A GENERAL FRAMEWORK FOR NONHOLONOMIC MECHANICS: NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS

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1 A GENERAL FRAMEWORK FOR NONHOLONOMIC MECHANICS: NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS DAVID IGLESIAS, JUAN C. MARRERO, D. MARTÍN DE DIEGO, AND DIANA SOSA Abstract. This paper presents a geometric description of Lagrangian and Hamiltonian systems on Lie affgebroids subject to affine nonholonomic constraints. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. It is shown that one can define an almost aff-poisson bracket on the constraint AV-bundle, which plays a prominent role in the description of nonholonomic dynamics. Moreover, these developments give a general description of nonholonomic systems and the unified treatment permits to study nonholonomic systems after or before reduction in the same framework. Also, it is not necessary to distinguish between linear or affine constraints and the methods are valid for explicitly time-dependent systems. Contents I. Introduction 2 II. Lagrangian and Hamiltonian formalism on Lie affgebroids 4 II.1. Lie algebroids 4 II.2. The prolongation of a Lie algebroid over a fibration 5 II.3. Lie affgebroids 6 II.4. The Lagrangian formalism on Lie affgebroids 8 II.5. The Hamiltonian formalism 10 II.6. The Legendre transformation and the equivalence between the Lagrangian and Hamiltonian formalisms 14 III. Affinely constrained Lagrangian systems 16 IV. Solution of Lagrange-d Alembert equations 19 IV.1. Projectors 24 IV.2. The constrained Poincaré-Cartan 2-section 27 V. Constrained Hamiltonian Systems and the nonholonomic bracket 28 V.1. Constrained Hamiltonian Systems 28 V.2. The nonholonomic bracket 35 VI. Examples 41 VI.1. Lagrangian systems with linear nonholonomic constraints on a Lie algebroid 41 VI.2. Standard affine nonholonomic Lagrangian systems Mathematics Subject Classification. 17B66, 37J60, 53D17, 70F25, 70H33. Key words and phrases. Lie algebroids, Lie affgebroids, Lagrangian Mechanics, Hamiltonian Mechanics, Nonholonomic Mechanics, Lagrange-d Alembert equations, Projectors, AV-bundles, Aff-Poisson brackets, Nonholonomic brackets. 1

2 2 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA VI.3. A homogeneous rolling ball without sliding on a rotating table with timedependent angular velocity 44 VII. Conclusions and Future work 47 Acknowledgments 48 References 48 I. Introduction Nonholonomic constraints is one of the more fascinating and applied topics actually. First, there are many open questions related with this subject: characterization of the integrability, construction of geometric integrators, stabilization, controllability... But morever, there is a wide range of applications of this kind of systems in engineering, robotics... (see [2, 6, 29] and references therein). During the last years, many authors have studied in detail the geometry of nonholonomic systems. Some of them developed a geometric formalism for the most typical nonholonomic systems, the ones determined by a mechanical Lagrangian, that is, L(v q ) = 1 2 g(v q, v q ) V (q), v q T q Q, where V : Q R is the potential energy on the configuration space Q, g is a Riemannian metric on Q and, additionally, the system is subjected to linear constraints on the velocities, expressed as a nonintegrable distribution on Q [6, 19]. The usual formalism for these systems was the use of adequate projections of the Levi-Civita connection associated to g to obtain the equations of motion of the system. Other authors preferred to work on the tangent bundle of the configuration space, which permits to introduce more general Lagrangians and different type of constraints (linear or nonlinear). Usually these systems are specified by a Lagrangian function L : TQ R and a constraint submanifold M of TQ [15, 18]. Moreover, also a description in terms of a nonholonomic bracket was introduced [5, 13, 36] with considerable applications to reduction. In this sense, reduction of nonholonomic systems was intensively studied u- sing different geometric techniques. For instance, introducing modified Ereshmann connections (nonholonomic connection) [3], using a symplectic distribution on the constraint submanifold [1, 35] and distinguishing the different casuistic depending on the position of the Lie group of symmetries acting on the system and the nonholonomic distribution or in a Poisson context [22] (see [6] and references therein). Moreover, using jet bundle techniques, many authors studied the geometry of nonholonomic systems admitting an extension to explicitly time-dependent nonholonomic systems [4, 14, 16, 30, 33, 34] and ready for extension to nonholonomic field theories [37]. Today we find an scenario with a very rich theory but with an important lack: a general framework unifying the different casuistic (unreduced and reduced equations, systems subjected to linear or affine constraints, time-dependent or time-independent systems...). The aim of the present paper is cover this lack and present a geometrical framework covering the different cases. Obviously, in order to reach this objective it will be necessary to use some new and sophisticated techniques, in particular, the concept of a Lie affgebroid (an affine version of

3 NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS 3 a Lie algebroid) and appropriate versions of Lagrangian and nonholonomic mechanics in this setting. The previous and other motivations coming from other fields (topology, algebraic geometry...) have recently caused a lot of interest in the study of Lie algebroids [21], which in our setting can be thought of as generalized tangent bundles, since it allows to consider, in a unified formalism, mechanical systems on Lie algebras, integrable distributions, tangent bundles, quotients of tangent bundles by Lie groups when an action is considered... In this sense, Lie algebroids can be used to give unified geometric descriptions of Hamiltonian and Lagrangian Mechanics. In [38], A. Weinstein introduced Lagrangian systems on a Lie algebroid E M by means of the linear Poisson structure on the dual bundle E and a Legendre-type map from E to E. Motivated by this description, E. Martínez [23] developed a geometric formalism on Lie algebroids extending the Klein s formalism in ordinary Lagrangian Mechanics on tangent bundles. This line of research has been followed in [17, 24]. In fact, a geometric description of Lagrangian and Hamiltonian dynamics on Lie algebroids, in terms of Lagrangian submanifolds of symplectic Lie algebroids, was given in [17]. More recently, in [7] (see also [8, 27, 28]) a comprehensive treatment of Lagrangian systems on Lie algebroids subject to linear nonholonomic constraints was developed. The proposed formalism allows us to treat in a unified way a variety of situations for time-independent Lagrangian systems subject to linear nonholonomic constraints (systems with symmetry, nonholonomic LL systems, nonholonomic LR systems,...). On the other hand, in [9, 26] an affine version of the notion of a Lie algebroid structure was introduced. The resultant geometric object is called a Lie affgebroid structure. Lie affgebroid structures may be used to develop a time-dependent version of Lagrange and Hamilton equations on Lie algebroids (see [10, 12, 25, 26, 31]). In fact, one may obtain Lagrange and Hamilton equations on Lie affgebroids using a cosymplectic formalism [12, 25]. In the same setting of Lie affgebroids, one also may obtain the Hamilton equations using the notion of an aff-poisson structure on an AV-bundle (see [10, 12]). An AV-bundle is an affine bundle of rank 1 modelled on the trivial vector bundle and an aff-poisson bracket on an AV-bundle may be considered as an affine version of a Poisson bracket on a manifold (see [9]). The aim of this paper is develop a geometric description of Lagrangian systems subject to affine constraints using the Lie affgebroid theory. The general geometric framework proposed covers the most interesting previous methods in the literature. The paper is organized as follows. In Section II, we recall several constructions (which will be useful in the sequel) about the geometric description of Lagrangian and Hamiltonian Mechanics on Lie affgebroids and the equivalence between both formalisms in the hyperregular case. In Section III, we introduce the Lagrange-d Alembert equations for an affine nonholonomic Lagrangian system on a Lie affgebroid. In Section IV, we discuss the existence and uniqueness of solutions for this type of systems. Moreover, we prove that in the regular case the nonholonomic dynamics can be obtained by different projections from the unconstrained dynamics. In Section V, we develop the Hamiltonian description and we discuss the equivalence between the Lagrangian and Hamiltonian formalism. We also introduce the nonholonomic bracket which gives the evolution of an observable for the nonholonomic dynamics. The nonholonomic bracket is an almost aff-poisson bracket (an aff-poisson bracket which doesn t satisfy, in general, the Jacobi identity ) on the constraint AV-bundle. In Section VI, we apply the results obtained in

4 4 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA the paper to two particular cases: a linear nonholonomic Lagrangian system on a Lie algebroid and an standard affine nonholonomic Lagrangian system on the 1-jet bundle of local sections of a fibration τ : M R. As a consequence, we directly deduce some results obtained in [7, 14, 16] (see also [4]). In addition, we also analyze (in the Lie affgebroid setting) the reduced equations for a well-known example of an affine nonholonomic mechanical system: a homogeneous rolling ball without sliding on a rotating table with time-dependent angular velocity. For this purpose, we will use a particular class of Lie affgebroids, namely, Atiyah affgebroids. The paper ends with our conclusions and a description of future research directions. II. Lagrangian and Hamiltonian formalism on Lie affgebroids In this section we recall some well-known facts concerning the geometry of Lie algebroids and Lie affgebroids. II.1. Lie algebroids. Let E be a vector bundle of rank n over the manifold M of dimension m and τ E : E M be the vector bundle projection. Denote by Γ(τ E ) the C (M)-module of sections of τ E : E M. A Lie algebroid structure ([, ] E, ρ E ) on E is a Lie bracket [, ] E on the space Γ(τ E ) and a bundle map ρ E : E TM, called the anchor map, such that if we also denote by ρ E : Γ(τ E ) X(M) the homomorphism of C (M)-modules induced by the anchor map then [X, fy ] E = f [X, Y ] E + ρ E (X)(f)Y, for X, Y Γ(τ E ) and f C (M). The triple (E, [, ] E, ρ E ) is called a Lie algebroid over M (see [21]). In such a case, the anchor map ρ E : Γ(τ E ) X(M) is a homomorphism between the Lie algebras (Γ(τ E ), [, ] E ) and (X(M), [, ]). If (E, [, ] E, ρ E ) is a Lie algebroid, one may define a cohomology operator which is called the differential of E, d E : Γ( k τe ) Γ( k+1 τe ), as follows (d E µ)(x 0,..., X k ) = k ( 1) i ρ E (X i )(µ(x 0,..., X i,..., X k )) i=0 + i<j( 1) i+j µ([x i, X j ] E, X 0,..., X i,..., X j,..., X k ), (1) for µ Γ( k τ E ) and X 0,..., X k Γ(τ E ). Moreover, if X Γ(τ E ), one may introduce, in a natural way, the Lie derivative with respect to X, as the operator L E X : Γ( k τ E ) Γ( k τ E ) given by L E X = i X d E + d E i X. If E is the standard Lie algebroid TM then the differential d E = d TM is the usual exterior differential associated with M. On the other hand, if E is a real Lie algebra g of finite dimension then g is a Lie algebroid over a single point and the differential d g is the algebraic differential of the Lie algebra. Now, suppose that (E, [, ] E, ρ E ) and (E, [, ] E, ρ E ) are Lie algebroids over M and M, respectively, and that F : E E is a vector bundle morphism over the map f : M M. Then (F, f) is said to be a Lie algebroid morphism if d E ((F, f) φ ) = (F, f) (d E φ ), for φ Γ( k (τ E ) ) and for all k. Note that (F, f) φ is the section of the vector bundle k E M defined by ((F, f) φ ) x (a 1,..., a k ) = φ f(x) (F(a 1),..., F(a k )),

5 NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS 5 for x M and a 1,..., a k E x. If (F, f) is a Lie algebroid morphism, f is an injective immersion and F Ex : E x E f(x) is injective, for all x M, then (E, [, ] E, ρ E ) is said to be a Lie subalgebroid of (E, [, ] E, ρ E ). If we take local coordinates (x i ) on M and a local basis {e α } of sections of E, then we have the corresponding local coordinates (x i, y α ) on E, where y α (a) is the α-th coordinate of a E in the given basis. Such coordinates determine local functions ρ i α, Cγ αβ on M which contain the local information of the Lie algebroid structure, and accordingly they are called the structure functions of the Lie algebroid. They are given by ρ E (e α ) = ρ i α x i and [e α, e β ] E = C γ αβ e γ, and they satisfy the following relations ρ j ρ i β α x j ρ i ρj α β x j = ρi γ Cγ αβ If f C (M), we have that and cyclic(α,β,γ) [ ρ i C ν ] βγ α x i + C µ βγ Cν αµ = 0. d E f = f x i ρi αe α, where {e α } is the dual basis of {e α }. Moreover, if θ Γ(τ E ) and θ = θ γe γ it follows that d E θ = ( θ γ x i ρi β 1 2 θ αc α βγ )eβ e γ. II.2. The prolongation of a Lie algebroid over a fibration. In this section, we will recall the definition of the Lie algebroid structure on the prolongation of a Lie algebroid over a fibration (see [11, 17]). Let (E, [, ] E, ρ E ) be a Lie algebroid of rank n over a manifold M of dimension m with vector bundle projection τ E : E M and π : M M be a fibration. We consider the subset T E M of E TM and the map τ π E : T E M M defined by T E M = {(b, v ) E TM /ρ E (b) = (Tπ)(v )}, τ π E (b, v ) = π M (v ), where Tπ : TM TM is the tangent map to π and π M : TM M is the canonical projection. Then, τe π : T E M M is a vector bundle over M of rank n + dim M m which admits a Lie algebroid structure ([, ] π E, ρπ E ) characterized by [(X π, U ), (Y π, V )] π E = ([X, Y ] E π, [U, V ]), ρ π E (X π, U ) = U, for all X, Y Γ(τ E ) and U, V vector fields which are π-projectable to ρ E (X) and ρ E (Y ), respectively. (T E M, [, ] π E, ρπ E ) is called the prolongation of the Lie algebroid E over the fibration π or the E-tangent bundle to M (for more details, see [11, 17]). An element Z T E M is said to be vertical if its projection onto the first factor is zero. Therefore, it is of the form (0, v x ), with v x a tangent vector to M at x which is π-vertical. Next, we consider a particular case of the above construction. Let E be a Lie algebroid over a manifold M with vector bundle projection τ E : E M and T E E be the prolongation of E over the projection τe : E M. T E E is a Lie algebroid over E and we can define a canonical section λ E of the vector bundle (T E E ) E as follows. If a E and (b, v) Ta E E then λ E (a )(b, v) = a (b). (2)

6 6 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA λ E is called the Liouville section associated with the Lie algebroid E. Now, one may consider the nondegenerate 2-section Ω E = d T E E λ E of (T E E ) E. It is clear that d T E E Ω E = 0. In other words, Ω E is a symplectic section. Ω E is called the canonical symplectic section associated with the Lie algebroid E. Using the symplectic section Ω E one may introduce a linear Poisson structure Π E on E, with linear Poisson bracket {, } E given by {F, G} E = Ω E (X ΩE F, XΩE G ), for F, G C (E ), (3) where X ΩE F and X ΩE G are the Hamiltonian sections associated with F and G, that is, i X Ω E F Ω E = d T E E F and i Ω X E Ω E = d T E E G. G Suppose that (x i ) are local coordinates on an open subset U of M and that {e α } is a local basis of sections of the vector bundle τ 1 E (U) U as above. Then, {ẽ α, ē α } is a local basis of sections of the vector bundle (τ τ E E ) 1 ((τe ) 1 (U)) (τe ) 1 (U), where τ τ E E the vector bundle projection and ( ẽ α (a ) = e α (τe (a )), ρ i ) α x i, ē α (a ) = a ( 0, y α a ). : T E E E is Here, (x i, y α ) are the local coordinates on E induced by the local coordinates (x i ) and the dual basis {e α } of {e α }. Moreover, we have that and [ẽ α, ẽ β ] τ E E = Cαβẽγ, γ [ẽ α, ē β ] τ E E = [ē α, ē β ] τ E E = 0, ρ τ E E (ẽ α )= ρ i α x i, ρτ E E (ē α)=, y α λ E (x i, y α ) = y α ẽ α, Ω E (x i, y α ) = ẽ α ē α Cγ αβ y γẽ α ẽ β, (4) Π E = ρ i α x i 1 y α 2 Cγ αβ y γ. (5) y α y β (For more details, see [17, 24]). II.3. Lie affgebroids. Let τ A : A M be an affine bundle with associated vector bundle τ V : V M. Denote by τ A + : A + = Aff(A, R) M the dual bundle whose fibre over x M consists of affine functions on the fibre A x. Note that this bundle has a distinguished section 1 A Γ(τ A +) corresponding to the constant function 1 on A. We also consider the bidual bundle τ : à M whose fibre at x M is the vector space Ãx = (A + x ). Then, A may be identified with an affine subbundle of à via the inclusion i A : A à given by i A(a)(ϕ) = ϕ(a), which is an injective affine map whose associated linear map is denoted by i V : V Ã. Thus, V may be identified with a vector subbundle of Ã. Using these facts, one can prove that there is a one-to-one correspondence between affine functions on A and linear functions on Ã. On the other hand, there is an obvious one-to-one correspondence between affine functions on A and sections of A +. A Lie affgebroid structure on A consists of a Lie algebra structure [, ] V on the space Γ(τ V ) of the sections of τ V : V M, a R-linear action D : Γ(τ A ) Γ(τ V ) Γ(τ V ) of the sections of A on Γ(τ V ) and an affine map ρ A : A TM, the anchor map, satisfying the following conditions: D X [Ȳ, Z ] V = [D X Ȳ, Z ] V + [Ȳ, D X Z ] V, D X+ Ȳ Z = D X Z + [Ȳ, Z ] V,

7 NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS 7 D X (fȳ ) = fd XȲ + ρ A(X)(f)Ȳ, for X Γ(τ A ), Ȳ, Z Γ(τ V ) and f C (M) (see [9, 26]). If ([, ] V, D, ρ A ) is a Lie affgebroid structure on an affine bundle A then (V, [, ] V, ρ V ) is a Lie algebroid, where ρ V : V TM is the vector bundle map associated with the affine morphism ρ A : A TM. A Lie affgebroid structure on an affine bundle τ A : A M induces a Lie algebroid structure ([, ], ρ ) on the bidual bundle à such that 1 A Γ(τ A +) is a 1-cocycle in the corresponding Lie algebroid cohomology, that is, d 1 A = 0. Indeed, if X 0 Γ(τ A ) then for every section X of à there exists a function f C (M) and a section X Γ(τ V ) such that X = fx 0 + X and ρ (fx 0 + X) = fρ A (X 0 ) + ρ V ( X), [fx 0 + X, gx 0 + Ȳ ] = (ρ V ( X)(g) ρ V (Ȳ )(f) + fρ A(X 0 )(g) gρ A (X 0 )(f))x 0 + [ X, Ȳ ] V + fd X0 Ȳ gd X0 X. Conversely, let (U, [, ] U, ρ U ) be a Lie algebroid over M and φ : U R be a 1-cocycle of (U, [, ] U, ρ U ) such that φ Ux 0, for all x M. Then, A = φ 1 {1} is an affine bundle over M which admits a Lie affgebroid structure in such a way that (U, [, ] U, ρ U ) may be identified with the bidual Lie algebroid (Ã, [, ], ρ ) to A and, under this identification, the 1-cocycle 1 A : à R is just φ. The affine bundle τ A : A M is modelled on the vector bundle τ V : V = φ 1 {0} M. In fact, if i V : V U and i A : A U are the canonical inclusions, then i V [ X, Ȳ ] V = [i V X, i V Ȳ ] U, i V D X Ȳ = [i A X, i V Ȳ ] U, ρ A (X) = ρ U (i A X), for X, Ȳ Γ(τ V ) and X Γ(τ A ). Let τ A : A M be a Lie affgebroid modelled on the Lie algebroid τ V : V M. Suppose that (x i ) are local coordinates on an open subset U of M and that {e 0, e α } is a local basis of sections of τ : à M in U which is adapted to the 1-cocycle 1 A, i.e., such that 1 A (e 0 ) = 1 and 1 A (e α ) = 0, for all α. Note that if {e 0, e α } is the dual basis of {e 0, e α } then e 0 = 1 A. Denote by (x i, y 0, y α ) the corresponding local coordinates on Ã. Then, the local equation defining the affine subbundle A (respectively, the vector subbundle V ) of à is y 0 = 1 (respectively, y 0 = 0). Thus, (x i, y α ) may be considered as local coordinates on A and V. The standard example of a Lie affgebroid may be constructed as follows. Let τ : M R be a fibration and τ 1,0 : J 1 τ M be the 1-jet bundle of local sections of τ : M R. It is well known that τ 1,0 : J 1 τ M is an affine bundle modelled on the vector bundle π = (π M ) V τ : V τ M, where V τ is the vertical bundle of τ : M R. Moreover, if t is the usual coordinate on R and η is the closed 1-form on M given by η = τ (dt) then we have the following identification J 1 τ = {v TM/η(v) = 1} (see, for instance, [32]). Note that V τ = {v TM/η(v) = 0}. Thus, the bidual bundle J 1 τ to the affine bundle τ 1,0 : J 1 τ M may be identified with the tangent bundle TM to M and, under this identification, the Lie algebroid structure on π M : TM M is the standard Lie algebroid structure and the 1-cocycle 1 J 1 τ on π M : TM M is just the 1-form η.

8 8 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA II.4. The Lagrangian formalism on Lie affgebroids. In this section, we will develop a geometric framework, which allows to write the Euler-Lagrange equations associated with a Lagrangian function L on a Lie affgebroid A in an intrinsic way (see [26]). Suppose that (τ A : A M, τ V : V M, ([, ] V, D, ρ A )) is a Lie affgebroid over M. Then, the bidual bundle τ : Ã M to A admits a Lie algebroid structure ([, ], ρ ) in such a way that the section 1 A of the dual bundle A + is a 1-cocycle. Now, we consider the Lie algebroid prolongation (T A, [, ] τa e A, ρ τa e A ) of the Lie algebroid (Ã, [, ], ρ ) over the fibration τ A : A M with vector bundle projection τ τa : T A A. If (x i ) are local coordinates on an open subset U of M and {e 0, e α } is a local basis of sections of the vector bundle τ 1 (U) U adapted to 1 A, then { T 0, T α, Ṽα} is a local basis of sections of the vector bundle (τ τa ) 1 (τ 1 1 A (U)) τa (U), where ( T 0 (a) = e 0 (τ A (a)), ρ i ) ( 0, Tα (a) = e α (τ A (a)), ρ i ) ( ) α, Ṽ α (a) = 0,, (6) x i a x i a y α a (x i, y α ) are the local coordinates on A induced by the local coordinates (x i ) and the basis {e α } and ρ i 0, ρ i α are the components of the anchor map ρ. We also have that [ T 0, T α ] τa e A = C γ 0α T γ, [ T α, T β ] τa e A = C γ αβ T γ, [ T 0, Ṽα] τa e A = [ T α, Ṽβ ] τa e A = [Ṽα, Ṽβ ] τa e A = 0, ρ τa A e ( T 0 ) = ρ i 0 x i, ρτa A e ( T α ) = ρ i α x i, ρτa A e (Ṽα) = y α, where C γ 0β and Cγ αβ are the structure functions of the Lie bracket [, ] with respect to the basis {e 0, e α }. Note that, if { T 0, T α, Ṽ α } is the dual basis of { T 0, T α, Ṽα}, then T 0 is globally defined and it is a 1-cocycle. We will denote by φ 0 the 1-cocycle T 0. Thus, we have that φ 0 (a)( b, X a ) = 1 A ( b), for ( b, X a ) T e A a A. One may also consider the vertical endomorphism S : T A T A, as a section of the vector bundle T A (T A) A, whose local expression is (see [26]) S = ( T α y α φ 0 ) Ṽα. (8) A section ξ of τ τa : T A A is said to be a second order differential equation (SODE) on A if φ 0 (ξ) = 1 and S ξ = 0. If ξ Γ(τ τa ) is a SODE then ξ = T 0 + y α Tα + ξ α Ṽ α, where ξ α are local functions on A, and ρ τa A e (ξ) = (ρ i 0 + yα ρ i α ) x i + ξα y α. Now, a curve γ : I R A in A is said to be admissible if ρ i A γ = (7) (τ A γ) or, A equivalently, (i A (γ(t)), γ(t)) T e γ(t) A, for all t I, i A : A Ã being the canonical inclusion. We will denote by Adm(A) the space of admissible curves on A. Thus, if γ(t) = (x i (t), y α (t)), for all t I, then γ is an admissible curve if and only if dx i dt = ρi 0 + ρi α yα, for i {1,..., m}.

9 NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS 9 It is clear that if ξ is a SODE then the integral curves of the vector field ρ τa A e (ξ) are admissible. On the other hand, let L : A R be a Lagrangian function. Then, we introduce the Poincaré- Cartan 1-section Θ L Γ((τ τa ) ) and the Poincaré-Cartan 2-section Ω L Γ( 2 (τ τa ) ) associated with L defined by From (7), (8) and (9), we obtain that Θ L = Lφ 0 + (d T A L) S, Ω L = d T A Θ L. (9) Θ L = (L y α L y α)φ 0 + L y T α α, ( Ω L = i ξ0 (d T A ( L L yα)) y γ (Cγ 0α + Cγ βα yβ ) ρ i L ) α x i θ α φ 0 2 L + y α y β θα ψ β + 1 ( ρ i 2 L β 2 x i y α 2 L ρi α x i y β + L ) y γ Cγ αβ θ α θ β, where θ α = T α y α φ 0, ψ α = Ṽ α ξ0 αφ 0 and ξ 0 = T 0 + y α Tα + ξ0 αṽα is an arbitrary SODE. Now, a curve γ : I = ( ǫ, ǫ) R A in A is a solution of the Euler-Lagrange equations associated with L if and only if γ is admissible and i (ia(γ(t)), γ(t))ω L (γ(t)) = 0, for all t. If γ(t) = (x i (t), y α (t)) then γ is a solution of the Euler-Lagrange equations if and only if dx i ( ) dt = ρi 0 + d L ρi α yα, dt y α = ρ i L α x i + (Cγ 0α + Cγ βα yβ ) L y γ, (11) for i {1,...,m} and α {1,...,n}. If we denote by C(A + ) the set of curves in A +, we can define the Euler-Lagrange operator δl : Adm(A) C(A + ) by (10) δl γ(t) (ã) = Ω L (γ(t))((i A (γ(t)), γ(t)), (ã, v γ(t) )), (12) for γ Adm(A) and ã Ãτ A A(γ(t)), where v γ(t) T γ(t) A is such that (ã, v γ(t) ) T e γ(t) A, for all t. It is easy to prove that the map δl doesn t depend on the chosen tangent vector v γ(t). From (10), we deduce that its local expression is δl = ( d dt ( L y α ) + ρ i L α x i + (Cγ 0α + Cγ βα yβ ) L ) y γ (e α y α e 0 ), where {e 0, e α } is the dual basis of {e 0, e α }. Then the Euler-Lagrange differential equations read as δl = 0. ( 2 L ) The Lagrangian L is regular if and only if the matrix (W αβ ) = y α y β is regular or, in a intrinsic way, if the pair (Ω L, φ 0 ) is a cosymplectic structure on T A, that is, { Ω L... Ω } {{ L φ } 0 }(a) 0, d T A Ω L = 0 and d T A φ 0 = 0, for all a A. (n Note that the first condition is equivalent to the fact that the map L : T A (T A) defined by L (X) = i X Ω L + φ 0 (X)φ 0 is an isomorphism of vector bundles.

10 10 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA Now, suppose that the Lagrangian L is regular and let R L Γ(τ τa ) be the Reeb section of the cosymplectic structure (Ω L, φ 0 ) characterized by the following conditions i RL Ω L = 0 and i RL φ 0 = 1. Then, R L is the unique Lagrangian SODE associated with L, that is, the integral curves of the vector field ρ τa e A (R L ) are solutions of the Euler-Lagrange equations associated with L. In such a case, R L is called the Euler-Lagrange section associated with L and its local expression is R L = T 0 + y α Tα + W αβ( ρ i L β x i (ρi 0 + yγ ρ i γ ) 2 L x i y β + (Cγ 0β + yµ C γ µβ ) L )Ṽα y γ, where (W αβ ) is the inverse matrix of (W αβ ). II.5. The Hamiltonian formalism. In this section, we will develop a geometric framework, which allows to write the Hamilton equations associated with a Hamiltonian section on a Lie affgebroid (see [12, 25]). Suppose that (τ A : A M, τ V : V M, ([, ] V, D, ρ A )) is a Lie affgebroid. Now, consider the prolongation (T V, [, ] τ V, ρ τ V ) of the bidual Lie algebroid (Ã, [, ], ρ ) over the fibration τv : V M. Let (x i ) be local coordinates on an open subset U of M and {e 0, e α } be a local basis of sections of the vector bundle τ 1 (U) U adapted to 1 A and [e 0, e α ] = C γ 0α e γ, [e α, e β ] = C γ αβ e γ, ρ (e 0 ) = ρ i 0 x i, ρ A e(e α ) = ρ i α x i. Denote by (x i, y 0, y α ) the induced local coordinates on à and by (xi, y 0, y α ) the dual coordinates on the dual vector bundle τ A + : A + M to Ã. Then, (xi, y α ) are local coordinates on V and {ẽ 0, ẽ α, ē α } is a local basis of sections of the vector bundle τ τ V A e ( ẽ 0 (ψ) = e 0 (τv (ψ)), ) ( ρi 0, ẽ α (ψ) = e α (τv (ψ)), ρi α x i ψ x i ψ : T V V, where ) ( ), ē α (ψ) = 0,. y α ψ Using this local basis one may introduce local coordinates (x i, y α ; z 0, z α, v α ) on T V. A direct computation proves that [ẽ 0, ẽ β ] τ V = C γ 0βẽγ, [ẽ α, ẽ β ] τ V = C γ αβẽγ, [ẽ 0, ē α ] τ V = [ẽ α, ē β ] τ V = [ē α, ē β ] τ V = 0, ρ τ V (ẽ 0 ) = ρ i 0 x i, ρτ V (ẽ α) = ρ i α x i, ρτ V (ē α) =, y α for all α and β. Thus, if {ẽ 0, ẽ α, ē α } is the dual basis of {ẽ 0, ẽ α, ē α } then for f C (V ). d T V f = ρ i f 0 x iẽ0 + ρ i f α x i ẽα + f ē α, y α d T V ẽ γ = 1 2 Cγ 0αẽ0 ẽ α 1 2 Cγ αβẽα ẽ β, d T V ẽ 0 = d T V ē γ = 0,

11 NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS 11 Let µ : A + V be the canonical projection given by µ(ϕ) = ϕ l, for ϕ A + x, with x M, where ϕ l Vx is the linear map associated with the affine map ϕ and h : V A + be a Hamiltonian section of µ. Now, we consider the Lie algebroid prolongation T A + of the Lie algebroid à over τ A + : A + M with vector bundle projection τ τ A + : T A + A + (see Section II.2). Then, we may introduce the map T h : T V T A + defined by T h(ã, X α ) = (ã, (T α h)(x α )), for A (ã, X α ) T e α V, with α V. It is easy to prove that the pair (T h, h) is a Lie algebroid morphism between the Lie algebroids τ τ V : T A e V V and τ τ A + : T A + A +. Next, denote by λ h and Ω h the sections of the vector bundles (T V ) V and Λ 2 (T V ) V given by λ h = (T h, h) (λ ), Ω h = (T h, h) (Ω ), (13) where λ and Ω are the Liouville section and the canonical symplectic section, respectively, associated with the Lie algebroid Ã. Note that Ω h = d T V λ h. On the other hand, let η : T V R be the section of (T V ) V defined by η(ã, X ν ) = 1 A (ã), (14) A for (ã, X ν ) T e ν V, with ν V. Note that if pr 1 : T V à is the canonical projection on the first factor then (pr 1, τv ) is a morphism between the Lie algebroids ττ V : T A e V V and τ : à M and (pr 1, τv ) (1 A ) = η. Thus, since 1 A is a 1-cocycle of τ : à M, we deduce that η is a 1-cocycle of the Lie algebroid τ τ V e A : T V V. Suppose that h(x i, y α ) = (x i, H(x j, y β ), y α ) and that {ẽ 0, ẽ α, ē α } is the dual basis of {ẽ 0, ẽ α, ē α }. Then η = ẽ 0 and, from (4), (13) and the definition of the map T h, it follows that Ω h = ẽ α ē α Cγ αβ y γẽ α ẽ β + (ρ i H α x i Cγ 0α y γ)ẽ α ẽ 0 + H ē α ẽ 0. (15) y α Thus, it is easy to prove that the pair (Ω h, η) is a cosymplectic structure on the Lie algebroid τ τ V : T A e V V. Remark II.1. Let T V V be the prolongation of the Lie algebroid V over the projection τv : V M. Denote by λ V and Ω V the Liouville section and the canonical symplectic section, respectively, of V and by (i V, Id) : T V V T V the canonical inclusion. Then, using (2), (13), (14) and the fact that µ h = Id, we obtain that (i V, Id) (λ h ) = λ V, (i V, Id) (η) = 0. Thus, since (i V, Id) is a Lie algebroid morphism, we also deduce that (i V, Id) (Ω h ) = Ω V. Now, let R h Γ(τ τ V e A ) be the Reeb section of the cosymplectic structure (Ω h, η) characterized by the following conditions i Rh Ω h = 0 and i Rh η = 1.

12 12 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA With respect to the basis {ẽ 0, ẽ α, ē α } of Γ(τ τ V e A ), R h is locally expressed as follows: R h = ẽ 0 + H ẽ α (C γ αβ y y H γ + ρ i H α α y β x i Cγ 0α y γ)ē α. (16) Thus, the vector field ρ τ V (R h ) is locally given by ρ τ V (R h ) = (ρ i 0 + H ρi α ) ( y α x i + ρ i H α x i + y γ(c γ 0α + H ) Cγ βα ) (17) y β y α and the integral curves of R h (i.e., the integral curves of ρ τ V (R h )) are just the solutions of the Hamilton equations for h, dx i dt = ρi 0 + H ρi α, y α dy α dt = ρ i H α x i + y γ(c γ 0α + Cγ βα H ), y β for i {1,...,m} and α {1,...,n}. Next, we will present an alternative approach in order to obtain the Hamilton equations. For this purpose, we will use the notion of an aff-poisson structure on an AV-bundle which was introduced in [9] (see also [10]). Let τ Z : Z M be an affine bundle of rank 1 modelled on the trivial vector bundle τ M R : M R M, that is, τ Z : Z M is an AV-bundle in the terminology of [10]. Then, we have an action of R on the fibres of Z. This action induces a vector field X Z on Z which is vertical with respect to the projection τ Z : Z M. On the other hand, there exists a one-to-one correspondence between the space of sections of τ Z : Z M, Γ(τ Z ), and the set {F h C (Z)/X Z (F h ) = 1}. In fact, if h Γ(τ Z ) and (x i, s) are local fibred coordinates on Z such that X Z = and h is s locally defined by h(x i ) = (x i, H(x i )), then the function F h on Z is locally given by F h (x i, s) = H(x i ) s, (18) (for more details, see [10]). Now, an aff-poisson structure on the AV-bundle τ Z : Z M is a bi-affine map which satisfies the following properties: {, } : Γ(τ Z ) Γ(τ Z ) C (M) i) Skew-symmetric: {h 1, h 2 } = {h 2, h 1 }. ii) Jacobi identity: {h 1, {h 2, h 3 }} V + {h 2, {h 3, h 1 }} V + {h 3, {h 1, h 2 }} V = 0, where {, } V is the affine-linear part of the bi-affine bracket. iii) If h Γ(τ Z ) then is an affine derivation. {h, } : Γ(τ Z ) C (M), h {h, h },

13 NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS 13 Condition iii) implies that, for each h Γ(τ Z ) the linear part {h, } V : C (M) C (M) of the affine map {h, } : Γ(τ Z ) C (M) defines a vector field on M, which is called the Hamiltonian vector field of h (see [10]). In [10], the authors proved that there is a one-to-one correspondence between aff-poisson brackets {, } on τ Z : Z M and Poisson brackets {, } Π on Z which are X Z -invariant, i.e., which are associated with Poisson 2-vectors Π on Z such that L XZ Π = 0. This correspondence is determined by {h 1, h 2 } τ Z = {F h1, F h2 } Π, for h 1, h 2 Γ(τ Z ). Note that the function {F h1, F h2 } Π on Z is τ Z -projectable, i.e., L XZ {F h1, F h2 } Π = 0 (because the Poisson 2-vector Π is X Z -invariant). Using this correspondence we will prove the following result. Theorem II.2. [12] Let τ A : A M be a Lie affgebroid modelled on the vector bundle τ V : V M. Denote by τ A + : A + M (resp., τv : V M) the dual vector bundle to A (resp., to V ) and by µ : A + V the canonical projection. Then: i) µ : A + V is an AV-bundle which admits an aff-poisson structure. ii) If h : V A + is a Hamiltonian section (that is, h Γ(µ)) then the Hamiltonian vector field of h with respect to the aff-poisson structure is a vector field on V whose integral curves are just the solutions of the Hamilton equations for h. Proof. i) It is clear that µ : A + V is an AV-bundle. In fact, if a + A + x, with x M, and t R then a + + t = a + + t1 A (x). Thus, the µ-vertical vector field X A + on A + is just the vertical lift 1 V A of the section 1 A Γ(τ A +). Moreover, one may consider the Lie algebroid τ : Ã = (A + ) M and the corresponding linear Poisson 2-vector Π A + on A +. Then, using the fact that 1 A is a 1-cocycle of τ : Ã = (A + ) M, it follows that the Poisson 2-vector Π A + is X A +-invariant. Therefore, Π A + induces an aff-poisson structure {, } on µ : A + V which is characterized by the condition {h 1, h 2 } µ = {F h1, F h2 } ΠA +, for h 1, h 2 Γ(µ). (19) One may also prove this first part of the theorem using the relation between special Lie affgebroid structures on an affine bundle A and aff-poisson structures on the AV-bundle AV ((A ) ) (see Theorem 23 in [10]). ii) From (18) and (19), we deduce that the linear map {h, } V : C (V ) C (V ) associated with the affine map {h, } : Γ(µ) C (V ) (that is, the Hamiltonian vector field of h) is given by {h, } V (ϕ) µ = {F h, ϕ µ} ΠA +, for ϕ C (V ). (20) Now, suppose that the local expression of h is On the other hand, using (5), we have that h(x i, y α ) = (x i, H(x j, y β ), y α ). (21) Π A + = ρ i 0 x i + ρ i α y 0 x i C γ 0α y y γ 1 α y 0 y α 2 Cγ αβ y γ. (22) y α y β

14 14 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA Thus, from (20), (21) and (22), we conclude that the Hamiltonian vector field of h is locally given by (ρ i 0 + ρ i H α ) ( y α x i + ρ i H α x i + y γ(c γ 0α + H ) Cγ βα ) y β y α which proves our result (see (17)). II.6. The Legendre transformation and the equivalence between the Lagrangian and Hamiltonian formalisms. Let L : A R be a Lagrangian function and Θ L Γ((τ τa ) ) be the Poincaré-Cartan 1-section associated with L. We introduce the extended Legendre transformation associated with L as the smooth map Leg L : A A + defined by Leg L (a)(b) = Θ L (a)(z), A for a, b A x, where z T e a A is such that pr 1 (z) = i A (b), pr 1 : T A Ã being the restriction to T A of the first canonical projection pr 1 : Ã TA Ã. The map Leg L is well-defined and its local expression in fibred coordinates on A and A + is Leg L (x i, y α ) = (x i, L L y αyα, L yα). (23) Thus, we can define the Legendre transformation associated with L, leg L : A V, by leg L = µ Leg L. From (23) and since µ(x i, y 0, y α ) = (x i, y α ), we have that leg L (x i, y α ) = (x i, L yα). (24) The maps Leg L and leg L induce the maps T Leg L : T A T A + and T leg L : T A T V defined by (T Leg L )( b, X a ) = ( b, (T a Leg L )(X a )), (T leg L )( b, X a ) = ( b, (T a leg L )(X a )), (25) for a A and ( b, X a ) T e A a A. Now, let { T 0, T α, Ṽα} (respectively, {ẽ 0, ẽ α, ē 0, ē α }) be a local basis of Γ(τ τa II.4 (respectively, of Γ(τ τ A + ) as in Section ) as in Section II.2) and denote by (xi, y α ; z 0, z α, v α ) (respectively, (x i, y 0, y α ; z 0, z α, v 0, v α )) the corresponding local coordinates on T A (respectively, T A + ). In addition, suppose that (x i, y α ; z 0, z α, v α ) are local coordinates on T V as in Section II.5. Then, from (23), (24) and (25), we deduce that the local expression of the maps T Leg L and T leg L is T Leg L (x i, y α ; z 0, z α, v α )=(x i, L L y αyα, L y α;z0, z α, z 0 ρ i 0( L x i 2 L x i y αyα ) +z α ρ i α( L x i 2 L x i y β yβ ) v α 2 L y α y β yβ, z 0 ρ i 2 L 0 x i y α +z β ρ i 2 L β x i y α + 2 L vβ y α y β ), T leg L (x i, y α ; z 0, z α, v α ) = (x i, L y α;z0, z α, z 0 ρ i 2 L 0 x i y α + zβ ρ i 2 L β x i y α +v β 2 L y α y β ). Thus, using (4), (10), (25) and (26), we can prove the following result. (26) (27)

15 NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS 15 Theorem II.3. [12] The pair (T Leg L, Leg L ) is a morphism between the Lie algebroids (T A, [, ] τa A e, ρ τa A e ) and (T A +, [, ] τ A +, ρτ A + ). Moreover, if Θ L and Ω L (respectively, λ and Ω ) are the Poincaré-Cartan 1-section and 2-section associated with L (respectively, the Liouville 1-section and the canonical symplectic section associated with Ã) then From (24), it follows (T Leg L, Leg L ) (λ ) = Θ L, (T Leg L, Leg L ) (Ω ) = Ω L. (28) Proposition II.4. [12] The Lagrangian L is regular if and only if the Legendre transformation leg L : A V is a local diffeomorphism. Next, we will assume that L is hyperregular, that is, leg L is a global diffeomorphism. Then, from (25) and Theorem II.3, we conclude that the pair (T leg L, leg L ) is a Lie algebroid isomorphism. Moreover, we can consider the Hamiltonian section h : V A + defined by h = Leg L leg 1 L, (29) the corresponding cosymplectic structure (Ω h, η) on the Lie algebroid τ τ V A e the Hamiltonian section R h Γ(τ τ V ). A e Using (13), (27), (28), (29) and Theorem II.3, we deduce that : T V V and Theorem II.5. [12] If the Lagrangian L is hyperregular then the Euler-Lagrange section R L associated with L and the Hamiltonian section R h associated with h satisfy the following relation R h leg L = T leg L R L. Moreover, if γ : I A is a solution of the Euler-Lagrange equations associated with L, then leg L γ : I V is a solution of the Hamilton equations associated with h and, conversely, if γ : I V is a solution of the Hamilton equations for h then γ = leg 1 L γ is a solution of the Euler-Lagrange equations for L. Note that if the local expression of the inverse of the Legendre transformation leg L : A V is leg 1 L (xi, y β ) = (x i, y α (x j, y β )), then the Hamiltonian section h is locally given by where the function H is h(x i, y α ) = (x i, H(x j, y β ), y α ), H(x i, y α ) = y β y β (x i, y α ) L(x i, y β (x i, y α )). Thus, Therefore, H x i = L (x j,y β ) x i, (x j,y α (x j,y β )) H = y β (x i, y α ). (30) y β (xi,y α) leg 1 L (xi, y α ) = (x i, H y α ). (31)

16 16 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA III. Affinely constrained Lagrangian systems We start with a free Lagrangian system on a Lie affgebroid A of rank n. Now, we plug in some nonholonomic affine constraints described by an affine subbbundle B of rank n r of the bundle A of admissible directions, that is, we have an affine bundle τ B : B M with associated vector bundle τ UB : U B M and the corresponding inclusions i B : B A and i UB : U B V, τ V : V M being the vector bundle associated with the affine bundle τ A : A M. If we impose to the admissible solution curves γ(t) the condition to stay on the manifold B, we arrive at the equations δl γ(t) = λ(t) and γ(t) B, where the constraint force λ(t) A + τ A(γ(t)) is to be determined. In the standard case (A = J 1 τ, τ : M R being a fibration), λ takes values in the annihilator of the constraint submanifold B. Therefore, in the case of a general Lie affgebroid, the natural Lagrange-d Alembert equations one should pose are δl γ(t) B τ B(γ(t)) and γ(t) B, where B = {ϕ A + /ϕ B 0} is the affine annihilator of B which is a vector subbundle of A + with rank equal to r. In more explicit terms, we look for curves γ(t) A such that - γ Adm(A), - γ(t) B τa(γ(t)), - there exists λ(t) B τ A(γ(t)) such that δl γ(t) = λ(t). If γ(t) is one of such curves, then (γ(t), γ(t)) is a curve in T A with i (γ(t), γ(t)) φ 0 = 1. Moreover, since γ(t) is in B, we have that γ(t) is tangent to B, that is, (γ(t), γ(t)) T B with i (γ(t), γ(t)) φ 0 = 1. Note that, since B is an affine subbundle of A, the Lie algebroid prolongation τ τb : T B B of the Lie algebroid à over the fibration τ B : B M is well-defined. Under some regularity conditions (to be made precise later on), we may assume that the above admissible curves are integral curves of a section X which we will assume that it is a SODE section taking values in T B. Then, from (12), we deduce that (i X Ω L )(b) B b, for all b B, where B is the vector bundle over B whose fibre at the point b B is B b = {α (T e A b A) / α(b, v b ) = 0, (b, v b ) T e A b A such that b B τa(b)}. Based on the previous arguments, we may reformulate geometrically our problem as the search for such a SODE X (defined at least on a neighborhood of B) satisfying i X(b) Ω L (b) B b, i X(b) φ 0 (b) = 1 and X(b) T e A b B, at every point b B. Now, we will prove the following result. Proposition III.1. If S is the vertical endomorphism then the vector bundles over B, B B and S (T B) B, are equal. In other words, Proof. First, we will see that the map B b = S (T e A b B), for all b B. S : (T e A b B) S (T e A b B)

17 NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS 17 is a linear isomorphism. In fact, suppose that α (T e A b B) and S α = 0. Then, we have that α(0, v b ) = 0, for all v b T b A such that (T b τ A )(v b ) = 0. Moreover, if (ã, u b ) T e A b A then, since T b τ A : T b B T τa(b)m is a linear epimorphism, it follows that there exists v b T b B satisfying Thus, using that α (T e A b B), we deduce that (T b τ A )(v b ) = ρ (ã). α(ã, u b ) = α(ã, v b ) + α(0, u b v b ) = 0. Therefore, α = 0. This implies that S : (T e A b B) S (T e A b B) is a linear isomorphism and dim S (T e A b B) = r. On the other hand, since T b τ A : T b B T τa(b)m is a linear epimorphism, we obtain that dim B b = r = dim S (T e A b B). Finally, we will prove that S A (T e b B) B b. A In fact, if α (T e b B), (b A, v b ) T e b A and b B τb(b), it follows that S(b, v b ) = (0, v b), with v b T b B. Consequently, (S α)(b, v b ) = 0. Proposition III.1 suggests us to introduce the following definition. Definition III.2. A nonholonomically constrained Lagrangian system on a Lie affgebroid A is a pair (L, B), where L : A R is a C -Lagrangian function, and i B : B A is a smooth affine subbundle of A, called the constraint affine subbundle. By a solution of the nonholonomically constrained Lagrangian system (L, B) we mean a section X Γ(τ τa ) which is a SODE on B and satisfies the Lagrange-d Alembert equations (i X Ω L ) B Γ(τ S (T B) ), (i X φ 0 ) B = 1, (32) X B Γ(τ τb ), where τ S (T B) is the vector bundle projection of S (T B) B. 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.

18 18 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA Remark III.3. We want to stress that a solution of the Lagrange-d Alembert equations needs to be defined only over B, but for practical purposes we consider it extended to A (or just to a neighborhood of B in A). We will not make any notational distinction between a solution on B and any of its extensions. Solutions which coincide on B will be considered as equal. In accordance with this convention, by a SODE on B we mean a section of T B which is the restriction to B of some SODE defined in a neighborhood of B. Now, we will analyze the form of the Lagrange-d Alembert equations in local coordinates. Let (x i ) be local coordinates on an open subset U of M and {e 0, e α } be a local basis of sections of the vector bundle τ 1 (U) U adapted to 1 A. Denote by (x i, y 0, y α ) the corresponding local coordinates on à and suppose that {µa α eα } is a local basis of the vector bundle UB M and that e B 0 is a local section of the affine bundle B M such that e 0 e B 0 = a α 0 e α. Then, using that an element e 0 (x)+y α e α (x) of A x belongs to B x if and only if (e 0 (x)+y α e α (x)) e B 0 (x) (U B) x, we deduce that the local equations defining the constrained subbundle B as an affine subbundle of A are Ψ a = µ a 0 + µa α yα = 0, for all a {1,...,r}, where µ a 0 = µa α aα 0. Observe that dt A Ψ a Γ((τ τa e A ) ) verifies that (d T A Ψ a )(b)(ã, X b ) = 0, for all (ã, X b ) T e A b B, and, therefore, (d T A Ψ a A )(b) (T e b B), for every b B and a {1,...,r}. On the other hand, using (1) and (7), we deduce that ( µ a 0 d T A Ψ a = ρ i 0 x i + yβ µa β x i ) φ 0 + ρ i α {φ 0, T α, Ṽ α } being the dual basis of { T 0, T α, Ṽα}. ( µ a 0 x i + yβ µa β x i ) T α + µ a αṽ α, Thus, since the matrix (µ a α) has maximum rank r, it follows that the sections {d T A Ψ a } a=1,...,r are linearly independent and, using that rank(t B) = r, we deduce that {(d T A Ψ a ) B } a=1,...,r is a local basis of sections of (T B) B. Therefore, using (8) and the fact that S : (T e A b B) S (T e A b B) is an isomorphism, for all b B, we obtain that {µ a α θα } a=1,...,r is a local basis of sections of S (T B) B, where θ α = T α y α φ 0. Consequently, the constrained motion equations (32) can be written as i X Ω L = λ a µ a α θα, i X φ 0 = 1, (d T A Ψ a )(X) = 0, along the points of B, where λ a are some Lagrange multipliers to be determined. Now, a section X Γ(τ τa ) is of the form X = z0 T0 + z α Tα + v α Ṽ α. If we assume that the Lagrangian L is regular, then the first two equations in (33) imply that any solution X has to be a SODE, that is, X = T 0 + y α Tα + X α Ṽ α and, then, the first equation in (33) becomes (see (10)) X α 2 L y α y β + (ρi 0 + yγ ρ i γ ) 2 L x i y β L ρi β x i (Cγ 0β + yν C γ νβ ) L y γ + λ aµ a β = 0. (33)

19 NONHOLONOMIC SYSTEMS ON LIE AFFGEBROIDS 19 Thus, if R L is the Euler-Lagrange section associated with L, X = R L W αβ λ a µ a βṽα and then, the third equation in (33) implies that (d T A Ψ a )(R L ) + λ b (d T A Ψ a )( W αβ µ b βṽα) = 0, for all a {1,..., r}. As a consequence, we get that there exists a unique solution of the Lagrange-d Alembert equations if and only if the matrix C ab (b) = ( W αβ µ b βµ a α)(b) (34) is regular, for all b B. From (33), we deduce that the differential equations for the integral curves of the vector field ρ τb A e (X) are the Lagrange-d Alembert differential equations, which read dx i ( ) d L dt y α dt = ρi 0 + ρi α yα, ρ i L α x i + (Cγ α0 + Cγ αβ yβ ) L y γ = λ aµ a α, µ a 0 + µ a αy α = 0, with i {1,...,m}, α {1,..., n} and a {1,...,r}. Remark III.4. In the above discussion, we obtained a complete set of differential equations that determine the dynamics. Now, we will analyze the form of the Lagrange-d Alembert equations in terms of adapted coordinates to B. Consider local coordinates (x i ) on a open set U of M and the local basis {φ A } A=1,...,n r of sections of U B. Complete it to a local basis of sections {e 0, φ A, φ a } of the vector bundle τ 1 (U) U adapted to 1 A. Then, in coordinates (x i, y 0, y A, y a ) adapted to this basis, the equations defining the constrained subbundle B as an affine subbundle of A (respectively, as an affine subbundle of Ã) are ya = 0 (respectively, y 0 = 1 and y a = 0). Thus, the Lagrange-d Alembert differential equations read as dx i dt = ρi 0 + ρ i Ay A, d dt ( L y A ) L ρi A x i + (Cγ A0 + Cγ AB yb ) L y γ = 0, y a = 0. (35) IV. Solution of Lagrange-d Alembert equations In this section, we will perform a precise global analysis of the existence and uniqueness of the solution of Lagrange-d Alembert equations. In what follows, we will assume that the Lagrangian L is regular at least in a neighborhood of B. Definition IV.1. A constrained Lagrangian system (L, B) is said to be regular if the Lagranged Alembert equations have a unique solution.

20 20 D. IGLESIAS, J. C. MARRERO, D. MARTÍN DE DIEGO, AND D. SOSA In order to characterize geometrically these nonholonomic systems which are regular, we define the vector subbundle F T A B B whose fibre at point b B is F b = 1 L (S A (T e b B) ), where L : T A (T A) is the vector bundle isomorphism defined by More explicitly, L (X) = i X Ω L + φ 0 (X)φ 0, for all X T A. (36) F b = {X T e A b A/exists χ a s.t. L (X) = χ a µ a αθ α b }. From the definition, it is clear that the rank of F is rank(f) = rank(t B) = rank(t A) rank(t B) = rank(b ) = r. If we consider the sections Z a Γ(τ τa ) such that L(Z a ) = µ a α θα, with a {1,..., r}, then {Z a } is a local basis of sections of F. Moreover, if R L is the Euler- Lagrange section associated with L, we have that (i Za Ω L )(R L ) + φ 0 (Z a )φ 0 (R L ) = µ a αθ α (R L ) = 0 which implies that φ 0 (Z a ) = 0. Therefore, Z a is completely characterized by the conditions i Za Ω L = µ a α θα and i Za φ 0 = 0. In addition, using these two conditions, we conclude that the local expression of Z a is the following Z a = W αβ µ a βṽα, (37) where (W αβ ) is the inverse matrix of (W αβ ). Thus, the matrix defined in (34) is just C ab = (d T A Ψ a )(Z b ) = W αβ µ a β µb α and we will denote it by C = (Cab ). A second important geometric object is the vector subbundle G T A B B whose annihilator at a point b B is S (T e A b B). We can consider the subbundle G T A B B, the orthogonal to G with respect to the cosymplectic structure (Ω L, φ 0 ), which is given by G b = {X T e A b A/(i X Ω L + φ 0 (X)φ 0 ) Gb 0}, for b B. Note that G b = 1 L (G b ) = F b, for all b B. Moreover, we obtain Proposition IV.2. G b is coisotropic in (T e A b A, Ω L (b), φ 0 (b)), for all b B. In other words, G b G b, for b B. Proof. In fact, using (37) and since F (respectively, G ) is locally generated by {Z a } a=1,...,r (respectively, {S (d T A Ψ a )} a=1,...,r ), we deduce that G b = F b G b, for all b B. Now, we introduce the section Λ L of the vector bundle 2 (T A) A defined by Λ L (α, β) = Ω L ( 1 L (α), 1 L (β)), (38) for α, β (T A). Λ L is the (algebraic) Poisson structure (on the vector bundle T A A) associated with the cosymplectic structure (Ω L, φ 0 ). We will denote by ΛL : (T A) T A the vector bundle morphism given by ΛL (α) = i α Λ L, for α (T A).