Reconstruction Phases for Hamiltonian Systems on Cotangent Bundles

Reconstruction phases describe the motions experienced by dynamical systems whose symmetry-reduced variables are undergo- ing periodic motion. A well known example is the non-trivial rotation experienced by a free rigid body after one period of oscillation of the body angular momentum vector. Here reconstruction phases are derived for a general class of Hamilto- nians on a cotangent bundle T ⁄ Q possessing a group of symmetries G, and in particular for mechanical systems. These results are presented as a synthesis of the known special cases Q = G and G Abelian, which are reviewed in detail.


Introduction
When the body angular momentum of a free rigid body undergoes one period of oscillation the body itself undergoes some overall rotation in the inertial frame of reference.This rotation is an example of a reconstruction phase, a notion one may formulate for an arbitrary dynamical system possessing symmetry, whenever the symmetry-reduced variables are undergoing periodic motion.Interest in reconstruction phases stems from problems as diverse as the control of artificial satellites [8] and wave phenomena [3,2].
This paper studies reconstruction phases in the context of holonomic mechanical systems, from the Hamiltonian point of view.Our results are quite general in the sense that non-Abelian symmetries are included; however certain singularities must be avoided.We focus on so-called simple mechanical systems (Hamiltonian='kinetic energy' + 'potential energy') but our results are relevant to other Hamiltonian systems on cotangent bundles T * Q.The primary prerequisite is invariance of the Hamiltonian with respect to the cotangent lift of a free and proper action on the configuration space Q by the symmetry group G. Our results are deduced as a special case of those in [6].
We do not study phases in the context of mechanical control systems and locomotion generation, as in [17] and [15]; nor do we discuss Hanay-Berry phases for 'moving' mechanical systems (such as Foucault's pendulum), as in [16].Nevertheless, these problems share many features with those studied here and our results may be relevant to generalizations of the cited works.
1.1.Limiting cases.The free rigid body is a prototype for an important class of simple mechanical systems, namely those for which Q = G.Those systems whose symmetry group G is Abelian constitute another important class, of which the heavy top is a prototype.Reconstruction phases in these two general classes have been studied before [16], [6].Our general results are essentially a synthesis of these two cases, but because the synthesis is rather sophisticated, detailed results are formulated after reviewing the special cases in Section 2. This introduction describes the new results informally after pointing out key features of the two prototypes.A detailed outline of the paper appears in 1.5 below.
1.2.The free rigid body.In the free (Euler-Poinsot) rigid body reconstruction phases are given by an elegant formula due to Montgomery [23].Both the configuration space Q and symmetry group G of the free rigid body can be identified with the rotation group SO(3) (see, e.g., [18,Chapter 15]); here we are viewing the body from an inertial reference frame centered on the mass center.Associated with each state x is a spatial angular momentum J(x) which is conserved.The body representation of angular momentum ν ∈ R 3 of a state x with configuration q ∈ SO( 3) is (1) ν = q −1 J(x) .
The body angular momentum ν evolves according to well known equations of Euler which, in particular, constrain solutions to a sphere O centered at the origin and having radius µ 0 , where µ 0 = J(x 0 ) is the initial spatial angular momentum.This sphere has a well known interpretation as a co-adjoint orbit of SO (3).Solutions to Euler's equations are intersections with O of level sets of the reduced Hamiltonian h : R 3 → R, given by h(ν) ≡ ν t ∈ O is periodic, in which case (1) implies that q T µ 0 = q 0 µ 0 , where T is the period.This means q T = gq 0 for some rotation g ∈ SO(3) about the µ 0 -axis.According to [23], the angle ∆θ of rotation is given by where S ⊂ O denotes the region bounded by the curve ν t (see figure) and dA O denotes the standard area form on the sphere O ⊂ R 3 .Astonishingly, it seems that (2) was unknown to 19th century mathematicians, a vindication of the 'bundle picture' of mechanics promoted in Montgomery's thesis [22].
1.3.The heavy top.Consider a rigid body free to rotate about a point O fixed to the earth (Fig. 2).The configuration space is Q ≡ SO(3) but full SO(3) spatial symmetry is broken by gravity (unless O and the center of mass coincide).A residual symmetry group G ≡ S 1 acts on Q according to θ • q ≡ R 3 θ q (θ ∈ S 1 ); here R 3 θ denotes a rotation about the vertical axis e 3 through angle θ.
PSfrag replacements The quotient space Q/G, known more generally as the shape space, is here identifiable with the unit sphere S 2 : for a configuration q ∈ SO(3) the corresponding 'shape' r ∈ S 2 is the position of the vertical axis viewed in body coordinates: (1) r = q −1 e 3 .
In the special Lagrange top case these equations are integrable (see, e.g., [4, §30]), but more generally they admit chaotic solutions.In any case, a periodic solution to the Euler-Poisson equations determines a periodic solution r t ∈ S 2 in shape space but the corresponding motion of the body q t ∈ SO(3) need not be periodic.However, if T is the period of the given solution to the Euler-Poisson equations, then (1) implies q T = R 3 ∆θ q 0 , for some angle ∆θ.Assume r t ∈ S 2 is an embedded curve having T as its minimal period.Then where Here S ⊂ S 2 denotes the region bounded by the curve r t , dA S 2 denotes the standard area form on S 2 , and I denotes the inertia tensor, about O, of the body in its reference configuration (q = id).Equation ( 2) follows, for instance, from results reviewed in 2.6 and 2.7, together with a curvature calculation along the lines of [16, pp. 48-50].
1.4.General characteristics of reconstruction phases.In both 1.2(2) and 1.3(2) the angle ∆θ splits into two parts known as the dynamic and geometric phases.The dynamic phase amounts to a time integral involving the inertia tensor. 1 The geometric phase is a surface integral, the integrand depending on the inertia tensor in the case of the heavy top but being independent of system parameters in the case of the free rigid body.Apart from this, an important difference is the space in which the phase calculations occur.In the heavy top this is shape space (which is just a point in the free rigid body).In the free rigid body one computes on momentum spheres, i.e., on co-adjoint orbits (which are trivial for the symmetry group S 1 of the heavy top).As we will show, phases in general mechanical systems are computed in 'twisted products' of shape space Q/G and co-adjoint orbits O, and geometric phases have both a 'shape' and 'momentum' contribution.The source of geometric phases is curvature.The 'shape' contribution comes from curvature of a connection A on Q, bundled over shape space Q/G, constructed using the kinetic energy.This is the so-called mechanical connection.The 'momentum' contribution to geometric phases comes from curvature of a connection α µ 0 on G, bundled over a co-adjoint orbit O, constructed using an Ad-invariant inner product on the Lie algebra g of G.We tentatively refer to this as a momentum connection.The mechanical connection depends on the Hamiltonian; the momentum connection is a purely Lie-theoretic object .This explains why system parameters appear explicitly in geometric phases for the heavy top but not in the free rigid body.
In arbitrary simple mechanical systems the dynamic phase is a time integral involving the so-called locked inertia tensor I. Roughly speaking, this tensor represents the contribution to the kinetic energy metric coming from symmetry variables.In a system of coupled rigid bodies moving freely through space, it is the inertia tensor about the instantaneous mass center of the rigid body obtained by locking all coupling joints [14, §3.3] 1.5.Paper outline.The new results of this paper are Theorems 3.4 and 3.5 (Section 3).These theorems contain formulas for geometric and dynamic phases in general Hamiltonian systems on cotangent bundles, 1 In the free rigid body one has 2T h(ν and in particular for simple mechanical systems.These results are derived as a special case of [6], of which Section 2 is mostly a review.Specifically, Section 2 gives the abstract definition of reconstruction phases, presents a phase formula for systems on arbitrary symplectic manifolds, and surveys the special limiting cases relevant to cotangent bundles.The mechanical connection A, the momentum connection α µ 0 , and limiting cases of the locked inertia tensor I are also defined.
Section 3 begins by showing how the curvatures of A and α µ 0 can be respectively lifted and extended to structures Ω A and Ω µ 0 on 'twisted products' of shape space Q/G and co-adjoint orbits O. On these products we also introduce the inverted locked inertia function ξ I .
The remainder of the paper is devoted to a proof of Theorems 3.4 and 3.5.Sections 4 and 5 review relevant aspects of cotangent bundle reduction, culminating in an intrinsic formula for symplectic structures on leaves of the Poisson-reduced space (T * Q)/G.Section 6 builds a natural 'connection' on the symplectic stratification of (T * Q)/G, and Sections 7 and 8 provide the detailed derivations of dynamic and geometric phases.Appendix A describes the covariant exterior calculus of bundle-valued differential forms, from the point of view of associated bundles.

Connections to other work.
Above what is explicitly cited here, our project owes much to [16].Additionally, we make crucial use of Cendra, Holm, Marsden and Ratiu's description of reduced spaces in mechanical systems as certain fiber bundle products [9].
In independent work, carried out from the Lagrangian point of view, Marsden, Ratiu amd Scheurle [19] obtain reconstruction phases in mechanical systems with a possibly non-Abelian symmetry group by directly solving appropriate reconstruction equations.Rather than identify separate geometric and dynamic phases, however, their formulas express the phase as a single time integral (no surface integral appears).This integral is along an implicitly defined curve in Q, whereas our formula expresses the phase in terms of 'fully reduced' objects.
The author thanks Matthew Perlmutter for helpful discussions and for making a preliminary version of [24] available.

Review
In the setting of Hamiltonian systems on a general symplectic manifold P , reconstruction phases can be expressed by an elegant formula involving derivatives of leaf symplectic structures and the reduced Hamiltonian, these derivatives being computed transverse to the symplectic leaves of the Poisson-reduced phase space P/G [6].This formula, recalled in Theorem 2.3 below, grew out of a desire to 'Poisson reduce' the earlier scheme of Marsden et al. [16, §2A], in which geometric phases were identified with holonomy in an appropriate principal bundle equipped with a connection.Familiarity with this holonomy interpretation is not a prerequisite for understanding and applying Theorem 2.3.
We are ultimately concerned with the special case of cotangent bundles P = T * Q, and in particular with simple mechanical systems, which are introduced in 2.4.After recalling the definition of the mechanical connection A in 2.5 we recall the formula for phases in the case of G Abelian (Theorem 2.6 & Addendum 2.7).After introducing the momentum connection α µ in 2.8 we write down phase formulas for the other limiting case, Q = G (Theorem & Addendum 2.9).

2.1.
An abstract setting for reconstruction phases.Assume G is a connected Lie group acting symplectically from the left on a smooth (C ∞ ) symplectic manifold (P, ω), and assume the existence of an Ad *equivariant momentum map J : P → g * .(For relevant background, see [14,1,18].)Here g denotes the Lie algebra of G. Assume G acts freely and properly, and that the fibers of J are connected.All these hypotheses hold in the case P = T * Q when we take G to act by cotangent-lifting a free and proper action on Q and assume Q is connected; details will be recalled in Section 3.
In general, P/G is not a symplectic manifold but merely a Poisson manifold, i.e., a space stratified by lower dimensional symplectic manifolds called symplectic leaves; see opi cited.In the free rigid body, for example, one has P = T * SO(3), G = SO(3), and P/G ∼ = so(3) * ∼ = R 3 .The symplectic leaves are the co-adjoint orbits, i.e., the spheres centered on the origin.
Let x t denote an integral curve of the Hamiltonian vector field X H on P corresponding to some G-invariant Hamiltonian H. Restrict attention to the case that the image curve y t under the projection π : P → P/G is T -periodic (T > 0).Then the associated reconstruction phase is the unique g rec ∈ G such that x T = g rec • x 0 ; see Fig. 3.

PSfrag replacements
The definition of the reconstruction phase g rec .
Noether's theorem (J(x t ) = constant) implies that y t , which is called the reduced solution, lies in the reduced space P µ 0 (see the figure), where and where µ 0 ≡ J(x 0 ) is the initial momentum.In fact, P µ 0 is a symplectic leaf of P/G (see Theorem 5.1) and the Ad * -equivariance of J implies g rec ∈ G µ 0 , where G µ 0 is the isotropy of the co-adjoint action at µ 0 ∈ g * .Invariance of H means H = h • π for some h : P/G → R called the reduced Hamiltonian; the reduced solution y t ∈ P µ 0 is an integral curve of the Hamiltonian vector field X hµ 0 corresponding to Hamiltonian h µ 0 ≡ h|P µ 0 .
2.2.Differentiating across symplectic leaves.We wish to define a kind of derivative in P/G transverse to symplectic leaves; these derivatives occur in the phase formula for general Hamiltonian systems to be recalled in 2.3 below.For this we require a notion of infinitesimal transverse.Specifically, if C denotes the characteristic distribution on P/G (the distribution tangent to the symplectic leaves), then a connection on the symplectic stratification of P/G is a distribution D on P/G complementary to C: TP = C ⊕ D. In that case there is a canonical two-form ω D on P/G determined by D, whose restriction to a symplectic leaf delivers that leaf's symplectic structure, and whose kernel is precisely D.
Below we concern ourselves exclusively with connections D defined in a neighborhood of a nondegenerate symplectic leaf, assuming D to be smooth in the usual sense of constant rank distributions.Then ω D is smooth also.
Fix a leaf P µ and assume D(y) is defined for all y ∈ P µ .Then at each y ∈ P µ there is, according to the Lemma below, a natural identification of the infinitesimal transverse D(y) with g * µ , denoted L(D, µ, y) : Now let λ be an arbitrary R-valued p-form on P/G, defined in a neighborhood of P µ .Then we declare the transverse derivative D µ λ of λ to be the g µ -valued p-form on P µ defined through ν, D µ λ(v 1 , . . ., v p ) = dλ L(D, µ, y)(ν), v 1 , . . ., v p where ν ∈ g * µ , v 1 , . . ., v p ∈ T y P µ and y ∈ P µ .Lemma and Definition.Let p µ : g * → g * µ denote the natural projection, and define T J −1 (µ) P ≡ ∪ x∈J −1 (µ) T x P .Fix y ∈ P µ and let v ∈ D(y) be arbitrary.Then for all w ∈ T J −1 (µ) P such that Tπ • w = v, the value of p µ dJ, w ∈ g * µ is the same.Moreover, the induced map v → p µ dJ, w : D(y) → g * µ is an isomorphism.The inverse of this isomorphism (which depends on D, µ and y) is denoted by L(D, µ, y) : We remark that the definition of L(D, µ, y) is considerably simpler in the case of Abelian G; see [6].

Reconstruction phases for general Hamiltonian systems.
Let g * reg ⊂ g * denote the set of regular points of the co-adjoint action, i.e., the set of points lying on co-adjoint orbits of maximal dimension (which fill an open dense subset).If µ 0 ∈ g * reg then g µ 0 is Abelian; see Appendix B. In that case G µ 0 is Abelian if it is connected.Now suppose, in the scenario described earlier, that a reduced solution y t ∈ P µ 0 bounds a compact oriented surface Σ ⊂ P µ 0 .
Theorem (Blaom [6]).If µ 0 ∈ g * reg and G µ 0 is Abelian, then the reconstruction phase associated with the periodic solution y t ∈ ∂Σ is g rec = g dyn g geom , where: Here h denotes the reduced Hamiltonian, D denotes an arbitrary connection on the symplectic stratification of P/G, ω D denotes the canonical two-form on P/G determined byD, and D µ 0 denotes the transverse derivative operator determined by D as described above.
The Theorem states that dynamic phases are time integrals of transverse derivatives of the reduced Hamiltonian while geometric phases are surface integrals of transverse derivatives of leaf symplectic structures.
We emphasize that while g dyn and g geom depend on the choice of D, the total phase g rec is, by definition, independent of any such choice.
For the application of the above to non-free actions see [6].
2.4.Simple mechanical systems.Suppose a connected Lie group G acts freely and properly on a connected manifold Q.All actions in this paper are understood to be left actions.A Hamiltonian H : T * Q → R is said to enjoy G-symmetry if it is invariant with respect to the cotangentlifted action of G on T * Q (see [1, p. 283] for the definition of this action).This action admits an Ad * -equivariant momentum map J : T * Q → g * defined through (1) , where ξ Q denotes the infinitesimal generator on Q corresponding to ξ.A simple mechanical system is a Hamiltonian H : T * Q → R of the form Q denotes the symmetric contravariant two-tensor on Q determined by some prescribed Riemannian metric • , • Q on Q (the kinetic energy metric), and V is some prescribed G-invariant function on Q (the potential energy).To ensure G-symmetry we are supposing that G acts on Q by • , • Q -isometries.

Mechanical connections.
In general, the configuration space Q is bundled in a topologically non-trivial way over shape space Q/G, i.e., there is no global way to separate shape variables from symmetry variables.However, fixing a connection on the bundle allows one to split individual motions.In the case of simple mechanical systems such a connection is determined by the kinetic energy, but in general there is no canonical choice.All the phase formulas we shall present assume some choice has been made.
Under our free and properness assumptions, the projection ρ : Q → Q/G is a principal G-bundle.So we will universally require that this bundle be equipped with a connection one-form A ∈ Ω 1 (Q, g).If a Ginvariant Riemannian metric on Q is prescribed (e.g., the kinetic energy in the case of simple mechanical systems) a connection A is determined by requiring that the corresponding distribution of horizontal spaces hor ≡ ker A are orthogonal to the ρ-fibers (G-orbits).In this context, A is called the mechanical connection; its history is described in [14,

§3.3]
As we shall recall in Section 4.2, a connection A on ρ : 2.6.Phases for Abelian symmetries.Let H : T * Q → R be an arbitrary Hamiltonian enjoying G-symmetry.When G is Abelian it is known that each reduced space P µ (µ ∈ g * , P = T * Q) is isomorphic to T * (Q/G) equipped with the symplectic structure It should be emphasized that the identification P µ ∼ = T * (Q/G) depends on the choice of connection A. See, e.g., [6] for the details.In the above equation is the usual projection; curv A denotes the curvature of A, viewed as a g-valued two-form on Q/G (see, e.g., [16, §4]).The value of the reduced Hamiltonian The Theorem below is implicit in [6].The special case in Addendum 2.7 is due to Marsden et al [16] (explicitly appearing in [6]).
denote the corresponding curve in shape space.Assume t → r t bounds a compact oriented surface S ⊂ Q/G.Assume r t and y t have the same minimal period T .Then the reconstruction phase associated with y t is g rec = g dyn g geom , where: and where ∂h/∂µ (µ , y ) ∈ g is defined through Here A denotes an arbitrary connection on Q → Q/G.

Locked inertia tensor (Abelian case).
In the special case of a simple mechanical system one may be explicit about the dynamic phase.To this end, define for each q ∈ Q a map Î(q) : g → g * through where ξ Q denotes the infinitesimal generator on Q corresponding to ξ.
Varying over all q ∈ Q, one obtains a function Î : Q → Hom(g, g * ).
When G is Abelian, Î is G-invariant, dropping to a function I : Q/G → Hom(g, g * ) called the locked inertia tensor (terminology explained in 1.4).As G acts freely on Q, Î(q) : g → g * has an inverse Î(q) −1 : g * → g leading to functions Î−1 : Q → Hom(g * , g) and Addendum.When H : T * Q → R is a simple mechanical system and A is the mechanical connection, then the dynamic phase appearing in the preceding Theorem is given by In particular, the reconstruction phase g rec is computed entirely in the shape space Q/G.
2.8.Momentum connections.In the rigid body example discussed in 1.2 (G = SO(3)), the angle ∆θ may be identified with an element of g µ 0 , where µ 0 ∈ g * ∼ = R 3 is the initial spatial angular momentum.This angle is the logarithm of the reconstruction phase g rec ∈ G µ 0 , there denoted g.Let ω − O denote the 'minus' version of the symplectic structure on O, viewed as co-adjoint orbit (see below).Then Equation 1.2(2) may alternatively be written As we shall see, this generalizes to arbitrary groups G, but it refers only to the µ 0 -component of the log phase.This engenders the following question, answered in the Proposition below: Of what g µ 0 -valued twoform on O is ω − O the µ 0 -component?For an arbitrary connected Lie group G equip g * with the 'minus' Lie-Poisson structure (see, e.g., [14, §2.8]).The symplectic leaves are the co-adjoint orbits; the symplectic structure on an orbit may be succinctly written Assuming g admits an Ad-invariant inner product, the bundle τ µ 0 : G → O ∼ = G/G µ 0 comes equipped with a connection one-form α µ 0 ≡ pr µ 0 , θ G ; here pr µ 0 : g → g µ 0 denotes the orthogonal projection.We shall refer to α µ 0 as the momentum connection on For simplicity, assume that µ 0 lies in g * reg and that G µ 0 is Abelian, as in 2.3.Then the curvature of α µ 0 may be identified with a g µ 0 -valued two-form on O = G • µ 0 denoted curv α µ 0 .
Proposition.Under the above conditions where g is any element of G such that µ = g • µ 0 , and which implies both the first part of the Proposition and the identity τ * µ 0 µ 0 , curv . This implies ω − O = − µ 0 , curv α µ 0 .2.9.Phases for Q = G.When Q = G, the Poisson manifold P/G = (T * G)/G is identifiable with g * and the reduced space P µ 0 is the coadjoint orbit O ≡ G • µ 0 , equipped with the symplectic structure ω − O discussed above.Continue to assume that g admits an Ad-invariant inner product.As we will show in Proposition 6.1, the restriction Here • denotes annihilator.The following result is implicit in [6].
Then the reconstruction phase associated with ν t is given by g rec = g dyn g geom , where: and where w(t) ∈ g µ 0 is defined through Here α µ 0 denotes the momentum connection on For a simple mechanical system on T * G the reduced Hamiltonian h : g * → R is of the form for some isomorphism I : g ∼ − → g * , the inertia tensor, which we may suppose is symmetric as an element of g * ⊗ g * .
Addendum ( [6]).Let G act on Hom(g * , g) via conjugation, so that . Then for a simple mechanical system one has , where pr µ 0 : g → g µ 0 is the orthogonal projection.Moreover the generalization 2.8(1) of Montgomery's rigid body formula holds.

Formulation of new results
According to known results reviewed in the preceding section, phases for simple mechanical systems are computed in shape space Q/G when G is Abelian, and on a co-adjoint orbit O = G • µ 0 when Q = G.For the general case, G non-Abelian and Q = G, we need to introduce the concepts of associated bundles and forms, and the locked inertia tensor for non-Abelian groups (3.1-3.3).In 3.4 and 3.5 we present the main results of the paper, namely explicit formulas for geometric and dynamic phases in Hamiltonian systems on cotangent bundles.
3.1.Associated bundles.Given an arbitrary principal bundle ρ : known as the associated bundle for O.As its fibers are diffeomorphic to O, it may be regarded as a 'twisted product' of Q/G and O.
Here the important examples will be the co-adjoint bundle g * Q and the co-adjoint orbit bundle We have seen that log geometric phases are surface integrals of the curvature curv A ∈ Ω 2 (Q/G, g) of the mechanical connection A, when G is Abelian, and of the curvature curv α µ 0 ∈ Ω 2 (O, g µ 0 ) of the momentum connection α µ 0 , when Q = G.For simple mechanical systems the log dynamic phase is a time integral of an inverted inertia tensor I −1 in both cases.To elaborate on the claims regarding the general case made in 1.4, we need to see how curv A, curv α µ 0 and I −1 can be viewed as objects on O Q .
A non-Abelian G forces us to regard curv A as an element of Ω 2 (Q/G, g Q ), i.e., as bundle-valued.See, e.g., Note A.6 and A.2(1) for the definition.The pull-back ρ * O curv A is then a two-form on O Q , but with values in the pull-back bundle ρ * O g Q .Pull-backs of bundles and forms are briefly reviewed in Appendix A.
On the other hand, curv α µ 0 is vector-valued because g µ 0 is Abelian under the hypothesis , which we now define more generally.
3.2.Associated forms.Let ρ : Q → Q/G be a principal bundle equipped with a connection A, and let O be a manifold on which G acts When R is replaced by a general vector space V on which G acts linearly, then the associated form This last remark applies, in particular, to curv α µ 0 .

Locked inertia tensor (general case).
When G is non-Abelian the map Î : Q → Hom(g, g * ) defined in 2.7 is G-equivariant if G acts on Hom(g, g * ) via conjugation.It therefore drops to a (bundle-valued) function I ∈ Ω 0 (Q/G, Hom(g, g * ) Q ), the locked inertia tensor: View the inclusion i O : O → g * as an element of Ω 0 (O, g * ).Then with the help of the associated form ) Here the wedge ∧ implies a contraction Hom(g * , g) ⊗ g * → g.

3.4.
Phases for simple mechanical systems.Before stating our new results, let us summarize with a few definitions.Put Recall here that A denotes a connection on Q → Q/G (the mechanical connection if H is a simple mechanical system), α µ 0 denotes the momentum connection on By construction, Ω A , Ω µ 0 and ξ I are all differential forms on O Q .The momentum curvature Ω µ 0 is g µ 0 -valued, and can therefore be integrated over surfaces S ⊂ O Q ; the forms Ω A and ξ I are ρ * O g Q -valued.To make them g µ 0 -valued requires an appropriate projection: Definition.Let G act on Hom(g, g µ 0 ) via g • σ ≡ Ad g •σ and let Pr µ 0 ∈ Ω 0 (O, Hom(g, g µ 0 )) denote the unique equivariant zero-form whose value at µ 0 is the orthogonal projection pr µ 0 : g → g µ 0 .
With the help of the associated form (Pr µ 0 ) Q and an implied contraction Hom(g, As we declare G to act trivially on g µ 0 , these forms are in fact identifiable with g µ 0 -valued forms as required.
For P = T * Q and G non-Abelian the reduced space P µ 0 can be identified with T Here ⊕ denotes product in the category of fiber bundles over Q/G (see Notation in 4.2).This observation was first made in the Lagrangian setting by Cendra et al. [9].We recall details in 4.2 and Proposition 5.1.A formula for the symplectic structure on P µ 0 has been given by Perlmutter [24].We derive the form of it we will require in 5.2.The value of the reduced Hamiltonian h µ 0 : where x ∈ T * q Q is any point satisfying T * A ρ • x = z and J(x) = µ.In the case of simple mechanical systems one has (1) Here V Q/G denotes the function on Q/G to which the potential V drops on account of its G-invariance, and (The second term above may be written intrinsically as 1/2 ((id , where (id g * ) Q is defined in 6.4.)The formula (1) is derived in 7.1.
Theorem.Let H : T * Q → R be a simple mechanical system, as defined in 2.4.Assume µ 0 ∈ g * reg , G µ 0 is Abelian, and let Assume z t ⊕η t and η t have the same minimal period T and assume t → η t bounds a compact oriented surface S ⊂ O Q .Then the corresponding reconstruction phase is g rec = g dyn g geom , where Here Ω A is the mechanical curvature, Ω µ 0 the momentum curvature, and ξ I the inverted locked inertia function, as defined above; A denotes the mechanical connection.
Notice that the phase g rec does not depend on the z t part of the reduced solution curve (z t , η t ), i.e., is computed exclusively in the space O Q .
3.5.Phases for arbitrary systems on cotangent bundles.We now turn to the case of general Hamiltonian functions on T * Q (not necessarily simple mechanical systems).To formulate results in this case, we need the fact, recalled in Theorem 4.2, that ( where ⊕ denotes product in the category of fiber bundles over Q/G (see Notation 4.2).This isomorphism depends on the choice of connection A on ρ : Theorem.Let H : T * Q → R be an arbitrary G-invariant Hamiltonian and h : T * (Q/G) ⊕ g * Q → R the corresponding reduced Hamiltonian.Consider a periodic reduced solution curve z t ⊕ η t ∈ P µ 0 ∼ = T * (Q/G) ⊕ O Q , as in the Theorem above.Then the conclusion of that Theorem holds, with the dynamic phase now given by Here is the isomorphism defined in 2.9.Theorems 3.4 and 3.5 will be proved in Sections 7 and 8.

Symmetry reduction of cotangent bundles
In this section and the next, we revisit the process of reduction in cotangent bundles by describing the symplectic leaves in the associated Poisson-reduced space.For an alternative treatment and a brief history of cotangent bundle reduction, see Perlmutter [24,Chapter 3].
In the sequel G denotes a connected Lie group acting freely and properly on a connected manifold Q, and hence on T * Q; J : T * Q → g * denotes the momentum map defined in 2.4(1); A denotes an arbitrary connection one-form on the principal bundle ρ : 4.1.The zero momentum symplectic leaf.The form of an arbitrary symplectic leaf P µ of (T * Q)/G will be described in Section 5.1 using a concrete model for the abstract quotient (T * Q)/G described in 4.2 below.However, the structure of the particular leaf P 0 = J −1 (0)/G can be described directly.Moreover, we shall need this description to relate symplectic structures on T * Q and T * (Q/G) (Corollary 4.3). Since , for each locally defined function f on Q/G.Here (ker Tρ) • denotes the annihilator of ker Tρ.In fact, 2.4(1) implies that (ker Tρ) • = J −1 (0), so that J −1 (0) is a vector bundle over Q, and we have the commutative diagram PSfrag replacements Notation.We will write J −1 (0) q ≡ J −1 (0) ∩ T * q Q = (ker T q ρ) • for the fiber of J −1 (0) over q ∈ Q.
From the definition of ρ • , it follows that ρ • maps J −1 (0) q isomorphically onto T * ρ(q) (Q/G).In particular, ρ • is surjective.It is readily demonstrated that the fibers of ρ • are G-orbits so that ρ • determines a diffeomorphism between T * (Q/G) and P 0 = J −1 (0)/G.Moreover, if ω Q/G denotes the canonical symplectic structure on T * (Q/G) and i 0 : J −1 (0) → T * Q the inclusion, then we have This formula is verified by first checking the analogous statement for the canonical one-forms on T * Q and T * (Q/G).

4.2.
A model for the Poisson-reduced space (T * Q)/G.Let hor = ker A denote the distribution of horizontal spaces on Q determined by A ∈ Ω 1 (Q, g).Then have the decomposition of vector bundles over Q (1) TQ = hor ⊕ ker Tρ , and the corresponding dual decomposition If A : T * Q → J −1 (0) denotes the projection along hor • , then the composite It the Hamiltonian analogue of the tangent map Tρ : TQ → T(Q/G).The momentum map J : Notation.If M 1 , M 2 and B are smooth manifolds and there are maps f 1 : M 1 → B and f 2 : M 2 → B, then one has the pullback manifold which we will denote by M 1 ⊕ B M 2 , or simply M 1 ⊕ M 2 .If f 1 and f 2 are fiber bundle projections then M 1 ⊕ M 2 is a product in the category of fiber bundles over B. In particular, in the case of vector bundles, M 1 ⊕ M 2 is the Whitney sum of M 1 and M 2 .In any case, we write an element of M 1 ⊕ M 2 as m 1 ⊕ m 2 (rather than (m 1 , m 2 )).
Noting that T * (Q/G) and g * Q are both vector bundles over Q/G, we have the following result following from an unravelling of definitions: The above model of (T * Q)/G is simply the dual of Cendra, Holm, Marsden and Ratiu's model of (TQ)/G [9].

Momentum shifting. Before attempting to describe the symplectic leaves of the Poisson-reduced space (T
In particular, we should understand the map T * A ρ : T * Q → T * (Q/G), which means first understanding the projection A : T * Q → J −1 (0) along hor • .
Let x ∈ T * q Q be given and define µ ≡ J(x).The restriction of J to T * q Q is a linear map onto g * (by 2.4(1)).The kernel of this restriction is J −1 (0) q and J −1 (µ) q ≡ J −1 (µ) ∩ T * q Q is an affine subspace of T * q Q parallel to J −1 (0) q ; see Fig. 4.
Describing the projection x → A (x) : T * q Q → J −1 (0) q along hor • q .Since J −1 (0) q and J −1 (µ) q are parallel, it follows from the decomposition 4.2(2) that J −1 (µ) q and hor • q intersect in a single point * , as indicated in the figure.We then have A (x) = x − * .Indeed, viewing the R-valued one-form µ, A as a section of the cotangent bundle T * Q → Q, one checks that the covector µ, A (q) ∈ T * q Q belongs simultaneously to J −1 (µ) and hor • , so that * = µ, A (q).We have therefore proven the following: This identity, Equation 4.1(1), and the above Lemma have the following important corollary, which relates the symplectic structures on the domain and range of the map T * A ρ : T * Q → T * (Q/G): Corollary.The two-forms (T * A ρ) * ω Q/G and ω + µ, (τ * Q ) * dA agree when restricted to J −1 (µ).

Symplectic leaves in Poisson reduced cotangent bundles
In this section we describe the symplectic leaves P µ ⊂ (T * Q)/G as subsets of the model described in 4.2.We then describe explicitly their symplectic structures.5.1.Reduced spaces as symplectic leaves.The following is a specialized version of the symplectic reduction theorem of Marsden, Weinstein and Meyer [20,21], formulated such that the reduced spaces are realized as symplectic leaves (see, e.g., [7, Appendix E]).
Theorem.Consider P , ω, G, J and P µ , as defined in 2.1, where µ ∈ J(P ) is arbitrary.Then: (1) P µ is a symplectic leaf of P/G (which is a smooth Poisson manifold).
(2) The restriction π µ : J −1 (µ) → P µ of π : P → P/G is a surjective submersion whose fibers are G µ -orbits in P , i.e., P µ is a realization of the abstract quotient J −1 (µ)/G µ .(3) If ω µ is the leaf symplectic structure of P µ , and i µ : J −1 (µ) → P the inclusion, then i * µ ω = π * µ ω µ .(4) P µ ∩ P µ = ∅ if and only if P µ = P µ , which is true if and only if µ and µ lie on the same co-adjoint orbit.Also, P/G = ∪ µ∈J(P ) P µ .(5 Proposition.Fix µ ∈ g * .Then, taking P ≡ T * Q and identifying P/G with T * (Q/G) ⊕ g * Q (Theorem 4.2), one obtains Here G • µ denotes the co-adjoint orbit through µ and the associated bundle O Q is to be viewed as a fiber subbundle of g * Q in the obvious way.
Proof.Under the given identification, the projection P → P/G is represented by the map π : Then the symplectic structure of the leaf where projections onto the first and second summands, and Because the restriction π µ : For the next part of the proof we need the following technical result proven at the end: So it readily follows from the lemma that A routine calculation of pullbacks shows that , where DA ∈ Ω 2 (Q, g) denotes the exterior covariant derivative of A. In deriving (4) we have used the fact that This identity simply states, in pullback jargon, that curv A is the twoform DA on Q, viewed as a g Q -valued form on the base Q/G.
Proof of the Lemma.We have u = d/dt x(t) | t=0 for some curve t → x(t) ∈ J −1 (µ), in which case , where q(t) ≡ τ * Q (x(t)).We can write q(t) = g(t) • q hor (t) for some Ahorizontal curve t → q hor (t) ∈ Q and some curve t → g(t) ∈ G with g(0) = id and with as required.

A connection on the Poisson-reduced phase space
To apply Theorem 2.3 to the case P = T * Q we need to choose a connection D on the symplectic stratification of P/G ∼ = T * (Q/G) ⊕ g * Q .Such connections were defined in 2.2.As we shall see, this more-orless amounts to choosing an inner product on g * (or g).Life is made considerably easier if this choice is Ad-invariant.(For example, in the case Q = G, which we discuss first, one might be tempted to use the inertia tensor I ∈ g * ⊗ g * to form an inner product.However, this seems to lead to intractable calculations of the phase.It also makes the geometric phase g geom more 'dynamic' and less 'geometric.')Fortunately, we will see that the particular choice of invariant inner product is immaterial.In 6.3 and 6.4 we discuss details needed to describe explicitly the transverse derivative operator D µ , and we also compute the canonical two-form ω D (both these depend on the choice of D).Recall that these will be needed to apply Theorem 2.3.
6.1.The limiting case Q = G.When Q = G, we have P/G ∼ = g * and the symplectic leaves are the co-adjoint orbits.A connection on the symplectic stratification of P/G is then distribution on g * furnishing a complement, at each point µ ∈ g * , for the space T µ (G • µ) tangent to the co-adjoint orbit G • µ through µ.As a subspace of g * this tangent space is the annihilator g • µ of g µ .Lemma.Let G be a connected Lie group whose Lie algebra g admits an Ad-invariant inner product.Then for all µ ∈ g * reg one has Here g * reg denotes the set of regular points of the co-adjoint action Proof.See Appendix B.
The following proposition constructs a connection E on the symplectic stratification of g * .
Proposition.Let G be a connected Lie group whose Lie algebra g admits an Ad-invariant inner product and equip g * with the corresponding Ad * -invariant inner product.Let E denote the connection on the symplectic stratification of g * obtained by orthogonalizing the distribution tangent to the co-adjoint orbits: Let forg E(µ) denotes the image of E(µ) under the canonical identification T µ g * ∼ = g * , i.e., forg E(µ) ⊂ g * is E(µ) ⊂ T µ g * with base point 'forgotten.'Then for all µ ∈ g * reg : (2) E(µ) is independent of the particular choice of inner product.
(3) The restriction ι µ : forg E(µ) → g * µ of the natural projection p µ : g * → g * µ is an isomorphism.(4) The orthogonal projection pr µ : g → g µ is independent of the choice of inner product and satisfies the identity (6) There exists a subspace V ⊂ g * containing µ and an open neighborhood S ⊂ V of µ such that T s S = E(s) for all s ∈ S.
Remark.One can choose the V in ( 6) to be G µ -invariant (see the proof below), so that S (suitably shrunk) is a slice for the co-adjoint action.This is provided, of course, that G has closed co-adjoint orbits.Although we do not assume that these orbits are closed, the reader may nevertheless find it helpful to think of S as a slice.We do not use (6) until Section 8.
Proof.In fact (3) is true for any space E(µ) complementary to T µ (G•µ), for this means which, on identifying the spaces with subspaces of g * , delivers the decomposition µ is the kernel of the linear surjection p µ : g * → g * µ , (3) must be true.The identity in ( 4) is an immediate corollary.
Because taking annihilator and orthogonalizing are commutable operations, we deduce from the above Lemma the formula (g holds.Claim (2) follows.Regarding (5), we have The second term in parentheses vanishes because g µ is Abelian (since µ ∈ g * reg ).The third and fourth terms vanish because they lie in [g, g µ ], which is the kernel of pr µ , on account of the Lemma.This kernel is evidently independent of the choice of inner product, which proves the first part of (4).
To prove (6), take which clearly contains µ.Since dim g µ = dim g ν if and only if ν ∈ g * reg , we conclude that , it follows that µ has a neighborhood S ⊂ V of µ such that S ⊂ g * reg and g s = g µ for all s ∈ S. For any s ∈ S we then have (8) forg where the first equality follows from (1).Equation (8) implies that E(s) = T s S, as required.
Henceforth E denotes the connection on the symplectic stratification of g * defined in the above Proposition.

The general case
] G , then the right-hand side of ( 1) is unchanged by a substitution by primed quantities, because E is G-invariant.This shows that the distribution D is well defined.It is a connection on the symplectic stratification of T * (Q/G) ⊕ g * Q because E is a connection on the symplectic stratification of g * , and because the symplectic leaf through a point z ⊕

Transverse derivatives.
To determine the transverse derivative operator D µ determined by D in the special case of cotangent bundles (needed to apply Theorem 2.3), we will need an explicit expression for the isomorphism L(D, y, µ) : (2) For each such y one has , where ι µ is defined by 6.1(3).
Proof.That each y ∈ P µ is of the form given in (1) follows from an argument already given in the proof of Proposition 5.1.Moreover, that proof shows that there exists x 0 ∈ J −1 (0) q such that ρ • (x 0 ) = z.We prove (2) by first computing the natural isomorphism D(y) , where M µ is the momentum shift defined in 4.3.Then x ∈ J −1 (µ).According to (1), an arbitrary vector v ∈ D(y) is of the form for some δ ∈ forg E(µ).We claim that the vector is a valid choice for the corresponding vector w in Lemma 2.2.Indeed, one has as required.We now compute The natural isomorphism D(y) Since L(D, y, µ) is the inverse of this map, this proves (2).
6.4.The canonical two-form determined by D. We now determine the canonical two-form ω D determined by D in the cotangent bundle case.According to Theorem 5.2, the symplectic structure of the leaf where pr 2) is given by ( 2) Here pr 1 and pr 2 denote the canonical projections T The form ω E denotes the canonical two-form on g * determined by E. The zero-form (id denotes the form associated with the identity map id g * : g * → g * , viewed as an element of Ω 0 (g * , g * ).(If one makes the identification The formula in ( 2) is easily verified by checking that ω D (v, • ) = 0 for v ∈ D, and by checking that the restriction of ω D to a leaf P µ coincides with the two-form on the right-hand side of (1).

The dynamic phase
For general G-invariant Hamiltonians H : T * Q → R the formula for g dyn in Theorem 3.5 follows from Theorem 2.3, Lemma 6.3, and the definition of D µ 0 given in 2.2.In this section we deduce the form taken by this phase in simple mechanical systems, as reported in Theorem 3.4.2)).Since J −1 (0) q = (ker T q ρ) • , it is not too difficult to see that (1) x ∈ J −1 (0 q , then x is the image under the isomorphism TQ ∼ − → T * Q of ξ Q (q), for some ξ ∈ g.For such ξ, and arbitrary η ∈ g, we compute where the first equality follows from 2.4(1).Since η ∈ g is arbitrary, it follows that ξ = Î−1 (q)(J(x)).We now conclude that (2) x ∈ hor • q ⇒ x, x * Q = ξ Q (q), ξ Q (q) Q = J(x), Î−1 (q)(J(x)) .An arbitrary element x ∈ T * q Q decomposes into unique parts along J −1 (0) q and hor • q , the first component being A (x). From (1) and (2) one deduces (3) x, where V Q/G denotes the function on Q/G to which V drops on account of its G-invariance.With the help of (3), one checks that H = h • π, i.e., h is the Poisson-reduced Hamiltonian.Substituting (4) into 3.5(1) delivers the formula (5) where pr µ 0 : g → g µ 0 denotes the orthogonal projection.
To establish the formula for g dyn in Theorem 3.4 it remains to show that (6) ( where ξ I ≡ ρ * O I −1 ∧ (i O ) Q .We will be ready to do so after providing the general definition of associated forms alluded to in 3.2.7.2.Associated forms (general case).Let V be a real vector space on which G acts linearly and O an arbitrary manifold on which G acts smoothly.Let λ be a V -valued k-form on O.For the sake of clarity, we will suppose k = 1; the extension to general k will be obvious.
Assuming that λ ∈ Ω 1 (O, V ) is equivariant in the sense that we will construct a bundle-valued differential form and ρ  * O denotes pullback.As always, we assume ρ : Q → Q/G is equipped with a connection one-form A.
We begin by noting that an arbitrary vector tangent to for some ξ ∈ g, some A-horizontal curve t → q hor (t) ∈ Q, and some curve t → ν(t As the reader is left to verify, the equivariance of λ ensures that Λ is well defined.Now ) and we claim that Λ is tensorial.
where the second quality follows from the equivariance of λ.What we have just shown is that for arbitrary u ∈ T(ρ * O Q), i.e., Λ is equivariant.Also, the generic tangent vector in (1) is vertical (in the principal bundle ρ * O Q → O Q ) if and only if d/dt [q hor (t), ν(t)] G | t=0 = 0.This is true if and only if d/dt ν(t) | t=0 = 0.It follows that Λ vanishes on vertical vectors.This fact and the forementioned equivariance establishes that Λ is tensorial.
, which is the sought after associated form λ Q .By construction one has the implicit formula , where q ≡ q hor (0) and ν ≡ ν(0).

The geometric phase
This section derives the formula for g geom reported in Theorem 3.4.We will carry out several computations, some of them somewhat involved.However, our objective throughout is clear: To apply the formula for g geom in 2.3 we must calculate the transverse derivative D µ 0 ω D of the leaf symplectic structures ω µ = ω D |P µ .To do so we must first compute dω D .Our preference for a coordinate free proof leads us to lift the computation to a bigger space, which we do with the help of the 'slice' S for the co-adjoint action delivered by 6.1 (6).
Using the fact that d is an antiderivation, that d commutes with pullbacks, and that dω Q/G = 0, we obtain from 6.4(2) (1) . Note here that we are using the exterior derivative in the generalized sense of bundle-valued forms, as defined with respect to the connection A; see A.5, Appendix A. The last term in parentheses is immediately dispensed with, for one has Bianchi's identity2 (2) To write down formulas for other terms in (1), it will be convenient to have an appropriate representation for vectors tangent to g * Q .Indeed, as the reader will readily verify, each such vector is of the form for some A-horizontal curve t → q hor (t) ∈ Q and some curve t → µ(t) ∈ g * .On occasion, and without loss of generality, we will take µ(t) to be of the form for some ξ ∈ g, µ ∈ g * and v ∈ forg E(µ) (see Proposition 6.1).
A straightforward computation gives where μ(0) ≡ d/dt µ(t) | t=0 ∈ g * .From this follows the formula = μ1 (0), DA( qhor 2 (0), qhor 3 (0)) + μ2 (0), DA( qhor 3 (0), qhor where D denotes exterior covariant derivative and qhor j (0) ≡ d/dt q hor j (t) | t=0 .To compute d(ω E ) Q is not so straightforward. 3The difficulty lies partly in the fact that the co-adjoint orbit symplectic structures, which ω E 'collects together,' are defined implicitly in terms of the infinitesimal generators of the co-adjoint action, and this action is generally not free.We overcome this by pulling (ω E ) Q back to a 'bigger' space where we can be explicit.We compute the derivative in the bigger space and then drop to g * Q .
Note that every vector tangent to Q × G × S is of the above form, and that From (6) and the definition of associated forms 3.2(1), we obtain b * (ω E ) Q u 1 , η 1 , ξ 1 , v 1 ; q, g, s , u 2 , η 2 , ξ 2 , v 2 ; q, g, s Now ω E is the canonical two-form on g * determined by E and according to 6.1(6), we have It follows from ( 7) that It is now that we see the reason for pulling (ω E ) Q back to Q × G × S. For if we define natural projections and denote by θ G ∈ Ω 1 (G, g) the right-invariant Mauer-Cartan form on G, then (8) may be written intrinsically as where we view π g We can now take d of both sides, obtaining where a single prime indicates pullback by π G , and a double prime indicates pullback by π Q .We expand and simplify (9) by invoking the following identities: If the primes are suppressed, then (10) and (11) are the Mauer-Cartan equations for G and the principal bundle Q resp., while (12) and ( 13) follow from Jacobi's identity.That we may add the primes follows from the fact that d commutes with pullbacks, and that pullbacks distribute over wedge products.After some manipulation, Equation ( 9 For future reference, we note here the easily computed formula (15) dπ g * ( u, η, ξ, v; q, g, s By (5), we have , µ , u 2 , 0, ξ 2 , v 2 ; q, id, µ , u 3 , 0, ξ 3 , v 3 ; q, id, µ ) .
We now substitute the formula for b * d(ω E ) Q in (14).In fact, since A ( u j , 0, ξ j , v j ; q, id, µ ) = 0 (j = 1, 2 or 3) , the only part on the right-hand side of ( 14) with a nontrivial contribution is and we obtain, with the help of (15), The second equality follows from Equations ( 1)-( 4) derived above; the last equality follows from 6.1(4).Since ν ∈ g * µ 0 in this computation is arbitrary, we conclude that = pr µ 0 DA( qhor 1 (0), qhor 2 (0)) .Comparing the right-hand side of ( 16) with the right-hand sides of ( 17) and ( 18), we deduce the intrinsic formula . The curve t → η t ∈ O Q in Theorem 3.4 is a closed embedded curve because it bounds the surface S. Because z t ⊕ η t and η t have the same minimal period, it follows that there exists a smooth map s : ∂S → T * (Q/G) ⊕ O Q such that s(η t ) = z t ⊕ η t .As pr 2 : T * (Q/G) ⊕ O Q → O Q is a vector bundle, the map s can be extended to a global section s : O Q → T * (Q/G) ⊕ O Q of pr 2 .This follows, for example, from [12,Theorem I.5.7].Define Σ ≡ s(S), so that pr 2 (Σ) = S and t → z t ⊕ η t is the boundary of Σ. Appealing to Theorem 2.3 and ( 19), we obtain which is the form of g geom given in Theorem 3.4.
according to g • (b ⊕ q) ≡ (b ⊕ g • q).One defines a map f : f * Q → Q by f (b ⊕ q) ≡ q and has the commutative diagram PSfrag replacements ρ The pullback f * V Q of an associated vector bundle V Q can be defined analogously but we will define it in a way making the pullback itself an associated bundle: This definition of f * V Q is equivalent to the forementioned alternative, for we have an isomorphism The map f : B → B defines a pullback operator on forms f * : where the pullback on the right-hand side is the usual one for vectorvalued forms.Making the identification (f • g) * V Q ∼ = g * (f * V Q ) indicated above, we have (f • g) * = g * • f * .A.4. Wedge products.The wedge product λ∧µ ∈ Ω p+q (B, (U ⊗V ) Q ) of forms λ ∈ Ω p (B, U Q ) and µ ∈ Ω q (B, V Q ) is defined through Suppose there is a natural, bilinear pairing (u, v) → u, v : U ×V → W that is equivariant in the sense that g •u, g •v = g • u, v .Then there is a G-invariant homomorphism U ⊗V → W allowing one to identify λ∧ μ with an element of Ω p+q tens (Q, W ); λ∧µ is correspondingly identified with an element of Ω p+q (B, W Q ).In the special case that G acts trivially on W (e.g., W = R), one has W Q ∼ = W × Q and there is a further identification Ω p+q (B, W Q ) ∼ = Ω p+q (B, W ).
A.5. Exterior derivatives.The exterior derivative dλ ∈ Ω p+1 (B, V Q ) of a form λ ∈ Ω p (B, V Q ) is defined through where D denotes exterior covariant derivative with respect to the connection A (see [12]).
A.6.Curvature.We next define the curvature form B V , which measures the degree to which Poincaré's identity d 2 = 0 fails for V Q -valued differential forms.By its equivariance, a tensorial zero-form F ∈ Ω 0 tens (Q, V ) satisfies the identity where ad V ξ denotes the infinitesimal generator of the linear action of G on V along ξ, viewed as an element of Hom(V, V ).From the definition of exterior covariant derivative, one deduces the identity DF = dF − A V ∧ F , where A V ∈ Ω 1 (Q, Hom(V, V )) is defined by It follows that D 2 F = −DA V ∧ F .Note that by the linearity of ξ → ad V ξ , we have The two-form DA V is tensorial (with G acting on Hom(V, V ) by conjugation), and so defines a two-form B V ∈ Ω 2 (B, Hom(V, V ) Q ) through 4BV = −DA V , allowing us to write D 2 F = BV ∧ F .Moreover, one can show that F in this identity can be replaced by an arbitrary, tensorial, V -valued p-form.One does so using the fact that such a form is an R-linear combination of products of the form ω ∧ F , for some ω ∈ Ω p tens (Q, R)

Figure 1 .
Figure 1.The dynamics of body angular momentum in the free rigid body.

5. 2 .
The leaf symplectic structures.The remainder of the section is devoted to the proof of the following key result, which is due (in a different form) to Perlmutter [24, Chapter 3]:Theorem.Let O denote the co-adjoint orbit through a point µ in the image of J, let ω − O denotes the 'minus' co-adjoint orbit symplectic structure on O (see 2.8), and let surjective submersion, by Theorem 5.1(2), to prove the above Theorem it suffices to verify the formula in 5.1(3).Appealing to the definition of π (Theorem 4.2) and Corollary 4.3, we compute(1)