian function of mechanical type, the system (L, D) is regular. Now, assume that is a Lie subalgebra of g, that S is a closed Lie subgroup of G with Lie algebra s and that the potential energy V is S-invariant.
Next, let us show that the nonholonomic LR system (L, D) may be reduced to a nonholonomic Lagrangian system on a right action Lie algebroid. In fact, consider the Riemannian homogeneous space M = S \ G and the standard transitive right action ψ of G on M = S \ G. Denote by Ψ : g → X(S \ G) the corresponding right infinitesimal action of g on S \ G. Then, Ψ induces a Lie algebroid structure on the trivial vector bundle pr 1 : On the other hand, using that the potential energy V is S invariant, we deduce that V induces a real function V : where π : G → S \ G is the canonical projection. Thus, we can introduce the Lagrangian functionL : S \ G × g → R on the action Lie algebroid pr 1 : Now, for every h ∈ G, we consider the subspace d(h) of g given by In particular, we have that Therefore, we can define a vector subbundle D of the Lie algebroid pr 1 : with h ∈ G and π(h) = h. Consequently, the pair ( L, D) is a nonholonomic Lagrangian system on the action Lie algebroid pr 1 : S \ G × g → S \ G.
In addition, we may prove the following result Proposition 6.1.
(1) If Φ : T G → S \ G × g is the map given by then Φ is a fiberwise bijective Lie algebroid morphism over π. Proof.
(1) Consider the standard (right) action r of G on itself As we know, the infinitesimal generator of r associated with an element ω of g is where ← − ω is the left-invariant vector field on G such that ← − ω (e) = ω. On the other hand, it is clear that the projection π : G → S \ G is equivariant with respect to the actions r and ψ. Thus, Therefore, if ρ : S/G × g → T (S \ G) is the anchor map of the Lie algebroid Furthermore, since we conclude that Φ is a Lie algebroid morphism over π.
In addition, it is obvious that if h ∈ G then is a linear isomorphism.
Next, we obtain the necessary and sufficient conditions for a curve ( h, ω) : I → S \ G × g to be a solution of the Lagrange-d'Alembert equations for the system ( L, D). Let ♭ <·,·> : g → g * be the linear isomorphism induced by the scalar product < ·, · >: g × g → R and I : g → g * be the inertia operator given by On the other hand, if h ′ ∈ S \ G we will denote by Ψh ′ : g → T e h ′ (S \ G) the linear epimorphism defined by In addition, if π(h ′ ) = h ′ , we identify the vector space D e h ′ with the vector subspace d(h ′ ) of g. Then, using (3.7), (6.7) and (6.11), we deduce that the curve ( h, ω) is a solution of the Lagrange-d'Alembert equations for the system ( L, D) if and only if˙ for all t, where D ⊥ e h(t) is the orthogonal complement of the vector subspace D e h(t) ⊆ g with respect to the scalar product < ·, · >. These equations will be called the reduced Poincaré-Chetaev equations.
We treat next a simple example of the above general situation.
The Veselova system. The most descriptive illustration of an LR system is the Veselova problem on the motion of a rigid body about a fixed point under the action of the nonholonomic constraint < ω, γ >= 0.
Here, ω is the vector of the angular velocity in the body frame, γ is a unit vector which is fixed in an space frame and < ·, · > denotes the standard scalar product in R 3 (see [66]). The Veselova system is an LR system on the Lie group G = SO(3) which is the configuration space of the rigid body motion. Thus, in this case, the Lie algebra g may be identified with R 3 and, under this identification, the Lie bracket [·, ·] g is the cross product × on R 3 . Moreover, the adjoint action of G = SO(3) on g ∼ = R 3 is the standard action of SO(3) on R 3 . This implies that < ·, · > is an Ad SO(3) -invariant scalar product on g ∼ = R 3 .
The vector subspace d of R 3 is just the orthogonal complement (with respect to < ·, · >) of a vector subspace < γ 0 > of dimension 1, with γ 0 a unit vector in R 3 , that is, is a Lie subalgebra of g ∼ = R 3 . Furthermore, the isotropy group S of γ 0 with respect to the adjoint action of G = SO(3), is a closed Lie subgroup with Lie algebra s. We remark that S is isomorphic to the circle S 1 .
Consequently, the corresponding homogeneous space M = S \ SO (3) is the orbit of the adjoint action of SO(3) on R 3 over the point γ 0 and, it is well-known that, such an orbit may be identified with the unit sphere S 2 . In fact, the map is a diffeomorphism (see, for instance, [50]). Under the above identification the (right) action of SO(3) on M = S \ SO(3) is just the standard (right) action of SO(3) on S 2 . Thus, our action Lie algebroid is the trivial vector bundle pr 1 : S 2 × R 3 → S 2 and the Lie algebroid structure on it is induced by the standard infinitesimal (right) action Ψ : R 3 → X(S 2 ) of the Lie algebra (R 3 , ×) on S 2 defined by Ψ(ω)(γ) = γ × ω, for ω ∈ R 3 and γ ∈ S 2 .
In the presence of a potentialṼ : γ → V (γ) the nonholonomic Lagrangian system ( L, D) on the Lie algebroid pr 1 : S 2 × R 3 → R 3 is given by where λ is the Lagrange multiplier. Since the system ( L, D) is regular, λ is uniquely determined. In fact, Eqs (6.12) and (6.13) are just the classical dynamical equations for the Veselova system (see [66]; see also [26,35]).

Semidirect product symmetry and left action Lie algebroids.
Here, we show how the reduction of some nonholonomic mechanical systems with semidirect product symmetry produces nonholonomic Lagrangian systems on left action Lie algebroids. Let us start by recalling the definition of a left action Lie algebroid (see [32]). Let (F, [·, ·] F , ρ F ) be a Lie algebroid over a manifold N and π : M → N be a smooth map. A left action of F on π : M → N is a R-linear map (T m π)(Ψ(X)(m)) = −ρ F (X(π(m))), for f ∈ C ∞ (N ), X, Y ∈ Sec(F ) and m ∈ M . If Ψ : Sec(E) → X(M ) is a left action of F on π : M → N and τ F : F → N is the vector bundle projection then the pullback vector bundle of F over π, for X, Y ∈ Sec(E) and m ∈ M . The triple (E, [·, ·] E , ρ E ) is called the left action Lie algebroid of F over π and it is denoted by F Ψ π (see [32]). Next, we consider a particular class of nonholonomic Lagrangian systems on left action Lie algebroids. Let V be a real vector space of finite dimension and · : G × V → V be a left representation of a Lie group G on V . We also denote by · : g × V → V the left infinitesimal representation of the Lie algebra g of G on V . Then, we can consider the semidirect Lie group S = G V with the multiplication The Lie algebra s of S is the semidirect product s = g V with the Lie bracket [·, ·] s : s × s → s given by for ω, ω ′ ∈ g andv,v ′ ∈ V . Here, [·, ·] g is the Lie bracket on g.
Moreover, we use the following notation. If v ∈ V then ρ v : g → V is the linear map defined by ρ v (ω) = ω · v, for ω ∈ g, and ρ * v : V * → g * is the dual map of ρ v : g → V . Now, let N be a smooth manifold. Then, it is clear that the product manifold F = s × T N is the total space of a vector bundle over N . Moreover, if (ω,v) ∈ s and X is a vector field on N then the pair ((ω,v), X) defines a section of the vector bundle τ F : F = s × T N → N . In fact, if {ω i } is a basis of g, {v j } is a basis of V and {X k } is a local basis of X(N ) then {((ω i , 0), 0), ((0,v j ), 0), ((0, 0), X k )} is a local basis of Sec(F ).
Next, suppose that v 0 is a point of V and that O v0 is the orbit of the action of G on V by v 0 , that is, Denote by π : M = N × O v0 → N the canonical projection on the first factor and by Ψ : Sec(F ) → X(M ) the left action of F on π, which is characterized by the following relation Then, we have the corresponding left action Lie algebroid τ E :  [ω α , ω β ] g = c γ αβ ω γ , ω α · u A = a B αA u B . Then, we have that c γ αβ a B γA = a C βA a B αC − a C αA a B βC . Next, we consider the local basis of sections {e i , e α , e A } of E given by Note that {e i , e α } is a local basis of sections of the constraint subbundle D. In addition, if (x i , v j ; y i , y α , y A ) are the local coordinates on E induced by the basis {e i , e α , e A }, it follows that and the rest of the fundamental Lie brackets are zero. Thus, a curve is a solution of the Lagrange-d'Alembert equations for the system (L, D) if and only ifẋ If we consider the local expression of the curve in the coordinates (x i , v j ;ẋ i , ω α , u A ) then, from (6.17), we deduce that the above equations are equivalent tȯ Nonholonomic Lagrangian systems, of the above type, on the left action Lie algebroid τ E : E = (s × T N ) × O v0 → M = N × O v0 may be obtained (by reduction) from an standard nonholonomic Lagrangian system with semidirect product symmetry.
In fact, let Q be the product manifold S × N and suppose that we have a Lagrangian functionL : T Q → R and a distributionD on Q whose characteristic spaceD ((g,v),n) ⊆ T g G×T v V ×T n N ≃ T g G×V ×T n N at the point ((g, v), n) ∈ S×N isD where v 0 is a fixed point of V . We can consider the natural left action of the Lie group S on Q and, thus, the left action A of the Lie subgroup H v0 = G v0 V of S on Q, where G v0 is the isotropy group of v 0 with respect to the action of G on V . The tangent lift T A of A is given by Using (6.19), it follows that the distributionD is invariant under the action T A of H v0 on T Q. Moreover, we will assume that the Lagrangian function is also H v0invariant. Therefore, we have a nonholonomic Lagrangian system (L,D) on the standard Lie algebroid T Q → Q which is H v0 -invariant. This type of systems were considered in [64].
Here, D is the vector subbundle of the vector bundle E whose fiber at the point (n, v) ∈ N × O v0 is given by (6.16). Proof.
(1) Suppose that ω 1 and ω 2 are elements of g, thatv 1 andv 2 are vectors of V and that X 1 and X 2 are vector fields on N . Then, we consider the vector fields Z 1 and Z 2 on Q defined by is the left-invariant vector field on G such that ← − ω 1 (e) = ω 1 (respectively, ← − ω 2 (e) = ω 2 ), e being the identity element of G.
A direct computation proves that Moreover, if ((ω 1 ,v 1 ), X 1 ) (respectively, ((ω 2 ,v 2 ), X 2 )) is the section of the vector bundle τ E : E → M induced by ω 1 ,v 1 and X 1 (respectively, ω 2 ,v 2 and X 2 ) then it is clear that Thus, using (6.14), it follows that On the other hand, we have that for (g, v, n) ∈ Q and (u g ,v, X n ) ∈ T g G × V × T n N ≃ T (g,v,n) Q. Therefore, from (6.15) and (6.21), we deduce that Consequently, using (6.22) and (6.23), we conclude that the pair (Φ, ϕ) is a Lie algebroid morphism. Note that one may choose a local basis {Z i } of vector fields on Q such that is a linear isomorphism.
The above theory may be applied to a particular example of a mechanical system: the Chaplygin Gyro (see [49,64]). This system consists of a Chaplygin sphere (that is, a ball with nonhomogeneous mass distribution) with a gyro-like mechanism, consisting of a gimbal and a pendulous mass, installed in it. The gimbal is a circle-like structure such that its center coincides with the geometric center of the Chaplygin sphere. It is free to rotate about the axis connecting the north and south poles of the Chaplygin sphere. The pendulous mass can move along the smooth track of the gimbal. For this particular example, the vector space V is R 3 , the Lie group G is SO(3) and the manifold N is R 2 . The action of SO(3) on R 3 is the standard one and v 0 = (0, 0, 1) is the advected parameter, see [64] for more details. We need some auxiliary properties of the splitting h and its prolongation. We first notice that h is an admissible map over the identity in M , because ρ E • h = id T M and T id M •ρ T M = id T M , but in general h is not a morphism. We can define the tensor K, a ker(ρ)-valued differential 2-form on M , by means of for every X, Y ∈ X(M ). It is easy to see that h is a morphism if and only if K = 0. In coordinates (x i ) in M , (x i , v i ) in T M , and linear coordinates (x i , y i , y A ) on E corresponding to a local basis {e i , e A } of sections of E adapted to the splitting h, we have that where Ω A ij are defined by [e i , e j ] = Ω A ij e A . Since h is admissible, its prolongation T h h is a well-defined map from T (T M ) to T E E. Moreover, it is an admissible map, which is a morphism if and only if h is a morphism. In what respect to the energy and the Cartan 1-form, we have that (T h h) ⋆ E L = EL and (T h h) ⋆ θ L = θL. Indeed, notice that by definition, On the other hand, for every Nevertheless, since h is not a morphism, and hence ( where J is the momentum map defined by L and Ker ρ and V, W ∈ T h(v) (T M ). The notation resembles the contraction of the momentum map J with the curvature tensor K. Instead of being symplectic, the map T h h satisfies Indeed, we have that and on a pair of projectable vector fields U, V projecting onto X, Y respectively, one can easily prove that from where the result follows by noticing that T h h • U is a projectable section and projects to h•X, and similarly is projectable and projects to K(X, Y ).
Let now Γ be the solution of the nonholonomic dynamics for (L, D), so that Γ satisfies the equation i Γ ω L − dE L ∈ D • and the tangency condition Γ D ∈ T D D. From this second condition we deduce the existence of a vector fieldΓ ∈ X(T M ) such that T h h•Γ = Γ• h. Explicitly, the vector fieldΓ is defined byΓ = T ρ ρ• Γ• h, from where it immediately follows thatΓ is a sode vector field on M .
Taking the pullback by T h h of the first equation we get ( Therefore, the vector fieldΓ is determined by the equations where T is the identity in T M considered as a vector field along the tangent bundle projection τ M (also known as the total time derivative operator). Equivalently we can write these equations in the form dΓθL − dL = J, K(T, · ) .
Note that ifā : I → T M is an integral curve ofΓ then a = h •ā : I → D is a solution of the constrained dynamics for the nonholonomic Lagrangian system (L, D) on E. Conversely, if a : I → D is a solution of the constrained dynamics then ρ • a : I → T M is an integral curve of the vector fieldΓ.
Finally we mention that extension of the above decomposition for non transitive Lie algebroids is under development.
Chaplygin systems and Atiyah algebroids. A particular case of the above theory is that of ordinary Chaplygin systems(see [4,7,17,37] and references there in). In such case we have a principal G-bundle π : Q → M = Q/G. Then, we may consider the quotient vector bundle E = T Q/G → M = Q/G and, it is well-known that, the space of sections of this vector bundle may be identified with the set of G-invariant vector fields on Q. Thus, using that the Lie bracket of two G-invariant vector fields is also G-invariant and the fact that a G-invariant vector field is π-projectable, we may define a Lie algebroid structure ([·, ·], ρ) on the vector bundle E = T Q/G → M = Q/G. The resultant Lie algebroid is called the Atiyah (gauge) algebroid associated with the principal bundle π : Q → M = Q/G (see [48]). Note that the canonical projection Φ : T Q → E = T Q/G is a fiberwise bijective Lie algebroid morphism. Now, suppose that (L Q , D Q ) is an standard nonholonomic Lagrangian system on T Q such that L Q is G-invariant and D Q is the horizontal distribution of a principal connection on π : Q → M = Q/G. Then, we have a reduced nonholonomic Lagrangian system (L, D) on E. In fact, L Q = L • Φ and Φ((D Q ) q ) = D π(q) , for all q ∈ Q. Moreover, ρ |D : D → T M = T (Q/G) is an isomorphism (over the identity of M) between the vector bundles D → M and T M → M . Therefore, we may apply the above general theory.
Next, we describe the nonholonomic Lagrangian system on the Atiyah algebroid associated with a particular example of a Chaplygin system: a two-wheeled planar mobile robot (see [17] and the references there in). Consider the motion of twowheeled planar mobile robot which is able to move in the direction in which it points and, in addition, can spin about a vertical axis. Let P be the intersection point of the horizontal symmetry axis of the robot and the horizontal line connecting the centers of the two wheels. The position and orientation of the robot is determined, with respect to a fixed Cartesian reference frame by (x, y, θ) ∈ SE(2), where θ ∈ S 1 is the heading angle, the coordinates (x, y) ∈ R 2 locate the point P and SE(2) is the group of Euclidean motions of the two-dimensional plane R 2 . Let ψ 1 , ψ 2 ∈ S 1 denote the rotation angles of the wheels which are assumed to be controlled independently and roll without slipping on the floor. The configuration space of the system is Q = T 2 × SE(2), where T 2 is the real torus of dimension 2.
The Lagrangian function L Q is the kinetic energy corresponding to the metric where m = m 0 + 2m 1 , m 0 is the mass of the robot without the wheels, J its momenta of inertia with respect to the vertical axis, m 1 the mass of each wheel, J 2 the axial moments of inertia of the wheels, and l the distance between the center of mass C of the robot and the point P . Thus, ). The constraints, induced by the conditions that there is no lateral sliding of the robot and that the motion of the wheels also consists of a rolling without sliding, areẋ sin θ −ẏ cos θ = 0, x cos θ +ẏ sin θ + cθ + Rψ 1 = 0, x cos θ +ẏ sin θ − cθ + Rψ 2 = 0, where R is the radius of the wheels and 2c the lateral length of the robot. The constraint distribution D is then spanned by , Note that if {ξ 1 , ξ 2 , ξ 3 } is the canonical basis of se (2), On the other hand, it is clear that Q = T 2 × SE(2) is the total space of a trivial principal SE(2)-bundle over M = T 2 . Moreover, the metric g Q is SE(2)invariant and D Q is the horizontal distribution of a principal connection on Q = Now, we consider the corresponding Atiyah algebroid Using the left-translations on SE(2), we have that the tangent bundle to SE(2) may be identified with the product manifold SE(2)×se (2) and, under this identification, the Atiyah algebroid is isomorphic to the trivial vector bundlẽ where τ T 2 : T T 2 → T 2 is the canonical projection. In addition, if ([·, ·], ρ) is the Lie algebroid structure onτ T 2 : T T 2 × se(2) → T 2 and { ∂ ∂ψ 1 , ∂ ∂ψ 2 , ξ 1 , ξ 2 , ξ 3 } is the canonical basis of sections ofτ T 2 : T T 2 × se(2) → T 2 then and the rest of the fundamental Lie brackets are zero. Denote by (ψ 1 , ψ 2 ,ψ 1 ,ψ 2 , ω 1 , ω 2 , ω 3 ) the (local) coordinates on T T 2 × se (2) induced by the basis { ∂ ∂ψ 1 , ∂ ∂ψ 2 , ξ 1 , ξ 2 , ξ 3 }. Then, the reduced Lagrangian L : and the constraint vector subbundle D is generated by the sections Since the system (L Q , D Q ) is regular on the standard Lie algebroid τ Q : T Q → Q, we deduce that the nonholonomic Lagrangian system (L, D) on the Atiyah algebroid τ T 2 : T T 2 × se(2) → T 2 is also regular. Now, as in Section 3, we consider a basis of sections ofτ T 2 : T T 2 × se(2) → T 2 which is adapted to the constraint subbundle D. This basis is The corresponding (local) coordinates on T T 2 ×se(2) are (ψ 1 , ψ 2 ,ψ 1 ,ψ 2 ,ω 1 ,ω 2 ,ω 3 ), where Therefore, using (3.7), we deduce that the Lagrange-d'Alembert equations for the system (L, D) arë where P , S and U are the real numbers On the other hand, the Lagrangian functionL : T T 2 → R on T T 2 is given bȳ

Nonlinearly constrained Lagrangian systems
We show in this section how the main results for linearly constrained Lagrangian systems can be extended to the case of Lagrangian systems with nonlinear nonholonomic constraints. This is true under the assumption that a suitable version of the classical Chetaev's principle in nonholonomic mechanics is valid (see e.g., [41] for the study of standard nonholonomic Lagrangian systems subject to nonlinear constraints).
Let τ : E → M be a Lie algebroid and M be a submanifold of E such that π = τ | M : M → M is a fibration. M is the constraint submanifold. Since π is a fibration, the prolongation T E M is well-defined. We will denote by r the dimension of the fibers of π : M → M , that is, r = dim M − dim M .
We define the bundle V → M of virtual displacements as the subbundle of τ * E of rank r whose fiber at a point a ∈ M is In other words, the elements of V are pairs of elements (a, b) ∈ E ⊕ E such that for every local constraint function φ.
We also define the bundle of constraint forces Ψ by Ψ = S * ((T E M) • ), in terms of which we set the Lagrange-d'Alembert equations for a regular Lagrangian function L ∈ C ∞ (E) as follows: which implies that a solution Γ of equations (7.1) is a sode section along M, that is, (SΓ − ∆)|M = 0. Note that the rank of the vector bundle (T E M) • → M is s = rank(E) − r and, since π is a fibration, the transformation S * : (T E M) • → Ψ defines an isomorphism between the vector bundles (T E M) • → M and Ψ → M. Therefore, the rank of Ψ is also s. Moreover, if a ∈ M we have Then, a direct computation proves that ζ ∈ V • a and S * (α a ) = ζ • T τ . Thus, we obtain Ψ a ⊆ { ζ • T τ | ζ ∈ V • a } and, using that the dimension of both spaces is s, we deduce (7.2) holds. Note that, in the particular case when the constraints are linear, we have V = τ * (D) and Ψ = D • .
Next, we consider the vector bundles F and T V M over M whose fibers at the point a ∈ M are Note that the dimension of T V a M is 2r and, when the constraints are linear, i.e., M is a vector subbundle D of E, we obtain where φ A are independent local constraint functions. Since π : M → M is a fibration, it follows that the matrix ( ∂φ A ∂y α ) is of rank s. Thus, if d is the differential of the Lie algebroid T E E → E, we deduce that {dφ A | M } A=1,...,s is a local basis of sections of the vector bundle (T E M) 0 → M. Note that ..,s is a local basis of sections of the vector bundle Ψ → M.
Next, we introduce the local sections {Z A } A=1,...,s of T E E → E defined by A direct computation, using (2.5), proves that where (W αβ ) is the inverse matrix of (W αβ = ∂ 2 L ∂y α y β ). Furthermore, it is clear that is regular. We are now ready to prove the following result.  (1) and (3) are equivalent. Moreover, proceeding as in the proof of Theorem 3.12, we deduce that the properties (2) and (3) (respectively, (2) and (4)) also are equivalent.
Remark 7.2 (Lagrangians of mechanical type). If L is a Lagrangian function of mechanical type, then, using Theorem 7.1, we deduce (as in the case of linear constraints) that the constrained system (L, M) is always regular.
for all a ∈ M. As in the case of linear constraints, we may prove the following. On the other hand, (4) in Theorem 7.1 is equivalent to (T E E)| M = T V M ⊕ (T V M) ⊥ and we will denote byP andQ the corresponding projectors induced by this decomposition, that is, Proof. Proceeding as in the proof of Lemma 3.11, we obtain that for all a ∈ M. Thus, it is clear that   Now, let (L, M) be a regular constrained Lagrangian system. In addition, suppose that f and g are two smooth functions on M and take arbitrary extensions to E denoted by the same letters. Then, as in Section 3.5, we may define the nonholonomic bracket of f and g as follows where X f and X g are the Hamiltonian sections on T E E associated with f and g, respectively.
Moreover, proceeding as in the case of linear constraints, one can prove thaṫ Thus, in the particular case when the restriction to M of the vector field ρ 1 (∆) on E is tangent to M, it follows thaṫ Alternatively, since T V M is an anchored vector bundle, we may consider the differentialdf ∈ Sec((T V M) * ) for a function f ∈ C ∞ (M). Thus, since the restriction ω L,M of ω L to T V M is regular, we have a unique sectionX f ∈ Sec(T V M) given by iX f ω L,M =df and it follows that   We will say that the constrained dynamics Γ (L ′ ,M ′ ) is the reduction of the constrained dynamics Γ (L,M) by the morphism Φ. As in the case of linear constraints (see Theorem 4.7), we also may prove the following result Theorem 7.9. Under the same hypotheses as in Theorem 7.8, we have that Now, let φ : Q → M be a principal G-bundle and τ : E → Q be a Lie algebroid over Q. In addition, assume that we have an action of G on E such that the quotient vector bundle E/G is defined and the set Sec(E) G of equivariant sections of E is a Lie subalgebra of Sec(E). Then, E ′ = E/G has a canonical Lie algebroid structure over M such that the canonical projection Φ : E → E ′ is a fiberwise bijective Lie algebroid morphism over φ (see Theorem 4.8).
Next, suppose that (L, M) is a G-invariant regular constrained Lagrangian system, that is, the Lagrangian function L and the constraint submanifold M are G-invariant. Then, one may define a Lagrangian function L ′ : E ′ → R on E ′ such that L = L ′ • Φ. Moreover, G acts on M and if the set of orbits M ′ = M/G of this action is a quotient manifold, that is, M ′ is a smooth manifold and the canonical projection Φ |M : M → M ′ = M/G is a submersion, then one may consider the constrained Lagrangian system (L ′ , M ′ ) on E ′ . Remark 7.10 (Quotient manifold). If M is a closed submanifold of E, then, using a well-known result (see [1,Theorem 4.1.20]), it follows that the set of orbits M ′ = M/G is a quotient manifold. • Since the orbits of the action of G on E are the fibers of Φ and M is G-invariant, we deduce that V a (Φ) ⊆ T a M, for all a ∈ M, which implies that Φ |Va : V a → V ′ Φ(a) is a linear isomorphism, for all a ∈ M. Thus, from Theorem 7.8, we conclude that the constrained Lagrangian system (L ′ , M ′ ) is regular and that where Γ (L,M) (resp., Γ (L ′ ,M ′ ) ) is the constrained dynamics for L (resp., L ′ ). In addition, using Theorem 7.9, we obtain that Φ : M → M ′ is an almost-Poisson morphism when on M and M ′ we consider the almost-Poisson structures induced by the corresponding nonholonomic brackets.
We illustrate the results above in a particular example in the following subsection.

7.4.
Example: a ball rolling on a rotating table. The following example is taken from [4,11,57]. A (homogeneous) sphere of radius r > 0, unit mass m = 1 and inertia about any axis k 2 , rolls without sliding on a horizontal table which rotates with constant angular velocity Ω about a vertical axis through one of its points. Apart from the constant gravitational force, no other external forces are assumed to act on the sphere. Choose a Cartesian reference frame with origin at the center of rotation of the table and z-axis along the rotation axis. Let (x, y) denote the position of the point of contact of the sphere with the table. The configuration space for the sphere on the table is Q = R 2 × SO(3), where SO(3) may be parameterized by the Eulerian angles θ, ϕ and ψ. The kinetic energy of the sphere is then given by T = 1 2 (ẋ 2 +ẏ 2 + k 2 (θ 2 +ψ 2 + 2φψ cos θ)), and with the potential energy being constant, we may put V = 0. The constraint equations areẋ − rθ sin ψ + rφ sin θ cos ψ = −Ωy, y + rθ cos ψ + rφ sin θ sin ψ = Ωx.
Since the Lagrangian function is of mechanical type, the constrained system is regular. Note that the constraints are affine, and hence not linear, and that the restriction to the constraint submanifold M of the Liouville vector field on T Q is not tangent to M. Indeed, the constraints are linear if and only if Ω = 0. Now, we can proceed from here to construct to equations of motion of the sphere, following the general theory. However, the use of the Eulerian angles as part of the coordinates leads to very complicated expressions. Instead, one may choose to exploit the symmetry of the problem, and one way to do this is by the use of appropriate quasi-coordinates (see [57]). First of all, observe that the kinetic energy may be expressed as T = 1 2 (ẋ 2 +ẏ 2 + k 2 (ω 2 x + ω 2 y + ω 2 z )), where ω x =θ cos ψ +φ sin θ sin ψ, ω y =θ sin ψ −φ sin θ cos ψ, ω z =φ cos θ +ψ, are the components of the angular velocity of the sphere. The constraint equations expressing the rolling conditions can be rewritten aṡ x − rw y = −Ωy, y + rω x = Ωx.
where R L is the vector field on M given by Note that R L = 0 if and only if Ω = 0. Now, it is clear that Q = R 2 ×SO(3) is the total space of a trivial principal SO(3)bundle over R 2 and the bundle projection φ : Q → M = R 2 is just the canonical projection on the first factor. Therefore, we may consider the corresponding Atiyah algebroid E ′ = T Q/SO(3) over M = R 2 . Next, we describe this Lie algebroid.

Conclusions and outlook
We have developed a geometrical description of nonholonomic mechanical systems in the context of Lie algebroids. This formalism is the natural extension of the standard treatment on the tangent bundle of the configuration space. The proposed approach also allows to deal with nonholonomic mechanical systems with symmetry, and perform the reduction procedure in a unified way. The main results obtained in the paper are summarized as follows: • we have identified the notion of regularity of a nonholonomic mechanical system with linear constraints on a Lie algebroid, and we have characterized it in geometrical terms; • we have obtained the constrained dynamics by projecting the unconstrained one using two different decompositions of the prolongation of the Lie algebroid along the constraint subbundle; • we have developed a reduction procedure by stages and applied it to nonholonomic mechanical systems with symmetry. These results have allowed us to get new insights in the technique of quasicoordinates; • we have defined the operation of nonholonomic bracket to measure the evolution of observables along the solutions of the system; • we have examined the setup of nonlinearly constrained systems; • we have illustrated the main results of the paper in several examples.
Current and future directions of research include the in-depth study of the reduction procedure following the steps of [4,8] for the standard case; the synthesis of socalled nonholonomic integrators [17,21,42] for systems evolving on Lie algebroids, and the development of a comprehensive treatment of classical field theories within the Lie algebroid formalism following the ideas by E. Martínez [54].