Lagrange-d'Alembert-Poincar\`e equations by several stages

The aim of this paper is to write explicit expression in terms of a given principal connection of the Lagrange-d'Alembert-Poincar\`{e} equations in several stages. This is obtained by using a reduced Lagrange-d'Alembert's Principle in several stages, extending methods introduced for the case of two stages by one of the authors and collaborators. The case of the Euler's disk is described as an illustrative example.


Introduction
The topic of nonholonomic mechanical systems is an old one and has been very active for the last few decades. It is related to applications, like robotics, locomotion, phases and others and also to questions of a purely mathematical interest.
The equations governing nonholonomic systems are Lagrange-d'Alembert Equations which are derived by the Lagrange-d'Alembert Principle. Reduction of Euler-Lagrange and Lagrange-d'Alembert Equations by a group of symmetries is a fundamental issue in mechanics. The basic facts on reduction by one stage are reviewed in detail in Appendix B.. This paper is devoted to a specific question, namely, to study the extension of the reduction theory by one stage developed in [10,11], for the case of several stages. In particular, one obtains explicit expressions of Lagrange-d'Alembert-Poincaré Equations by n stages, written in terms of a given principal connection. As in [10,11], we will use the idea of reducing the variational principle rather than reducing the equations, including a detailed study of the geometry of variations. The question of writing explicit equations for the case of several stages, where one has a chain of normal subgroups of the given symmetry group G such that G = N 0 ⊲ N 1 ⊲ ... ⊲ N n+1 = {e} rather than just none or one, is clearly natural to complete the theory and deal with a wider class of examples. We describe the illustrative example of Euler's Disk, obtaining reduced equations by two stages. These equations are the ones previously obtained in [9], which were solved by using hypergeometric functions, which means some kind of integrability.
The paper is reasonably self-contained and the results that constitute the basic background are clearly stated in the appendices A and B.
In section 2 we describe one of the main ingredients for the paper, namely, an explicit formula for the Lie bracket by several stages. In section 3 we study an important particular case of the general theory: the Euler-d'Alembert-Poincaré equations by stages. In sections 4 and 5 we obtain the main results of the paper, that is, the Lagrange-Poincaré and the Lagrange-d'Alembert-Poincaré equations by several stages. In section 6 we describe the example of Euler's disk.
Some future studies are in order, like establishing the connection with Hamiltonian reduction by stages [18] and, more generally, Dirac-Weinstein reduction by stages [12].

The Lie bracket by stages
In this section we calculate explicit expressions for the Lie bracket by several stages. For doing this, we need some background on reduced covariant derivatives and associated connections on associated bundles, which we recall in Appendix A.. The formulas that we obtain are fundamental for writing reduced equations of motion by several stages, which is the main purpose of the paper.

An explicit formula for the Lie bracket by two stages
In this section we will use results and notation from [11], see also Appendix A.. Let a given principal bundle π : Q → Q/G with structure group G, and choose a Riemannian invariant metric on Q. Let N be a normal subgroup of G, then for each q ∈ Q we have a decomposition of T Q as an orthogonal direct sum T q Q = Ver N (T q Q) ⊕ H N (q), where H N (q) is the orthogonal complement of V er N (T q Q). Then, this collection of H N (q) defines a connection on the principal bundle Q with structure group N. Let A N be the corresponding 1-form connection. From the definition of A N it follows easily that for each g ∈ G and every v q ∈ T q Q we have that A N (gv q ) = Ad g A N (v q ).
Next, we use the notation and results on quotient horizontal and quotient vertical connections, summarized in Appendix A.. Given [q 0 , ξ 0 ] G ∈ g, let X 0 be the G-invariant vertical vector field defined by X 0 (q 0 ) = ξ 0 q 0 , which we can prove that it is well defined, and satisfies A (X 0 (q 0 )) = ξ 0 .
Let W → Q be a vector bundle where G acts by vector bundle isomorphisms and let w ∈ Γ G (W ), which is identified with [w] G . We define the quotient, or reduced, vertical connection [∇ (A,V ) ] G,[q 0 ,ξ 0 ] G [w] G by which, in the present context, is equivalent to the one given in [11] (see equation 7.5 in Appendix A.).
The following proposition has been proven in [11], Corollary 6.3.11. Proposition 2.1. Consider a Lie group G, N a normal subgroup of G and K = G/N. We sometimes think of G as a principal bundle π N : G → K with structure group N acting on the left, and also, as a principal bundle G → G/G ≡ {[e G ] G }, where e G is the neutral element of G, the base is a single point and the structure group G acts on the left by left translations. Similarly, we consider sometimes K as a principal bundle where the base is a single point and the structure group K acts on the left by left translations, K → K/K ≡ {[e K ] K }, where e K is the neutral element of K. Let g, n and k be the Lie algebras of G, N and K respectively. We call e N = e G the neutral element of N. The adjoint bundleg → {[e G ] G } of G → {[e G ] G }, is naturally identified with g, say g ≡g, by ξ ≡ [e G , ξ] G . Likewise there is a natural identification k ≡k given by κ ≡ [e G , κ] G .
Let an arbitrary identification g ≡ k ⊕ n as linear spaces. Choose a G-invariant Riemannian metric on G, which is determined by its restriction to g. Let A N be the principal connection defined on the principal bundle G → K with structure group N in the way described at the beginning of this section, which has the property that A N (gv q ) = Ad g A N (v q ), for every g, q ∈ G, v q ∈ T q G. Then g = k A N ⊕ n where k A N is the horizontal lift of k = T e K K in the bundle G → K, at the point e G ∈ G. Note that, by definition, the subspaces k A N and n of g are orthogonal.
Then, for the given identification g ≡ k ⊕ n, the bracket on the Lie algebra g can be written in terms of the brackets on the Lie algebra n and the Lie algebra k, and also in terms of ∇ (A N ,V ) and B A N using the formula, given in [11], Formula (2) should be interpreted as follows. By definition n → K is a vector bundle on K which is a left principal bundle K → {[e K ] K } with structure group K over a single point. We have an action of K on n covering the action of Then the right hand side of (2) should be interpreted as being . Now, according to the general definition given in [11] that we recall in formula (57) in Appendix B., B A N is a n-valued 2-form on K given by B A N (k)(k, δk) = [g, B(g)(ġ, δg)] N , where k = [g] N ,k = T π Nġ and δk = T π N δg. It can be checked that B A N is K-invariant. Then In order to obtain a more explicit expression of the Lie bracket [κ 1 ⊕ η 1 , κ 2 ⊕ η 2 ] in terms of the decomposition g = k⊕n, we shall see explicitly how to compute [ B A N ] G/N (κ 1 , κ 2 ) and [∇ (A N ,V ) ] G/N,κ η and identify them with elements of n, using the decomposition g = k⊕n and the identifications η ↔ σ η and κ ↔ [e K , κ] K as we have explained in the last proposition.
. This field is a Ginvariant horizontal field on the principal bundle G → K with principal connection A N .
Using the general formula (56) in Appendix B. we have that where we have used the definition of the bracket in g and taken into account that the X κ i are left invariant vector fields on G such that In order to simplify the notation, we define the bilinear form a N : k × k → n/K ≡ n given by the following formula, and we can write Calculation of [∇ (A N ,V ) ] G/N,κ η. Using the definition of the quotient vertical connection we have where X 0 is the uniquely determined invariant vertical vector field on the principal bundle K → {e K } with structure group K, such that A K (X 0 (e K )) = κ, where A K is the uniquely determined principal connection on the principal bundle K → {e K }, see Appendix A..

Notation.
For the rest of this paper we will often consider some horizontal lift with respect to some connection, at the neutral element e = e G of G, like, for instance, κ A N (e G ). In those cases we will often avoid writting e G , for simplicity.
We are going to use the following lemma.
Proof. Let [q(s), ξ ( s)] G be any curve ing. Then we have (see definition 7.2 in Appendix A.), Also we have So, considering the case where the last identity follows from the definition of the action of K = G/N over n.
Now, applying the definition of covariant derivative in the adjoint bundle, we have ,ġ(t)), Ad g(t) η + (Ad g(t) η)˙ t=t 0 N K .
If we choose g(t) horizontal with respect to A N such that g(t 0 ) = e, we obtain the formula where κ A N is the horizontal lift of κ in e G to G.
We define the bilinear form b N : k × n → n/K ≡ n given by Then we have an explicit formula for the Lie bracket or equivalently, using the defined bilineal forms, 2.2 The Lie bracket by several stages. Now we will obtain a formula that generalizes (12) for the case of several stages, in fact, we will show that it can be done by a repeated application of (12). This will be one of the main results of this paper. Another important result will be the application of this generalization for reducing a nonholonomic system by several stages.
For each j = 1, ..., n + 1, consider the left principal bundle N j−1 → N (j−1,j) = N j−1 /N j with structure group N j and with connection A N j , defined in a similar way as it was done for the principal bundle G → G/N with structure group N with connection A N , at the beginning of section 2, using the same G-invariant metric, which will be, naturally, N j−1 -invariant.
Consider the map b (N j−1 ,N j ) : n (j−1,j) × n j → n j , j = 1, .., n + 1, defined by , where η A N j is the horizontal lift of η in the bundle N j−1 → N (j−1,j) at the neutral element. Note that η A N j ∈ n j−1 and Similarly, we consider a (N j−1 ,N j ) : n (j−1,j) × n (j−1,j) → n j , j = 1, ..., n + 1, defined by On the other hand, the map a (N j−1 ,N j ) is related directly with the curvature of the connection A N j , as follows Then we have the following formula for the Lie bracket on g : where by definition, for i = 0, ..., n and j = 0, ..., n, b Proof. We shall prove this formula by induction.
If n = 1 then formula (14) becomes (1,2) ), and this expression coincides with equation (12) for the case Now, we shall suppose that formula (14) is valid in the case in which we have n − 1 normal subgroups and we shall see that the formula holds when the number of subgroups is n.
Using the formula (12) that we have obtained in the case of one normal subgroup, with N 1 ✁ G whose Lie algebra can be written as Using the inductive hypothesis we obtain  Replacing this expression in (15) and considering that b Replacing the index p by i we can group terms together and we obtain  In other words, In particular, for n = 2 and n = 3, we obtain the following expressions and

The Euler-d'Alembert-Poincaré equations by several stages
For Euler-Poincaré equations and Euler-d'Alembert-Poincaré equations, see [19] and [10]. These equations are the particular case of Lagrange-Poincaré equations and Lagranged'Alembert-Poincaré equations, respectively, for the case in which Q = G is the trivial bundle whose base is a single point. In this section we are going to study the Eulerd'Alembert-Poincaré equations by several stages and in section 5 we will study the case of a trivial bundle Q = X × G. From this, the case of a general bundle Q will follow by glueing together principal bundle charts.

The Euler-Poincaré equations by several stages.
We are going to use the notation of theorem 2.3.
Let v(t) = n i=0 η (i,i+1) (t) ∈ g be a curve and consider a variation of this curve In order to write Euler-Poincaré equations we need to calculate From the previous identities we can deduce the following equalities, for 0 ≤ i ≤ n : Since we are assuming that g ≡ n i=0 n (i,i+1) we have g * ≡ n j=1 n * (j,j+1) .
The terms a Then we have the Euler-Poincaré equations by stages for 0 ≤ i ≤ n and v ∈ g, or equivalently, where η (j,j+1) ∈ n (j,j+1) .

The Euler-d'Alembert-Poincaré equations by several stages.
In this section we are going to use some results and notation from [10]. Also, we are going to use the notation presented in Appendix B., for the particular case Q = G.
Let g ≡ n (0,1) ⊕ n (1,2) ⊕ ... ⊕ n (n−1,n) ⊕ n (n,n+1) as before, and let D be a left invariant distribution on G, which represents the nonholonomic constraint. In particular, D e ⊂ g. The vertical bundle V is such that V e = T e G ≡ g. We shall also assume the dimension hypothesis (see equation 55 in Appendix B.).
To write the Euler-d'Alembert-Poincaré equations by several stages we will consider an easy particular case first, and the general case later.
Particular case. We are going to assume that Since Q = G we have that S = D and, therefore, The Euler-d'Alembert-Poincaré equations are obtained in the following way: Let us consider a curve with ω(t) ∈ S such that ω(t 0 ) = ω(t 1 ) = 0. Then we can write Then, as before, we have It follows that β (j,j+1) ∈ g * and ξ = n k=0 ξ (k,k+1) ∈ g, then the Euler-Poincaré equationṡ are satisfied for all curves ξ (j,j+1 Finally, for this particular case, we obtain the following Euler-d'Alembert-Poincaré equations by n stages, which are obtained taking for each i = 0, ..., n, ε (i,i+1) arbitrary, while ε (j,j+1) = 0, for i = j, General case. Now suppose that D = S ⊆ g ≡ n (0,1) ⊕ n (1,2) ⊕ ...⊕ n (n−1,n) ⊕ n (n,n+1) is an arbitrary subspace. Let m = dim(g) and d i = dim(n (i,i+1) ), for 0 ≤ i ≤ n. Consider a base {e 1 , ..., e m } of g adapted to the previous decomposition of g. Namely, consider {e 1 , ..., e m } such that the first d 0 vectors of the base belong to n (0,1) , the following d 1 vectors belong to n (1,2) , the following d 2 to n (2,3) and so on up to the last d n vectors which should belong to n (n,n+1) .
In this case we will write On the other hand, we can define the arbitrary subspace S ⊆ g as being the kernel of a linear function f : g → R m−dim(S) . In coordinates we have that points (z 1 , z 2 , ..., z m ) belonging to S are those which satisfy the system of equations f j (z 1 , z 2 , ..., z m ) = 0, for j = 1, ..., m − dim(S).
From this system we can obtain m − dim(S) coordinates in terms of the rest dim(S) independent coordinates. More precisely, we can represent (not necessarily in a unique way) S as being the graph of a linear map, as follows. We can choose a set of indexes A ⊆ {1, ..., m}, with dim(S) elements, and linear functionals such that points of S are precisely (z i , ϕ j ( z i )), with i ∈ A and j ∈ {1, ..., m} − A, where z i with i ∈ A are independent coordinates.
In order to obtain a system of implicit scalar differential equations in terms ofμ ij , µ ij and ε i j , which will be linear inμ ij , one may proceed as follows. Taking advantage of the fact that the variables ε i r i , for 1 ≤ r i ≤ s i , 0 ≤ i ≤ n, are independent, we can choose their values arbitrarily. Then to obtain differential (implicit) equations of motion we only need to fix the value of each one of them to be arbitrary while the rest are fixed to be 0 in system (32). Moreover, since the dependence of the ε i r i is linear we may simply fix the value of each one of them to be 1 while the rest are fixed to be 0.
To the system of implicit differential equations so obtained we should add the algebraic restrictions ∂l/∂v i = µ i . This gives the Euler-d'Alembert-Poincare equations in coordinates.
Remark. The particular case S ≡ S (0,1) ⊕ S (1,2) ⊕ ... ⊕ S (n−1,n) ⊕ S (n,n+1) corresponds to the case in which all the variables are independent, that is, there are no variables h i j .
In the local case Q = X × G, we have g ≡ X × g through the identification [(x, g), ξ] G ≡ (x, Ad g −1 ξ), in particular we have [(x, e), ξ] G ≡ (x, ξ). Then the isomorphism α A is given by . Now let us calculate theg-valued two formB. By definition, q , δx h q the horizontal lifts ofẋ and δx at the point q ∈ π −1 (x). Of course,ẋ h q , δx h q can be replaced by any pair of elementsq, δq that satisfies the conditions T πq =ẋ and T πδq = δx.
Since Q = X × G, we have Now we need to calculate the covariant derivatives on the vector bundles g and g * : In the local case, after the identification [(x, e), ξ] G ≡ (x, ξ), we have Let α(t) = (x(t), α(t)) ∈ g * with the identification g * ≡ X × g * given by [(x, e), α] G ≡ (x, α).
The local vertical Lagrange-Poincaré equation. As in Appendix B. for a reduced Lagrangian l : T (Q/G) ⊕ g → R, we have the vertical Lagrange-Poincaré equations D Dt In the local case, we denote l(x,ẋ, [e, ξ] G ) ≡ l(x,ẋ, (x, ξ)) ≡ l(x,ẋ, ξ) and we can identify ∂l/∂v ≡ ∂l/∂ξ. Then, using the formula (33), we obtain D Dt Then the vertical Lagrange-Poincaré equation becomes The local horizontal Lagrange-Poincaré equations. In the general case we have Now we need to calculate this in the local case. For this we proceed as in Appendix B.. Now let v(t, s) be a horizontal deformation of the curve v(t) such that v(t, 0) = v(t), and let x(t, s) be the base point of v(t, s), then x(t, 0) = x(t).
Since (x(t, s), ξ(t, s)) is horizontal for each t, using the formula (33) we have Finally the horizontal Lagrange-Poincaré equations are So, the Lagrange-Poincaré system of vertical and horizontal equations is Local Lagrange-Poincaré equations by several stages. Consider Q = X × G and the identification g ≡ X × g given by [(x, e), ξ] G ≡ (x, ξ) and [(x, g), ξ] G ≡ (x, e, Ad g −1 ξ).
In the case in which the number of stages is two, these vertical equations by stages are: The local horizontal Lagrange-Poincaré equations by several stages. Recall that the horizontal Lagrange-Poincaré equations are Let ξ ∈ g such that ξ = n 0=1 η (i,i+1) and ∂l ∂ξ = n j=0 β (j,j+1) , as in (40)

Local Lagrange-d'Alembert-Poincaré equations by several stages
Now we consider the following situation: Let Q = X × G be a principal trivial bundle with structure group G. Let G in the condition of the theorem 2.3 in the page 7. As before, we have the identification of the adjoint bundle g ≡ X × g. We shall consider first a particular case and then the general case, as we did in section 3.2.
We proceed as before in the general case in section 3.2 to obtain the equations of motion.
6 Example of reduction by stages 6

.1 The Lagrange-d'Alembert-Poincaré equations by stages for Euler's disk
Let us consider an Euler's disk of radius r and thickness re rolling on a rough horizontal plane and having only one point of contact with it, in other words, the horizontal and vertical positions of the disk is excluded from the configuration space. We will use the description and notation of reference [9], however here we shall apply different methods. The configuration space for Euler's disk is and a point q ∈ Q is written as q = (θ, ϕ, ψ, x). Let denote A an orthogonal 3 × 3 matrix representing the instantaneous orientation of the disk, therefore Ae 3 has e 3 component belonging to (0, 1). The angle θ is the angle from the axis e 3 to the vector z = Ae 3 . The vector y is the unit vector directed from the point of contact with the plane to the origin of the system (Ae 1 , Ae 2 , Ae 3 ). The vector u is defined by u = z × y being tangent to the disk at the point of contact x and having the direction of the motion of this point on the plane. The unit vector u has the expression u = (− cos ϕ, − sin ϕ, 0) which defines the angle ϕ. The angle ψ is the angle from the vector −y to the vector Ae 1 where the positive sense for measuring the angle ψ on the plane of the disk is the counterclockwise sense, as viewed from z.
The symmetry group that we are going to consider is G = SO(2) × R 2 ≡ S 1 × R 2 . Since the group is abelian, we can consider that the group G acts on the left on the configuration space Q by the action (α, a)(θ, ϕ, ψ, x) = (θ, ϕ, ψ + α, x + a).
In this context, we are going to apply the techniques of reduction by stages to the example of Euler's disk. We shall write the Lagrange-d'Alembert-Poincaré equations by stages for this example.
We choose N 1 = R 2 as a normal subgroup of the symmetry group G = S 1 × R 2 , and N 2 = {e} as a normal subgroup of N 1 . So, following the notation of the theorem 2.3 in page 7, we have N (0,1) = N 0 /N 1 = G/N 1 = S 1 and N (1,2) = N 1 /N 2 = R 2 . Then we can write the chain of normal subgroups as follows We note g, n 1 and n 2 the Lie algebras of the groups G, N 1 and N 2 respectively. And n (0,1) , n (1,2) denote the Lie algebras of the groups N (0,1) and N (1,2) .
This system of equations is the same that we have obtained in [9], where we developed reduction in one stage.

A. Reduced connections
We shall recall some known facts on associated bundles and covariant derivatives. Proofs can be found in [11].
Let a left principal bundle π : Q → Q/G and A a principal connection. Consider a left action ρ : G × M → M of the Lie group G on a manifold M. The associated bundle with standard fiber M is, by definition, Q × G M = (Q × M )/G, where the action of G on Q × M is given by g(q, m) = (gq, gm). The class (or orbit) of (q, m) is denoted [q, m] G . The projection π M : Q × G M → Q/G is defined by π M ([q, m] G ) = π(q) and it is easy to check that is well defined and is a surjective submersion.
Let [q 0 , m 0 ] G ∈ Q × G M and let x 0 = π(q 0 ) ∈ Q/G. Let x(t), t ∈ [a, b], be a curve on Q/G and let t 0 ∈ [a, b] be such that x(t 0 ) = x 0 . The parallel transport of this element [q 0 , m 0 ] G along the curve x(t) is defined to be the curve [q, m] G (t) : is the horizontal lift of the curve x(t) with initial condition q 0 . For t, t + s ∈ [a, b], we adopt the notation τ t t+s : π −1 M (x(t)) → π −1 M (x(t + s)) for the parallel transport map along the curve x(s) of any point [q(t), m(t)] G G . From now on, we assume that M is a vector space and ρ is a linear representation. In this case, the associated bundle with standard fiber M is naturally a vector bundle. We will use the identification T M = M × M. We shall sometimes use the notation ρ ′ (ξ) for the second component of the infinitesimal generator of an element ξ ∈ G, that is, ξm = (m, ρ ′ (ξ)(m)). Thus, we have a linear representation of the Lie algebra G on the vector space M , ρ ′ : G → End(M ) (the linear vector fields on M are identified with the space of linear maps of M to itself).
, m(t)] G ) = π(q(t)) its projection on the base Q/G, and let τ t t+s , denote the parallel transport along Thus, the covariant derivative of [q(t), m(t)] G is an element of π −1 M (x(t)). Note that if [q(t), m(t)] G is a vertical curve, then the covariant derivative in the associated bundle is just the fiber derivative, since its base point is constant. That is, where m ′ (t) is the time derivative of m. By definition, a connection (sometimes called an affine connection) ∇ on a vector bundle τ : V → Q is a map ∇ : X ∞ (Q) × Γ(V ) → Γ(V ), say (X, v) → ∇ X v, having the following properties: where X[f ] denotes the derivative of f in the direction of the vector field X.
Given a connection on V, the parallel transport of a vector v 0 ∈ τ −1 (q 0 ) along a curve q(t) in Q, t ∈ [a, b], such that q(t 0 ) = q 0 for a fixed t 0 ∈ [a, b], is the unique curve v(t) such that v(t) ∈ τ −1 (q(t)) for all t, v(t 0 ) = v 0 , and which satisfies ∇q (t) v(t) = 0 for all t. The operation of parallel transport establishes for each t, s ∈ [a, b], a linear map T t t+s : τ −1 (q(t)) → τ −1 (q(t + s)) associated to each curve q(t) in Q. Then we can define the operation of covariant derivative on curves v(t) in V similar to that in the previous definition 7.1; that is, Observe that the connection ∇ can be recovered from the covariant derivative (and thus from the parallel transport operation). Indeed, ∇ is given by where for each q 0 ∈ Q, each X ∈ X ∞ (Q) and each v ∈ Γ(V ), we have, by definiton, that q(t) is any curve in Q such thatq(t 0 ) = X(q 0 ) and v(t) = v(q(t)) for all t. This property establishes, in particular, the uniqueness of the connection associated to the covariant derivative D/Dt. The notion of a horizontal curve v(t) on V is defined by the condition that its covariant derivative vanishes. A vector tangent to V is called horizontal if it is tangent to a horizontal curve. Correspondingly, the horizontal space at a point v ∈ V is the space of all horizontal vectors at v.
In the case of an associated bundle we recall from [11] the following formula that gives the relation between the covariant derivative of the affine connection and the principal connection: This gives an affine connection on Q × G M , called∇ A . More precisely, let ϕ : Q/G → Q × G M be a section of the associated bundle and let X(x) ∈ T x (Q/G) be a given vector tangent to Q/G at x. Let x(t) be a curve in Q/G such thatẋ(0) = X(x); thus, ϕ (x(t)) is a curve in Q × G M . The covariant derivative of the section ϕ with respect to X at x is then, by definition,∇ Notice that we only need to know ϕ along the curve x(t) in order to calculate the covariant derivative.
Definition 7.2. The associated bundle with standard fiber g, where the action of G on g is the adjoint action, is called the adjoint bundle, and is sometimes denoted Ad(Q). We will use the notationg := Ad(Q) in this paper. We letπ G :g → Q/G denote the projection The following properties hold. Let π : Q → Q/G be a principal bundle with structure group G, as before. The tangent lift of the action of G on Q defines an action of G on T Q and so we can form the quotient (T Q)/G =: T Q/G. There is a well defined map τ Q /G : T Q/G → Q/G induced by the tangent of the projection map π : Q → Q/G and given by [v q ] G → [q] G . The vector bundle structure of T Q is inherited by this bundle.
(c) The rules where λ ∈ R, v q , u q ∈ T q Q, and [v q ] G and [u q ] G are their equivalence classes in the quotient T Q/G, define a vector bundle structure on T Q/G having base Q/G. The fiber (T Q/G) x is isomorphic, as a vector space, to T q Q, for each x = [q] G .
is a well defined vector bundle isomorphism. The inverse of α A is given by An action ρ : G × V → V of a Lie group G on a vector bundle τ : V → Q with extra structure [ , ] (a Lie algebra structure on each fiber of V, in such a way that V is a Lie algebra bundle), ω (a V -valued 2-form on Q) and ∇ (a covariant derivative D/Dt for curves in V related in the standard way to a connection ∇ on V ) , is a vector bundle action such that, for each g ∈ G, ρ g : V → V is a morphism that commutes with the structures given by [ , ], ω, ∇ in the vector bundles. Definition 7.3. Let τ : V → Q be a vector bundle and let D/Dt be the covariant derivative along curves associated to a connection ∇ on V . Let ρ : G × V → V be a vector bundle action covering the action ρ 0 : G × Q → Q which we assume that it is a principal bundle. Let A be a principal connection on the principal G-bundle Q → Q/G. Let v(t) be any curve in V and let q(t) = τ (v(t)) for all t. Choose t 0 and let q 0 = q(t 0 ). Let g q (t) and q h (t) be such that q h (t) is a horizontal curve, q(t) = g q (t)q h (t), and g q (t 0 ) = e. Then we define and let q(t) be any curve in Q such thatq(t 0 ) = X(q 0 ). Define q h (t) and g q (t) as in Definition 7.3. Then In particular, ∇ Then Let τ : V → Q be a given vector bundle and let ρ : G × V → V denote a given G-action on V. Then the quotient V /G carries a naturally defined vector bundle structure over the base Q/G, say τ /G : The projection π G (V ) : V → V /G is a surjective vector bundle homomorphism covering π, and the restriction π G (V )|τ −1 (q) : τ −1 (q) → (τ /G) −1 ([q] G ) is a linear isomorphism for each q ∈ Q. Let G × V → V be a vector bundle action and let π G (V ) : V → V /G be the vector bundle homomorphism described above. Then (π G (V )) * : Γ(V /G) → Γ G (V ) is a linear isomorphism. Consider the horizontal invariant bundle, that is, the vector space of all horizontal invariant vector fields on Q along π −1 (x) given by I H G (T Q) x := {X : π −1 (x) → T Q : X(q) ∈ HorT q Q, ∀q ∈ Q, g * X = X}.
Let Y 0 ∈ I V G (T Q) such that β A (Y 0 ) = [q 0 , ξ 0 ] G , where the map is the well defined Lie algebra isomorphism given by where Y ∈ I V G (T Q) x , x ∈ Q/G, and q ∈ π −1 (x) is arbitrary. Then Here we use the standard notation in the calculus of variations, namely, δ t 1 t 0 L(q,q)dt = ∂ ∂λ λ=0 t 1 t 0 L (q(t, λ),q(t, λ)) dt.
Using the Lagrange-d'Alembert Principle, equations of motion, called Lagrange-d'Alembert equations, can be written in a local chart as follows (q,q) ∈ D q .
The form of these equations is independent of the choice of coordinates, which is one of the great advantages of Lagrange-d'Alembert Principle.
Nonholonomic Systems with Symmetry. Now we shall assume that π : Q → Q/G is a principal bundle with structure group G and we denote by V the vertical distribution, that is, V q = T q π −1 ([q] G ) , for each q ∈ Q, which is obviously an integrable distribution whose integral manifolds are the group orbits π −1 ([q] G ).
Following [6] let us consider the following condition, for simplicity: (A1) DIMENSION ASSUMPTION. For each q ∈ Q we have the equality It is easy to see that, under assumption (A1), the dimension of the space S q = D q ∩ V q does not depend on q ∈ Q, and moreover, the collection of spaces S q , q ∈ Q, is a subbundle of D, of V, and of T Q.
Also we consider the following condition: It follows immediately from (A1) and (A2) that S is a G-invariant distribution.
The Nonholonomic Connection. It is known, and easy to prove, that there is always a G-invariant metric on Q. See, for example, [11]. In many important physical examples there is a natural way of choosing such a metric, representing, for instance, the inertia tensor of the system (see [6]). Let us choose an invariant metric on Q. Then, under condition (A1), we can define uniquely a principal connection form A : T Q → g such that the horizontal distribution Hor A T Q satisfies the condition that, for each q, the space Hor A T q Q coincides with the orhogonal complement H q of the space S q in D q . This connection is called the nonholonomic connection.
For each q ∈ Q, let us denote U q the orthogonal complement of S q in V q . Then it is easy to see that U is a distribution and we have the Whitney sum decomposition T Q = H ⊕ S ⊕ U .
We obviously have D = H ⊕ S and V = S ⊕ U .
Under the additional assumption (A2), all three distributions H, S and U are Ginvariant, therefore we can write, The Geometry of the Reduced Bundles. Recall from [11] that there is a vector bundle isomorphism to the connection∇ A ong such that (x(0),v(0)) = (x 0 ,v 0 ) , and let (x(s), u(s)) be the horizontal lift of x(s) with respect to the connection ∇ such that (x(0), u(0)) = (x 0 ,ẋ 0 ). (Notice that in general, (x(s), u(s)) is not the tangent vector (x(s),ẋ(s)) to x(s)). Thus, (x(s), u(s),v(s)) is a horizontal curve with respect to the connection C = ∇ ⊕∇ A naturally defined on T (Q/G) ⊕g in terms of the connection ∇ on T (Q/G) and the connection∇ A ong.
We shall often write whenever there is no danger of confusion.
The covariant derivative on a given vector bundle, for instanceg, induces a corresponding covariant derivative on the dual bundle, in our caseg * . More precisely, let α(t) be a curve ing * . We define the covariant derivative of α(t) in such a way that for any curvev(t) ong such that both α(t) andv(t) project on the same curve x(t) on Q/G, we have Recall that the curvature B ≡ B A of A is given by where X, Y are vector fields on Q/G and X h 1 , X h 2 are their horizontal lifts, respectively. The curvature 2-form B ≡ B A of the connection A induces a g-valued 2-form B ≡ B A on Q/G given by B(x)(δx,ẋ) = [q, B(q)(δq,q)] G , where for each (x,ẋ) and (x, δx) in T x (Q/G), (q,q) and (q, δq) are any elements of T q Q such that π(q) = x, T π(q,q) = (x,ẋ) and T π(q, δq) = (x, δx) (see [11] for the proof). The g-valued 2-form B on Q/G will be called the reduced curvature form .