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VARIATIONAL PRINCIPLES FOR LIE–POISSON AND HAMILTON–POINCARÉ EQUATIONS

As is well-known, there is a variational principle for the 
Euler–Poincare equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principle 
for the Lie–Poisson equations on g*, the dual of g, and also to 
generalize this construction. 
The more general situation is that in which the original configuration 
space is not a Lie group, but rather a configuration manifold Q 
on which a Lie group G acts freely and properly, so that Q → Q/G 
becomes a principal bundle. Starting with a Lagrangian system on TQ 
invariant under the tangent lifted action of G, the reduced equations on 
(TQ)/G, appropriately identified, are the Lagrange–Poincare equations. 
Similarly, if we start with a Hamiltonian system on T*Q, invariant under 
the cotangent lifted action of G, the resulting reduced equations on 
(T*Q)/G are called the Hamilton–Poincare equations. 
Amongst our new results, we derive a variational structure for the 
Hamilton–Poincare equations, give a formula for the Poisson structure 
on these reduced spaces that simplifies previous formulas of Montgomery, 
and give a new representation for the symplectic structure on the associated 
symplectic leaves. We illustrate the formalism with a simple, but 
interesting example, that of a rigid body with internal rotors.


Introduction
This paper presents some advances in geometric mechanics and in particular variational principles and reduction for systems with symmetry. It is a great pleasure to dedicate this paper to Vladimir Arnold, since his pioneering paper [Arn1] has influenced the development of this theory in an absolutely fundamental way.
The present paper assumes that the reader is familiar with geometric mechanics; the relevant background can be found in [Arn3] and [MR]. The paper [CMR2] is a basic ingredient for the present discussion. It and [CMR1] can be consulted for additional information and history.
The usual way in which Lagrangian reduction proceeds is to begin with Hamilton's principle for a system on a configuration manifold Q and with a symmetry group G acting on Q and then to drop this variational principle to the quotient Q/G to derive a reduced variational principle. This theory has its origins in specific examples such as fluid mechanics (see, for example, [Arn2], [Bre]), while the systematic theory of Lagrangian reduction was begun in [MS1], [MS2] and further developed in [CMR2]. In the case Q = G, the reduced equations are the Euler-Poincaré equations and the associated Euler-Poincaré reduction theorem is well-known (see, for example, [MR]). In the general case, the reduced equations associated to this construction are called the Lagrange-Poincaré equations and their geometry has been fairly well developed.
This reduction of Hamilton's principle is often regarded as a Lagrangian analog of Poisson reduction on the Hamiltonian side. However, a more faithful analog is the reduction of Hamilton's phase space principle. The development of this theory is one of the main objectives of the present paper. In the case Q = G, this collapses to a variational principle for the Lie-Poisson equations on g * .
It should be stressed that in this paper we do not set any momentum maps equal to constants; that is, we are outside the realm of symplectic and Routh reduction. For variational principles in this context, we refer to [MRS].

The Lie-Poisson Case
It is well understood the sense in which Lie-Poisson dynamics on g * , the dual of a Lie algebra g, are Hamiltonian relative to the Lie-Poisson bracket on functions on g * . Here we show how these same equations can be derived from a variational principle for Lie-Poisson dynamics, which is a reduction of a certain form of Hamilton's phase space variational principle.
Notation and Setting. Let G be a Lie group and let L : T G → R be a given Lagrangian. Let FL : T G → T * G be the fiber derivative, that is, the Legendre transformation. Assume that FL is a diffeomorphism, that is, L is hyperregular. Let g be the Lie algebra of G, regarded as T e G, the tangent space to G at the identity element e. Assume that L is invariant under the tangent lift of left translations and let l : g → R be the reduced Lagrangian, given by l = L|g.
It is well-known that the reduced Euler-Lagrange equations, called the Euler-Poincaré equations, can be derived by a very simple and effective reduced variational method; see, for example, [MR]. To explain this reduced variational principle, let g(t) be a curve in G with fixed endpoints g 0 = g(t 0 ) and g 1 = g(t 1 ), and let v(t) be the body velocity defined by v(t) = g −1 (t)ġ(t), where the notation g −1 (t)ġ(t) stands for T L g(t) −1ġ(t).
Recall that Hamilton's principle is defined by the requirement that a curve g(t) be a critical point of the action t1 t0 L(g(t),ġ(t)) dt for variations δg(t) such that δg(t i ) = 0 for i = 0, 1. It is well-known that this is equivalent to the Euler-Lagrange equations, which we write symbolically as See the discussion later in Section 5 and, for example, [MR] for how to write these equations intrinsically on the second order bundle. Because of invariance, Hamilton's principle holds if and only if v(t) is a critical point of the reduced action t1 t0 l(v(t)) dt for variations that are restricted (or, if one prefers, are constrained ) to be of the form δv(t) =η(t) + [v(t), η(t)], where η(t) is a curve in g such that η(t i ) = 0, for i = 0, 1. This is proved by tracing through the variations δv that are induced by variations δg. In fact, η is the body representation of the variations on the group, that is, η = g −1 δg (see [MR] for details).
By applying the usual integration by parts argument one sees that the reduced Hamilton principle is equivalent to the Euler-Poincaré equations: d dt where δl/δv ∈ g * is the usual differential of l evaluated at v, that is, δl/δv = dl(v).
As the next theorem describes, these equations, together with the reconstruction equation v = g −1ġ (2) are equivalent to the Euler-Lagrange equations on G. The Euler-Poincaré equations (1) together with this reconstruction equations (2) are called the Euler-Arnold equations.
Since L is hyperregular and invariant, the Hamiltonian H : T * G → R which is obtained from the Legendre transformation by the usual formula H(g, p) = p ·ġ − L(g,ġ), where p = FL(g,ġ), is well defined; in this expression for H, as usual, one uses hyperregularity to expressġ as a function of (g, p). If L is hyperregular and invariant, so is H. That is, the inverse Legendre transformation FH = (FL) −1 : T * G → T G is also a diffeomorphism and H is invariant under the left action of G. If we define h : g * → R by h = H|g * , it is clear that which is called the reduced Legendre transformation.
It is well-known that Hamilton's equations are equivalent to the phase space version of Hamilton's principle, namely δ t1 t0 (p ·ġ − H(g, p)) dt = 0, where variations are taken amongst curves in T * G and where the variations δg, δp satisfy δg(t i ) = 0, for i = 0, 1, and δp arbitrary.
The Lie-Poisson equations for a Hamiltonian h : g * → R, where h = H|g * , are normally derived by using the method of Lie-Poisson reduction, that is, as a special case of Poisson reduction. In this context, the Lie-Poisson bracket is seen as the reduction of the canonical cotangent bracket on T * G. See, for example, [MR] for these derivations. The Lie-Poisson equations arė Four Action Principles and Four Sets of Equations. Sticking with the special case of systems on Lie groups for the moment, one has the following elementary, but important result.
Theorem 2.1. With the above notation and hypotheses of hyperregularity, the following conditions are equivalent.
(iv) The Lie-Poisson Variational Principle. The curve (v(t), µ(t)) ∈ g × g * is a critical point of the action where η(t) is a curve in g such that η(t i ) = 0, for i = 0, 1, and where the variations δµ are arbitrary.
Proof. We have already remarked that Hamilton's principle (i) is equivalent to the Euler-Lagrange equations (v). We have also remarked that Euler-Poincaré reduction theory shows that (i) and (ii) are equivalent and that the standard arguments in the calculus of variations shows that (ii) and (vi) are equivalent. It is standard that the Euler-Lagrange equations (v), under the assumption of hyperregularity, are equivalent to Hamilton's equations (vii). It is also standard that Hamilton's phase space principle (iii) is equivalent to Hamilton's equations (vii). Another standard item is that under hyperregularity, the Euler-Poincaré equations (vi) are equivalent to the Lie-Poisson equations (viii). To complete the proof, one can show directly by the usual arguments in the calculus of variations that the variational principle (iv) is equivalent to the Lie-Poisson equations (viii).
While this technically gives a complete proof of the theorem, it does not provide much insight into where the interesting Lie-Poisson variational principle, item (iv), comes from. Notice that this principle involves twice as many variables as does the Euler-Poincaré variational principle, just as Hamilton's phase space principle involves varying curves in T * G, while Hamilton's principle involves curves varying in G, a space of half the dimension.
To get this insight, we first recall that Hamilton's phase space principle states that δ (p ·ġ − H(g, p)) dt = 0. ( In this principle, note that the pointwise function in the integrand, namely is defined on T G ⊕ T * G, regarded as a bundle over G, the base space common to T G and T * G. The group G acts on T G⊕T * G by simultaneously left translating on each factor by the tangent and cotangent lift. Explicitly, the action of an element h ∈ G is given by The map F is invariant under this action of G as is easily checked, assuming invariance of H. Thus, the function F drops to the quotient, namely to the function f : When the function F is taken into the phase space variational principle, one is varying curves (g(t), p(t)) and one of course insists that the slotġ actually is the time derivative of g(t). This restriction induces in a natural way a restriction on the variations of v = g −1ġ and these restrictions are computed exactly as in the Euler-Poincaré theory to be given by δv(t) =η(t) + [v(t), η(t)], where η(t) is a curve in g such that η(t i ) = 0, for i = 0, 1. In fact, this computation gives the relation η = g −1 δg.
In summary, the preceding argument gives a direct verification of the equivalence of (iii) and (iv). This point of view will be important in the generalizations to follow.
One of the main goals of the paper is to generalize the preceding theorem, replacing the spaces T G and T * G with T Q and T * Q for a general configuration space Q on which a Lie group G acts. This main result is given in Theorem 8.1 below.

Lagrange-Poincaré Bundles
Now we will generalize the preceding construction to the case in which we have an action of a Lie group G on a configuration manifold Q, in such a way that, with this action, Q becomes a principal bundle. As suggested by the preceding arguments, the bundle T * Q ⊕ T Q as a bundle over Q should play a key role as this is the domain of definition of the function appearing in Hamilton's phase space action principle.
One of our first goals will be to recall some results on Lagrangian reduction from [CMR2], namely we shall describe the Lagrange-Poincaré bundles, the geometry of variations and the Lagrange-Poincaré equations. A key point in doing this is to choose a principal connection A on the bundle π : Q → Q/G and, using it, decompose arbitrary variations of curves in Q into vertical and horizontal components. This gives rise, correspondingly, to two reduced equations, namely, vertical Lagrange-Poincaré equations, corresponding to vertical variations, and horizontal Lagrange-Poincaré equations, corresponding to horizontal variations, which are Euler-Lagrange equations on Q/G with an additional term involving the curvature B of A. Using this, the geometric description of reduced horizontal and reduced vertical variations, and Hamilton's phase space action principle, generalizing Theorem 2.1 to obtain the main result Theorem 8.1, is fairly straightforward.
In this section we recall some results from differential geometry on principal and associated bundles. We will need to do this mainly to establish our notations and conventions.
Horizontal and Vertical Spaces. As indicated above, we let π : Q → Q/G be a left principal bundle. Recall that a (principal ) connection A on Q is a Lie algebra valued one form A : T Q → g with the properties (i) A(ξq) = ξ for all ξ ∈ g; that is, A takes infinitesimal generators of a given Lie algebra element to that element; note that we denote the infinitesimal action of ξ ∈ g on Q at q ∈ Q by concatenation, as ξq, and (ii) A(g · v) = Ad g (A(v)), where Ad g denotes the adjoint action of G on g and gv denotes the lifted action of g ∈ G on v ∈ T G.
The restriction of the connection A to the tangent space T q Q is denoted A q . Recall that connections may be characterized by giving their vertical and horizontal spaces defined at q ∈ Q by Ver q = Ker T q π, Hor q = Ker A q .
Thus, Ver(T Q) = q∈Q Ver q is the subbundle of vectors tangent to the group orbits. The vertical and horizontal components of a vector v q will be denoted Ver(v q ) and Hor(v q ) respectively. By definition, This provides a Whitney sum decomposition T Q = Hor(T Q) ⊕ Ver(T Q) where Hor(T Q) = q∈Q Hor q and Ver(T Q) are the horizontal and vertical subbundles of T Q; both are invariant under the action of G. A vector is called horizontal if its vertical component is zero, i. e., if A(v q ) = 0, and it is called vertical if its horizontal component is zero, i. e., if T q π(v q ) = 0. Note that T q π : Hor q → T π(q) (Q/G) is an isomorphism.
Curvature. The curvature of A will be denoted B A or simply B. By definition, it is the Lie algebra valued two form on Q defined by where d denotes the exterior derivative.
Cartan Structure Equations. The Cartan structure equations state that for arbitrary vector fields u, v on Q (not necessarily horizontal), where the bracket on the right hand side is the Lie bracket in g. We write this equation for short as Horizontal Lifts. Given a vector X ∈ T x (Q/G), and q ∈ π −1 (x), the horizontal lift X h q of X at q is the unique horizontal vector in T q Q that projects via T π to the vector X(x); that is, X h q ∈ (T q π) −1 (X). We denote by X h the vector field along π −1 (x) formed by all horizontal lifts of X at points of π −1 (x). Let , the horizontal lift of x(t), which at t = t 0 coincides with q 0 , is uniquely determined by requiring all its tangent vectors to be horizontal. This curve is denoted x h q0 and is defined on [a, b]. Consider a curve q(t), where t ∈ [a, b], and choose t 0 ∈ [a, b]. Then there is a unique horizontal curve q h (t) such that q h (t 0 ) = q(t 0 ) and π(q h (t)) = π(q(t)) for all t ∈ [a, b]. Therefore, we can define a curve g q (t) ∈ G, for t ∈ [a, b] by the decomposition Evidently g q (t 0 ) is the identity. Also, notice that if x(t) = π(q(t)) and q 0 = q(t 0 ) then q h (t) = x h q0 (t). One can check (see, for example, [CMR2]) that for any curve q(t), t ∈ [a, b] in Q we have Associated Bundles. Consider a left representation ρ : G × M → M of the Lie group G on a vector space M . Recall that the associated vector bundle with standard fiber M is, by definition, 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 or simply [q, m]. 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 it is well defined and is a surjective submersion.
Parallel Transport in Associated Bundles. 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 in 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) = x h q0 (t), m 0 G . 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 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). Here we are using the identification T M = M × M , appropriate for vector spaces. Thus, we are thinking of the infinitesimal generator as a map ρ : g → End(M ) (the linear vector fields on M are identified with the space of linear maps of M to itself). Thus, we have a linear representation of the Lie algebra g on the vector space its projection on the base Q/G, and let, as above, τ t t+s , where t, t + s ∈ [a, b], denote parallel transport along x(t) from time t to time t + s.
, m(t + s)] G and so we get the well-known fact that the covariant derivative of a vertical curve in the associated bundle is just the fiber derivative, that is, where m (t) is the time derivative of m.
Affine Connections in Vector Bundles. Recall from, for example, [KN], that the covariant derivative for curves in a given vector bundle τ : V → S is related to the notion of an affine connection ∇ by where, for each s 0 ∈ S, each X ∈ X ∞ (S) (the smooth vector fields on S), and each v ∈ Γ(V ) (the space of sections of V ), and s(t) is any curve in S such thaṫ Affine Connections on Associated Bundles. The following formula gives the relation between the covariant derivative of the affine connection and the principal connection.
The previous definition of the covariant derivative of a curve in the associated vector bundle Q × G M thus leads to an affine connection on Q × G M . Let us call this connection∇ A or simply∇. 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 Notice that we only need to know ϕ along the curve x(t) in order to calculate the covariant derivative.
The Adjoint Bundle. 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 denoted g := Ad(Q). We letπ G :g → Q/G denote the projection given byπ Let [q(s), ξ(s)] G be any curve ing. Then one checks that (again see [CMR2]) In addition, the adjoint bundle is a Lie algebra bundle; that is, each fiberg x , x ∈ Q/G, ofg carries a natural Lie algebra structure defined by The Bundle T Q/G. 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] Q . The vector bundle structure of T Q is inherited by this bundle. One can express reduced variational principles in a natural way in terms of this bundle without any reference to a connection on Q. It is, however, also interesting to introduce an (arbitrarily chosen) connection on Q relative to which one can more concretely realize the space T Q/G. This is also useful for writing the reduced Euler-Lagrange equations, called Lagrange-Poincaré equations.
One of the main tools needed for realizing the structure of the bundle is the map In fact (see [CMR2]), α A is a well defined vector bundle isomorphism with inverse given by

The Geometry of Variations
Spaces of Curves. The space of all (smooth) curves from a fixed time interval I = [t 0 , t 1 ] to Q will be denoted Ω(Q). Given a map f : for q(t) ∈ Ω(Q 1 ). For given q i ∈ Q, i = 0, 1, by definition, Ω(Q; q 0 ) and Ω(Q; q 0 , q 1 ) are, respectively, the spaces of curves q(t) on Q such that q(t 0 ) = q 0 and q(t i ) = q i , i = 0, 1. If π : Q → S is a bundle, q 0 ∈ Q, and π(q 0 ) = x 0 , then Ω(Q; x 0 ) denotes the space of all curves in Ω(Q) such that π(q(t 0 )) = x 0 . The space Ω(Q; x 1 ) is defined in an analogous way. Similarly, Ω(Q; x 0 , q 1 ) is the space of all curves in Ω(Q) such that π(q(t 0 )) = x 0 and q(t 1 ) = q 1 . The spaces of curves Ω(Q; q 0 , x 1 ), Ω(Q; x 0 , x 1 ), etc. are defined in a similar way. If V → Q and W → Q are vector bundles then Ω(V ) → Ω(Q) and Ω(W ) → Ω(Q) are vector bundles in a natural way and there is a natural identification Ω Variations of curves q(t) belonging to Ω(Q; q 0 ) or Ω(Q; q 0 , q 1 ) satisfy the corresponding fixed endpoints conditions, namely, δq(t 0 ) = 0 or δq(t i ) = 0 for i = 0, 1, respectively. Let τ : V → Q be a vector bundle and let v(t, λ) be a deformation in V of a curve v(t) in V . If τ (v(t, λ)) = q(t) does not depend on λ we will call v(t, λ) a V -fiber deformation of v(t), or simply, a fiber deformation of v(t). For each t, the variation δv(t) = ∂v(t, λ) ∂λ λ=0 may be naturally identified with an element, also called δv(t), of τ −1 (q(t)). In this case, the curve δv in V is, by definition, a V -fiber variation of the curve v, or, simply, a fiber variation of the curve v.
Horizontal and Vertical Variations. Consider a curve q ∈ Ω(Q; q 0 ). A vertical variation δq of q satisfies, by definition, the condition δq(t) = Ver(δq(t)) for all t.
The Structure of Vertical Variations. Given a curve q ∈ Ω(Q; q 0 , q 1 ), let v = A(q,q) ∈ g. Variations δq of q(t) induce corresponding variations δv ∈ g in the obvious way: Consider the decomposition q = g q q h introduced in (5). A vertical deformation q(t, λ) can be written as q(t, λ) = g q (t, λ)q h (t). The corresponding variation Now we introduce some important notation. Define the curve Notice that, by construction, Lemma 4.1. For any vertical variation δq = ηq of a curve q ∈ Ω(Q; q 0 , q 1 ) the corresponding variation δv of Proof. For completeness, we will give the proof in the case that G is a matrix group. The more general case can be treated using the appendix to [BKMR] 1 . By (6) As we saw in Theorem 2.1, one uses the constrained variations δξ =η + [ξ, η] for computing the corresponding variational principle. The above construction of v, η is not computing the same objects. These constrained variations are, instead, special instances of the construction of covariant variations, to be introduced shortly in Definition 4.3. In the second remark following Lemma 4.4, we shall explicitly remark on how the constructions of variations for the Euler-Poincaré equations and those for the Lagrange-Poincaré case are related.
The Structure of Horizontal Variations. Using the relation v = A(q,q) and differentiating with the help of the Cartan structure equations, one finds that variations δv corresponding to horizontal variations δq of a curve q ∈ Ω(Q; q 0 , q 1 ) are given as follows.
1 Another direct proof in the general case was told to us by Marco Castrillon.
The Covariant Variation on the Adjoint Bundle. Any curve in Q, q ∈ Ω(Q; q 0 , q 1 ) induces a curve ing in a natural way, namely, , where x(t) = π(q(t)) for all t. We want to study variations δ[q, v] G corresponding to vertical and also to horizontal variations δq of q.
While vertical variations δq give rise to vertical variations δ[q, v] G , horizontal variations δq need not give rise to horizontal variations δ[q, v] G . The deviation of any variation δ[q, v] G from being horizontal is measured by the covariant variation δ A [q, v] G (t), defined as follows. .
Vertical Variations and the Adjoint Bundle. We first consider the case of vertical variations. Using Lemma 4.1 and (8), one finds the following.
corresponding to a vertical variation δq = ηq is given by Remarks. 1. In view of (9), we can write 2. Let us now show that the formula δv =η +[v, η] for the constrained variations for the Euler-Poincaré equations coincides with the construction of the covariant variation given in Definition 4.3. Given a Lie group G, we regard it as a principal bundle over a point, that is, we take G = Q. The identification of g with T (Q/G)⊕g in this case is given by v → [e, v] G . This equivalence defines δv ≡ δ A [e, v] G and the preceding lemma shows that δv =η + [v, η], which is the same type of variation one has for the Euler-Poincaré equations.
The Reduced Curvature Form. In preparation for the consideration of variations δ A [q, v] G (t) corresponding to horizontal variations, one has the following.
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). This is proved readily by showing that the right hand side does not depend on the choice of (q,q) and (q, δq) using equivariance properties of the curvature.
Horizontal Variations and the Adjoint Bundle. Now we are ready to describe covariant variations δ A [q, v] G (t) corresponding to horizontal variations δq. By (8), Since δq is horizontal, we have A(q, δq) = 0. Using this and Lemmas 4.2 and 4.5, we obtain the following result.

The Euler-Lagrange and Euler-Poincaré Operators
Now we have the tools needed to carry out the reduction of the Euler-Lagrange equations by means of reduction of Hamilton's principle.

Reduced Spaces of Curves.
In what follows we often identify the bundles T Q/G and T (Q/G) ⊕g, using the isomorphism α A (see (10)). This leads to other natural identifications as well. For instance, the reduced set of curves [Ω(Q; q 0 , q 1 )] G is the set of curves [q] G (t) = [q(t)] G on Q/G such that the curve q(t) belongs to Ω(Q; q 0 , q 1 ). This reduced set of curves is naturally identified with the set of curves [q(t),q(t)] G in T Q/G such that q(t i ) = q i , for i = 0, 1, and in turn, this is identified, via the map with the set of curves T π(q(t),q(t)) ⊕ [q(t), A(q(t),q(t))] G in T (Q/G) ⊕g, such that q(t i ) = q i , for i = 0, 1. The image of this reduced set of curves will be denoted Ω(α A )([Ω(Q; q 0 , q 1 )] G ).
The Reduced Lagrangian. Let L : T Q → R be an invariant Lagrangian, that is, L(g(q,q)) = L(q,q) for all (q,q) ∈ T Q and all g ∈ G. Because of this invariance, we get a well defined reduced Lagrangian l : T Q/G → R satisfying l([q,q] G ) = L(q,q).
As we will see in detail in this section, the evolution of the reduced system will be a critical point, say a curve [q] G in the reduced set of curves [Ω(Q; q 0 , q 1 )] G , of the reduced action However, variations of curves in the reduced family of curves are not of the usual sort found in Hamilton's principle, and so the equations of motion in the bundle T Q/G cannot be written in a direct way.
We will use the description of vertical and horizontal variations given in the preceding section to derive equations of motion in a suitably reduced bundle. Equations corresponding to vertical variations will be called the vertical Lagrange-Poincaré equations, and equations corresponding to horizontal variations will be called the horizontal Lagrange-Poincaré equations.
Identification of Bundles. We shall allow a slight abuse of notation, namely we will consider l as a function defined on T (Q/G) ⊕g or T Q/G interchangeably, using the isomorphism α A . Also, we often use a slight abuse of the variable-notation for a function, namely we will write l(x,ẋ,v) to emphasize the dependence of l on (x,ẋ) ∈ T (Q/G) andv ∈g.
The Second Order Tangent Bundle T (2) Q. As will be clear below, the second order bundle plays a fundamental role in the study of the Euler-Lagrange equations. Forq ∈ Q, elements of T (2) q Q are equivalence classes of curves in Q; namely, two given curves q i (t), i = 1, 2, such that q 1 (t 1 ) = q 2 (t 2 ) =q are equivalent if in any local chart q 1 and q 2 agree up to and including their second derivatives. The equivalence class of the curve q(t) atq = q(t) will be denoted [q] (2) q or sometimes 2 by (q,q,q). There are natural fiber bundle structures T (2) Q → T Q → Q. The bundle T (2) Q → T Q is, in fact, a vector bundle in a natural way.
Using the projection π G (Q) : Q → Q/G, we obtain a bundle map This bundle map induces a bundle map (2) q .

The class of the element [q]
(2) q in the quotient T (2) Q/G will be denoted [q] (2) q G . More generally, it is easy to see that for any map f : M → N we have a naturally induced map (2) f (q) . In particular, a group action ρ : G×Q → Q can be naturally lifted to a group action (2) q = [ρ g • q] (2) ρ(g,q) .
We will often write ρ (2) g
We remark as an aside that T (2) G carries a natural Lie group structure. 3 If [g] (2) g , and [h] (2) h are classes of curves g and h in G, we define the product [g] (2) (2) h as being the class [gh] (2) gh at the pointḡh of the curve gh. The Lie algebra T e T (2) G of 2 The second order bundle T (2) Q is also denotedQ by some authors (see, for example, [MPS], [MR] and references therein).
3 Recall that T (1) G = T G is the semidirect product group G g whose Lie algebra is the semidirect product g g, where the second factor is regarded as the representation space of the adjoint action. This semidirect product Lie algebra is, as a vector space, equal to g ⊕ g.
T (2) G can be naturally identified, as a vector space, with g⊕g⊕g, which, therefore, carries a unique Lie algebra structure such that this identification becomes a Lie algebra isomorphism.
There is also a natural identification of T (2) e G with g ⊕ g, which is useful in what follows. Let us assume that G is a group of matrices. Let g(t) be any curve such that g(0) = e and let ξ(t) =ġ(t)g −1 (t). Thenξ(t) = −ġ(t)g −1 (t)ġ(t)g −1 (t) +g(t)g −1 (t).

Then the identification T
(2) e G ≡ g ⊕ g is given by [g] (2) e ≡ ξ 1 ⊕ ξ 2 . Also, T (2) Q is a principal bundle with structure group T (2) G in a natural way. More precisely, if [g] (2) g ∈ T (2) G is the class of a curve g in G and [q] (2) q ∈ T (2) q (Q) is the class of a curve q in Q we let [g] (2)

g [q]
(2) q ∈ T (2) gq Q denote the class [gq] (2) gq of the curve gq at the pointḡq. In particular, if ξ ∈ g ⊕ g and [q] (2) q ∈ T (2) Q are given, there is a well defined element ξ[q] (2) q ∈ T (2) Q. More precisely, let ξ = ξ 1 ⊕ ξ 2 , and let g(t) be any curve in G such that ξ = [g] (2) e , according with the identification described above. Then, we have As with the map A : T Q → g, we let A 2 : T (2) Q → g ⊕ g be defined by (2) q = v(t) ⊕v(t).
For any given ξ = ξ 1 ⊕ ξ 2 ∈ g ⊕ g and q ∈ Q, we can easily see, using the definition, that A 2 (ξq) = ξ. Similarly, as with α A , we get an isomorphism defined by It is a well defined bundle isomorphism, because We have also used the natural isomorphismg ⊕g ≡ g ⊕ g, given by the natural identification Ad g ξ 1 ⊕ Ad g ξ 2 ≡ Ad g (ξ 1 ⊕ ξ 2 ).
The inverse of α A2 is given by (2) (2),h x,q , where the meaning of [x] (2),h x,q has been explained above. We refer to [CMR2] for additional properties and discussion of higher order bundles.
Euler-Lagrange Operator. Next we introduce some notation and recall some basic results concerning Euler-Lagrange operators. The fundamental relationship between the variational and differential-equation description of the evolution of a given system is given by the following well-known result.
Theorem 5.1 (Euler-Lagrange). Let L : T Q → R be a given Lagrangian on a manifold Q and let S(L)(q) = t1 t0 L(q,q) dt be the action of L defined on Ω(Q; q 0 , q 1 ). Let q(t, λ) be a deformation of a curve q(t) in Ω(Q; q 0 , q 1 ) and let δq(t) be the corresponding variation. Then, by definition, δq(t i ) = 0 for i = 0, 1.
There is a unique bundle map EL (L) : T (2) Q → T * Q such that, for any deformation q(t, λ), keeping the endpoints fixed, we have where, as usual, The 1-form bundle-valued map EL (L) is called the Euler-Lagrange operator.
In local coordinates EL (L) has the following classical expression: in which it is understood that one regards the second term on the right hand side as a function on the second order tangent bundle by formally applying the chain rule and then replacing everywhere dq/dt byq and dq/dt byq. The Euler-Lagrange equations can, of course, be written simply as EL (L)(q,q,q) = 0.
Theorem 5.2. Let G be a Lie group, L : T G → R a left G-invariant Lagrangian and l : g → R its reduction. There is a unique bundle map EP(l) : g ⊕ g → g * such that, for any deformation v(t, λ) = g(t, λ) −1ġ (t, λ) ∈ g induced on g by a deformation g(t, λ) ∈ G of g(t) ∈ Ω(G; g 0 , g 1 ) keeping the endpoints fixed, and thus δg(t i ) = 0, for i = 0, 1, we have where, as usual, The map EP(l) is called the Euler-Poincaré operator and its expression is given by δl δv where, as before, it is to be understood that the time derivative on the second term is performed formally using the chain rule and that the expression dv/dt is replaced throughout byv.
The Euler-Poincaré equations can be written simply as EP(l)(v,v) = 0. Formula (14) represents the most general variation δv of v induced by an arbitrary variation δg via left translation. As in Theorem 2.1, η = g −1 δg and so the condition δg = 0 at the endpoints is equivalent to the condition η = 0 at the endpoints.

The Lagrange-Poincaré Operator
In this section we introduce the Lagrange-Poincaré operator using the same type of technique of reduction of variational principles that was used in the preceding section to define the Euler-Lagrange and the Euler-Poincaré operators.
Reducing the Euler-Lagrange Operator. The map EL (L) : T (2) Q → T * Q, being G-equivariant, induces a quotient map [EL (L)] G : T (2) Q/G → T * Q/G, which depends only on the reduced Lagrangian l : T Q/G → R; that is, we can identify [EL (L)] G with an operator EL (l). This is called the reduced Euler-Lagrange operator and it does not depend on any extra structure on the principal bundle Q. However, to write the explicit expressions, which are also physically meaningful, we use the additional structure of a principal connection A on the principal bundle Q → Q/G to identify the quotient bundle and T * Q/G with T * (Q/G) ⊕g * using the bundle isomorphisms α A2 from Lemma 13 and α A from (10), and also a connection ∇ on Q/G to concretely realize the reduced Euler-Lagrange operator; this will naturally lead us to the Lagrange-Poincaré operator.
Geometry of Reduced Variations. A general variation δv(t) of a given curvē v(t) ing is constructed as follows: choose a family of curvesv(t, s) ing such that v(t, 0) =v(t) and define This δv(t) is, for each t, an element of Tg. However, it turns out that we will not need these kinds of general variations δv subsequently. Instead, we are interested in the special kind of deformationsv(t, s) of the curvev(t) for which the projectioñ π G (v(t, s)) = x(t, s) does not depend on s, that is, deformations that take place only in the fiber ofg over x(t) =π G (v(t)); thus, for each fixed t, the curve s → v(t, s) is a curve in the fiber over x(t). Then, sinceg is a vector bundle, the variation δv(t) induced by such a deformationv(t, s), is naturally identified with a curve ing, also called δv(t); this is ag-fiber variation, according to the notation introduced in Section 4. Below, δv will always mean ag-variation, unless explicitly stated otherwise. Examples ofg-fiber variations δv are the covariant variations δ Av considered in Definition 4.3, but, of course, there are more general variations of this type. We encountered such an example when reviewing the Euler-Poincaré equations. In that case, Q = G, the connection A : T G → g is given by right translation, and , for η(t) a curve in g vanishing at the endpoints.
In the Euler-Poincaré case, any deformation of a curve v(t) is a deformation along the fiber, because in this situation the base ofg is a point. However, it is not true that any curve in g is induced by a variation δg that vanishes at the endpoints; the latter are only the curves of the typeη(t) + [v(t), η(t)], for η(t) an arbitrary curve in g vanishing at the endpoints.
In the study of the Lagrange-Poincaré operator, we will use variations of curves in Q/G ⊕g (the first summand means the vector bundle over Q/G with zero dimensional fiber). For a given curve x(t) ⊕v(t) in Q/G ⊕g, and a given arbitrary deformation x(t, λ) ⊕v(t, λ), with x(t, 0) ⊕v(t, 0) = x(t) ⊕v(t), the corresponding covariant variation δx(t) ⊕ δ Av (t) is, by definition, It is clear that δ Av is ag-fiber variation ofv. The most important example of a covariant variation δx(t) ⊕ δ Av (t) is the one to be described next. Let q(t, s) be a deformation of a curve q(t) = q(t, 0) in Q. This induces a deformation x(t, s) ⊕v(t, s) of the curve x(t) ⊕v(t) by taking x(t, s) = [q(t, s)] G andv(t, s) = [q(t, s), A(q(t, s),q(t, s))] G , whereq(t, s) represents the derivative with respect to t. Using (8) and Definition 4.3, it follows that the covariant variation corresponding to this deformation of x(t) ⊕v(t) is given by is an element ofg for each t, with η(t) ∈ g an arbitrary curve vanishing at the endpoints. This is a special kind of covariant variation. It is precisely to these kinds of variations that we will apply the usual techniques of the calculus of variations in the next theorems to derive the Lagrange-Poincaré operator and equation. The previous formula may be rewritten as which emphasizes the similarity with the Euler-Poincaré case.
Lagrange-Poincaré Operator. We are now ready to state a theorem that introduces the Lagrange-Poincaré operator. Its proof will be contained in the proof of Theorem 6.4.
Notice that, after all the identifications described at the beginning of the present paragraph, the operator LP(l) coincides with the operator [EL (L)] G . In the following paragraph we introduce some additional structure, namely, an arbitrary connection ∇ on the manifold Q/G. This will also help us write explicit expressions of Hor(LP)(l) and Ver(LP)(l). For simplicity we shall assume, as in [CMR2], that this connection is torsion free; however, as in [GSS], it is easy to relax this condition.
The problem of computing the Lagrange-Poincaré equations can be done using any connection, as we remarked earlier and, in addition, the problem can be localized to any local trivialization of the bundle Q → Q/G. Because of this, one may choose the vector space or trivial connection associated with such a local trivialization of the bundle. Of course we are not assuming that the bundle has a global flat connection.
Explicit formulas for Hor(LP)(l) and Ver(LP)(l) in coordinates using any connection can be calculated from what we have developed and are given in [CMR2]. Doing so, one arrives at the coordinate formulas given in [MS2]. We also mention that it is possible to derive these equations from [CIM] in a straightforward way.
Reduced Covariant Derivatives. Calculating intrinsic formulas for the horizontal and vertical Lagrange-Poincaré operators Hor(LP)(l) and Ver(LP)(l) can be made more explicit by giving meaning to the partial derivatives ∂l ∂x , ∂l ∂ẋ , and ∂l ∂v .
Sinceg and T (Q/G) are vector bundles, we may interpret the last two derivatives in a standard (fiber derivative) way as being elements of the dual bundles T * (Q/G) andg * respectively, for each choice of (x,ẋ,v) in T (Q/G) ⊕g. In other words, for given (x 0 ,ẋ 0 ,v 0 ) and (x 0 , x ,v ) we define To define the derivative ∂l/∂x, one uses the chosen connection ∇ on the manifold Q/G, as we will explain next. Let (x 0 ,ẋ 0 ,v 0 ) be a given element of T (Q/G) ⊕g. For any given curve x(s) on Q/G, let (x(s),v(s)) be the horizontal lift of x(s) with respect to the connection∇ A ong (see (7)) 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 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 Likewise ,

Reduced Variational Principles and the Lagrange-Poincaré Equations.
The main result in this section is the following theorem. Its proof also contains the proof of the preceding theorem.
Theorem 6.4. Under the hypothesis of Theorem 6.1, the vertical Lagrange-Poincaré operator is given by or simply, and the horizontal Lagrange-Poincaré operator is given by or simply, Proof. To compute the vertical and horizontal Lagrange-Poincaré operator, it suffices to consider variations δ Av of a curve x(t) ⊕v(t) corresponding to vertical and horizontal variations δq of a curve q ∈ Ω(Q, q 0 , q 1 ). The computations below will show that these variations suffice to give us the variational principle in the directions of the two summands in δx ⊕η ∈ T (Q/G) ⊕g. First, consider variations δ Av of a curve x(t) ⊕v(t) corresponding to vertical variations δq of a curve q. We have According to Lemma 4.4 withv = [q, v] G we obtain, for all curves η(t) ∈ g such that η(t i ) = 0 for i = 1, 2, the equation Arbitrariness of η then yields arbitrariness ofη = [q, η] G , so we get Now consider variations δx ⊕ δ Av corresponding to horizontal variations δq. Then we have, for all δx with δx(t i ) = 0, for i = 0, 1 Integration by parts and Lemma 4.6 withv = [q, v] G gives Integration by parts of the term (∂l/∂ẋ)δẋ is justified by showing that which can be done, for example, by using Gaussian coordinates relative to the connection ∇ at each point x(t). Arbitrariness of δx then yields Remarks. 1. The operators EL (l), Hor(LP)(l) and Ver(LP)(l) depend on the (principal) connection A on the principal bundle Q but not on the connection ∇ on Q/G. It is only the explicit expressions of Hor(LP)(l) and Ver(LP)(l) that appear in Theorem 6.4 that depend on ∇. As we have remarked previously, in local coordinates it is often convenient to choose ∇ to be simply the usual Euclidean, or vector space, connection. 2. Important particular cases of Theorems 6.1 and 6.4 occur when G = Q and also when G = {e}. If G = Q then the operator Hor(LP)(l) is 0 and Ver(LP)(l) is the Euler-Poincaré operator, as we saw before. Thus, in a sense, the vertical Lagrange-Poincaré operator in the bundleg is a covariant version of the usual Euler-Poincaré operator on a Lie algebra. If G = {e} then Ver(LP)(l) is 0, L = l and Hor(LP)(l) = EL (L) is the usual Euler-Lagrange operator.
Explicit Covariant Description of the Lagrange-Poincaré Operator. We begin by describing the identification of T (2) Q with T Q ⊕ T Q using a connection on Q, whose proof is straightforward.
Using the diffeomorphism γ 2 as an identification, for a given Lagrangian L : T Q → R, the Euler-Lagrange operator Now we shall use this description of EL to write a more explicit covariant expression of the Euler-Lagrange operator Let τ : V → Q and ϕ : F → Q be given vector bundles and denote by Λ and Σ the Christoffel symbols of the given connections in the bundles V and F, respectively. We shall call D Dt the corresponding covariant derivative in any of the bundles V or F. Recall that are to be understood in the covariant sense, as explained before. Also recall that, if there is no danger of confusion, we often omit the superscript C, and write the previous covariant derivatives simply as ∂f ∂x and ∂v ∂x .
We need the following general lemma.
Lemma 6.6. Let f : V → F be a given fiber-preserving map. Let v(t) be any given curve in V. Then we have Proof. Let us choose arbitrary local trivializations of V and F. Then we have where, in this case, since we are working in a local trivialization, the meaning of ∂f ∂x is the usual one. Sincė , v h (t) is horizontal, and τ (v(t)) = τ (v h (t)) for all t. Then, by definition, we have On the other hand, since v h is horizontal, its covariant derivative is 0, so we havė We also have By using the previous expressions we easily obtain the assertion of the lemma. Now let us apply the previous lemmas to find expressions for the Euler-Lagrange operator. Take V = T Q, v = (q,q), F = T * Q, and f (q,q) = ∂L ∂q .
Then, using the previous lemmas, the Euler-Lagrange operator (15) at a point where, this time, as we often do, we have eliminated the superscript C. Now, we shall show how to write the Lagrange-Poincaré operator in an explicit covariant way. Using the connection in the bundle T (Q/G) we have a natural identification and therefore, also an identification To use the previous lemmas, take V = T (Q/G) ⊕g, F = T * (Q/G) ⊕g * , and where l is the reduced Lagrangian. Recall that since we have connections in the vector bundles T (Q/G) andg, we have the Whitney sum connection in T (Q/G) ⊕g and also the dual connection in T * (Q/G) ⊕g * . Then the term D Dt ∂l ∂v (x,ẋ,v) that appears in the Vertical Lagrange-Poincaré operator can be written as The term D Dt ∂l ∂ẋ that appears in the Horizontal Lagrange-Poincaré operator is written as Collecting formulas, we have proved the following theorem.
Theorem 6.7. Using the above notations, the Lagrange-Poincaré operator the identifications α A and α A * . Namely, we have, Reducing Hamilton's Principle. For a given Hamiltonian H : T * Q → R Hamilton's phase space principle states that δ (p ·q − H(q, p)) dt = 0 with suitable boundary conditions, which is equivalent to Hamilton's equations of motion. In this principle, note that the pointwise function in the integrand, namely is defined on T Q ⊕ T * Q, regarded as a bundle over Q, the base space common to T * Q and T Q. The group G acts on T Q⊕T * Q by simultaneously left translating on each factor by the tangent and cotangent lift. Explicitly, the action of an element h ∈ G is given by The map F is invariant under this action of G as is easily checked, assuming invariance of H. Thus, the function F drops to the quotient, namely to the function f : T Q/G ⊕ T * Q/G → R, or, equivalently, f : T (Q/G) ⊕g ⊕ T * (Q/G) ⊕g * → R given by f (x,ẋ,v)⊕(x, y,μ) = y,ẋ + μ,v −h(x, y,μ), where h is the reduction of H from T * Q to T * (Q/G) ⊕g * .
When the function F is used in the phase space variational principle, one is varying curves (q(t), p(t)) and one of course insists that the slotq actually is the time derivative of q(t). This restriction induces in a natural way a restriction on the variations of the curve [q(t),q(t)] G = (x(t),ẋ(t),v(t)) and these restrictions are computed as in Theorem 6.4. Thus, with the boundary conditions δx(t i ) = 0 andη(t i ) = 0, for i = 0, 1. On the other hand, arbitrary variations δp induce arbitrary fiber variations δy and δμ.
Using the same kind of argument, based on reducing the action and the variations, that we have used to derive the Lagrange-Poincaré reduced variational principle and equations, we can easily show that Hamilton's phase space variational principle can also be reduced. In fact, we obtain the reduced equations of motion by applying the usual integration by parts argument to the action t1 t0 ( y,ẋ + μ,v − h(x, y,μ)) dt, with variations with the restrictions explained above. In this way, we obtain the following equations of motion, called Hamilton-Poincaré equations: One should be aware that the derivative ∂h/∂x must be interpreted in a covariant sense, similar to the partial derivative ∂l/∂x as defined in the previous section; this time one uses the covariant derivatives in the bundles T * (Q/G) andg * induced by duality by the corresponding covariant derivatives in T (Q/G) andg.
If we write y = ∂l ∂ẋ andμ = ∂l ∂v , and take into account the equality ∂l ∂x = − ∂h ∂x , then the Hamilton-Poincaré equations become the Lagrange-Poincaré equations.

The Hamilton-Poincaré Variational Principle
We are now in a position to summarize what we have obtained in the following result that generalizes Theorem 2.1.
Theorem 8.1. With the above notation and hypotheses of hyperregularity, the following conditions are equivalent.
(iv) The Hamilton-Poincaré Variational Principle. The reduced curve is a critical point of the action t1 t0 ( y,ẋ + μ,v − h(x, y,μ)) dt, with variations where δx(t) andη(t) satisfy the same conditions as in (ii) and δy and δμ are arbitrary fiber variations.
The Reduced Poisson Structure. With the Hamilton-Poincaré equations established, it is not difficult to obtain the reduced bracket in the Poisson manifold T * Q/G = T * (Q/G) ⊕g * , in terms of the structure of Lagrange-Poincaré bundles. Using this and the Hamilton-Poincaré equations, the formula for the Poisson bracket follows easily.
The proof of this result is considerably simpler than that given in [MMR] and [Mon] (see also [LMMR]).
The Symplectic Leaves of T * (Q/G)⊕g. For each µ ∈ g * , let O µ be the coadjoint orbit through µ in g * . Let us define the subset S µ ⊂ T * Q by S µ = {α q : J(α q ) ∈ O µ }, where J : T * Q → g * is the standard cotangent bundle momentum mapping for the action of G on Q. It is clear that S µ is an invariant subset under the cotangent action of G on T * Q. By the well-known equivalence of point and orbit reduction (see [OR] for the case when the coadjoint orbit is not an embedded submanifold of g * ), the reduced spacesS µ = S µ /G are the symplectic leaves of the Poisson manifold T * (Q/G) ⊕g * . To describeS µ more precisely, let us define, for each µ ∈ g * , the bundleÕ µ ⊂g * , byÕ µ = (Q × O µ )/G. Then, as explained for instance in [MP], it is easy to prove using the previous definitions, thatS µ = T * (Q/G) × XÕµ .
We now show how to write a covariant formula for the symplectic structure ω µ ofS µ , using the Hamilton-Poincaré equations. A generic tangent vector toS µ at a point (x, y) ⊕ν is given, according to the Hamilton-Poincaré equations, by ẋ, Dy Dt , advν ≡ (ẋ, p, advν) .
Proof. In order to prove that the preceding expression is, in fact, the symplectic form, one can simply observe that ω µ ((x, y)⊕ν) ∂f 1 ∂y , ∂f 1 ∂x , adv 1ν , ∂f 2 ∂y , ∂f 2 ∂x , adv 2ν = {f 1 , f 2 }((x, y)⊕ν), wherev i = ∂f i ∂ν , for i = 1, 2. Closedness and nondegeneracy of ω µ are proved as follows. To prove nondegeneracy, note that, by construction, ω µ ((x, y) ⊕ν)(X g , X f ) = dg · X f where f, g : T * X ⊕g * → R are arbitrary functions, so that at a given point ((x, y) ⊕ν), X f represents an arbitrary tangent vector to the symplectic leaf. Since we can choose g such that dg · X f is nonzero at the given point whenever X f is nonzero, we see that ω µ is nondegenerate. It is standard that the Jacobi identity for the Poisson bracket gives closedness of ω µ .
Using the previous expressions one sees that ω µ is the sum of three 2-forms, corresponding to the three terms of the reduced Poisson bracket. The one corresponding to the third term is the usual Kostant-Kirillov-Souriau symplectic form in the fiberg * x ofg * at x ∈ Q/G, the one corresponding to the first term is the canonical symplectic form on T * (Q/G), and the one corresponding to the second term is a "magnetic term" involving the curvature.

Example: The Rigid Body with Rotors
Following [MS2], the configuration space of a rigid body with rotors whose axes are parallel to the three principal axes of inertia of the rigid body, is the principal bundle π : SO(3) × S 1 × S 1 × S 1 → S 1 × S 1 × S 1 , with structure group SO(3) acting on the left on the first factor. Since it is a trivial bundle, we can choose the trivial principal connection A. The reduced cotangent bundle is easily seen to be so(3) * × R 3 * × S 1 × S 1 × S 1 → S 1 × S 1 × S 1 .
Let ν r = ∂l ∂Ω r and y r = ∂l ∂θ r The Hamiltonian of the system is obtained by the Legendre transformation and the reduced Hamiltonian is given by h(ν, y) = 1 2 3 r=1 (ν r − y r ) 2 I r + 1 2 3 r=1 y 2 r K r .
This reduced Hamiltonian is in agreement with the reduced Lagrangian, as given in [MS2], via the Legendre transformation, after one identifies vectors and covectors in so(3) ≡ R 3 via the Euclidean metric. Since the connection is trivial, covariant derivatives are the same as usual derivatives. We have ∂h ∂ν r = ν r − y r I r .
Then, using the Hamilton-Poincaré equations we can conclude that y is a constant of the motion. We observe that the vector whose components are ν r − y r I r is Ω. Thus, the Hamilton-Poincaré equations can be written using the standard identification so(3) ≡ R 3 , as follows:ẏ = 0, ∂h ∂ν r = Ω, These equations are equivalent to the Lagrange-Poincaré equations given in [MS2] via the reduced Legendre transformation, as is easy to check. The variational description of the Hamilton-Poincaré equations in this example is given by t1 t0 ( y,θ + ν.Ω − h(ν, y)) dt, with variations restricted by δΩ =η + [Ω, η], δη(t i ) = 0, δθ(t i ) = 0, for i = 0, 1, and δν, δy arbitrary.
Future Work. We plan to develop the geometry of the bundle T Q ⊕ T * Q, how it fits into the spaces T * T Q, T T * Q, and the generalized Legendre transform (allowing one to treat degenerate Lagrangians) introduced by [Tul].