Mechanical Systems: Symmetries and Reduction

Glossary

Coadjoint orbit:

The orbit of an element of the dual of the Lie algebra under the natural action of the group.

Cotangent bundle:

A mechanical phase space that has a structure that distinguishes configurations and momenta. The momenta lie in the dual to the space of velocity vectors of configurations.

Equivariance:

Equivariance of a momentum map is a property that reflects the consistency of the mapping with a group action on its domain and range.

Free action:

An action that moves every point under any nontrivial group element.

KKS (Kostant-Kirillov-Souriau) form:

The natural symplectic form on coadjoint orbits.

Lie group action:

A process by which a Lie group, acting as a symmetry, moves points in a space. When points in the space that are related by a group element are identified, one obtains the quotient space.

Magnetic terms:

These are expressions that are built out of the curvature of a connection. They are so named because terms of this form occur in the equations of a...

Appendix: Principal Connections

In preparation for the next section which gives a brief exposition of the cotangent bundle reduction theorem, we now give a review and summary of facts that we shall need about principal connections. An important thing to keep in mind is that the magnetic terms in the cotangent bundle reduction theorem will appear as the curvature of a connection.

Principal Connections Defined

We consider the following basic set up. Let Q be a manifold and let G be a Lie group acting freely and properly on the left on Q. Let

$$ {\pi}_{Q,G}:Q\to Q/G $$

denote the bundle projection from the configuration manifold Q to shape space S = Q/G. We refer to π _Q , G  : Q→Q/Gas a principal bundle.

One can alternatively use right actions, which is common in the principal bundle literature, but we shall stick with the case of left actions for the main exposition.

Vectors that are infinitesimal generators, namely those of the form ξ _Q (q) are called vertical since they are sent to zero by the tangent of the projection map π _Q,G .

Definition 1

A connection, also called a principal connection on the bundle π _Q , G  : Q→Q/G, is a Lie algebra valued 1-form

$$ \mathcal{A}:\mathrm{TQ}\to \mathfrak{g} $$

where $\mathfrak{g}$ denotes the Lie algebra of G, with the following properties:

• 1. the identity $\mathcal{A}$(ξ _Q (q)) = ξ holds for all ξ ∈ $\mathfrak{g}$; that is, $\mathcal{A}$ takes infinitesimal generators of a given Lie algebra element to that same element, and

• 2. we have equivariance: $\mathcal{A}\left({T}_q{\Phi}_g(v)\right)={\mathrm{Ad}}_g\left(\mathcal{A}(v)\right)$

for all v ∈ T _q Q, where Φ _g : Q → Q denotes the given action for g ∈ G and where ${\mathrm{Ad}}_g$ denotes the adjoint action of G on $\mathfrak{g}$.

A remark is noteworthy at this point. The equivariance identity for infinitesimal generators noted previously (see Eq. 7), namely,

$$ {T}_q{\Phi}_{\mathfrak{g}}\left({\xi}_Q(q)\right)={\left({\mathrm{Ad}}_{\mathfrak{g}}\xi \right)}_Q\left(\mathfrak{g}\cdot q\right) $$

shows that if the first condition for a connection holds, then the second condition holds automatically on vertical vectors.

If the G-action on Q is a right action, the equivariance condition (ii) in Definition 1 needs to be changed to $\mathcal{A}\left({T}_q{\Phi}_g(v)\right)={\mathrm{Ad}}_{g^{-1}}\left(\mathcal{A}(v)\right)$ for all g ∈ G and v ∈ T _q Q.

Associated One-Forms

Since $\mathcal{A}$ is a Lie algebra valued 1-form, for each q ∈ Q, we get a linear map $\mathcal{A}$(q): T _q Q → $\mathfrak{g}$ and so we can form its dual $\mathcal{A}{(q)}^{\ast }:{\mathfrak{g}}^{\ast}\to {T}_q^{\ast }Q$. Evaluating this on μ produces an ordinary 1-form:

$$ {\alpha}_{\mu }(q)=\mathcal{A}{(q)}^{\ast}\left(\mu \right). $$

(24)

This 1-form satisfies two important properties given in the next Proposition.

Proposition 5

For any connection $\mathcal{A}$ and μ ∈ ${\mathfrak{g}}^{\ast }$, the corresponding 1-form α _μ defined by Eq. 24 takes values in J ⁻¹(μ) and satisfies the following G-equivariance property:

$$ {\Phi}_g^{\ast }{\alpha}_{\mu }={\alpha}_{{\mathrm{Ad}}_g^{\ast}\mu }. $$

Notice in particular, if the group is Abelian or if μ is G-invariant, (for example, if μ = 0), then α _μ is an invariant 1-form.

Horizontal and Vertical Spaces

Associated with any connection are vertical and horizontal spaces defined as follows.

Definition 3

Given the connection $\mathcal{A}$, its horizontal space at q ∈ Q is defined by

$$ {H}_q=\left\{{v}_q\in {T}_qQ|\mathcal{A}\left({v}_q\right)=0\right\} $$

and the vertical space at q ∈ Q is, as above,

$$ {V}_q=\left\{{\xi}_Q(q)|\xi \in \mathfrak{g}\right\}. $$

The map

$$ {v}_q\mapsto {\mathrm{ver}}_q\left({v}_q\right):= {\left[\mathcal{A}(q)\left({v}_q\right)\right]}_Q(q) $$

is called the vertical projection, while the map

$$ {v}_q\mapsto {\mathrm{hor}}_q\left({v}_q\right):= {v}_q-{\mathrm{ver}}_q\left({v}_q\right) $$

is called the horizontal projection.

Because connections map infinitesimal generators of a Lie algebra elements to that same Lie algebra element, the vertical projection is indeed a projection for each fixed q onto the vertical space and likewise with the horizontal projection.

By construction, we have

$$ {v}_q={\mathrm{ver}}_q\left({v}_q\right)+{\mathrm{hor}}_q\left({v}_q\right) $$

and so

$$ {T}_qQ={H}_q\oplus {V}_q $$

and the maps hor _q and ver _q are projections onto these subspaces.

It is sometimes convenient to define a connection by the specification of a space H _q declared to be the horizontal space that is complementary to V _q at each point, varies smoothly with q and respects the group action in the sense thatH _g ⋅ q  = T _q Φ _g (H _q ). Clearly this alternative definition of a principal connection is equivalent to the definition given above.

Given a point q ∈ Q, the tangent of the projection map π _Q,G restricted to the horizontal space H _q gives an isomorphism between H _q and T _[q](Q/G). Its inverse ${\left[{\left.{T}_q{\pi}_{Q,G}\right|}_{H_q}\right]}^{-1}:{T}_{\pi_{Q,G}(q)}\left(Q/G\right)\to {H}_q$ is called the horizontal lift to q ∈ Q.

The Mechanical Connection

As an example of defining a connection by the specification of a horizontal space, suppose that the configuration manifold Q is a Riemannian manifold. Of course, the Riemannian structure will often be that defined by the kinetic energy of a given mechanical system.

Thus, assume that Q is a Riemannian manifold, with metric denoted ≪ , ≫ and that G acts freely and properly on Q by isometries, so π _Q,G : Q → Q/G is a principal G-bundle.

In this context we may define the horizontal space at a point simply to be the metric orthogonal to the vertical space. This therefore defines a connection called the mechanical connection.

Recall from the historical survey in the introduction that this connection was first introduced by Kummer (1981) following motivation from Smale (1970) and Abraham and Marsden (2008). See also Guichardet (1984), who applied these ideas in an interesting way to molecular dynamics. The number of references since then making use of the mechanical connection is too large to survey here.

In Proposition 7 we develop an explicit formula for the associated Lie algebra valued 1-form in terms of an inertia tensor and the momentum map. As a prelude to this formula, we show the following basic link with mechanics. In this context we write the momentum map on TQ simply as J: TQ → ${\mathfrak{g}}^{\ast }$.

Proposition 6

The horizontal space of the mechanical connection at a point q ∈ Q consists of the set of vectors v _q  ∈ T _q Q such that J(v _q ) = 0.

For each q ∈ Q, define the locked inertia tensor 𝕀(q) to be the linear map $\mathbb{I}(q):\mathfrak{g}\to {\mathfrak{g}}^{\ast }$ defined by

$$ \left\langle \mathbb{I}(q)\eta, \zeta \right\rangle =\left\langle \left\langle {\eta}_Q(q),{\zeta}_Q(q)\right\rangle \right\rangle $$

(25)

for any η, ζ ∈ $\mathfrak{g}$. Since the action is free, 𝕀(q) is nondegenerate, so Eq. 25 defines an inner product. The terminology “locked inertia tensor” comes from the fact that for coupled rigid or elastic systems, 𝕀(q) is the classical moment of inertia tensor of the rigid body obtained by locking all the joints of the system. In coordinates,

$$ {I}_{ab}={g}_{ij}{K}_a^i{K}_b^j, $$

(26)

where ${\left[{\xi}_Q(q)\right]}^i={K}_a^i(q){\xi}^a$ define the action functions ${K}_a^i$.

Define the map $\mathcal{A}$: TQ → $\mathfrak{g}$ which assigns to each v _q  ∈ T _q Q the corresponding angular velocity of the locked system:

$$ \mathcal{A}(q)\left({v}_q\right)=\mathbb{I}{(q)}^{-1}\left(\mathbf{J}\left({v}_q\right)\right), $$

(27)

where L is the kinetic energy Lagrangian. In coordinates,

$$ {\mathcal{A}}^a={I}^{ab}{g}_{ij}{K}_b^i{v}^j $$

(28)

since ${J}_a\left(q,p\right)={p}_i{K}_a^i(q)$.

We defined the mechanical connection by declaring its horizontal space to be the metric orthogonal to the vertical space. The next proposition shows that $\mathcal{A}$ is the associated connection one-form.

Proposition 7

The $\mathfrak{g}$-valued one-form defined by Eq. 27 is the mechanical connection on the principal G-bundle π _Q,G : Q → Q/G.

Given a general connection $\mathcal{A}$ and an element μ ∈ ${\mathfrak{g}}^{\ast }$, we can define the μ-component of $\mathcal{A}$ to be the ordinary one-form α _μ given by

$$ {\displaystyle \begin{array}{l}\, {\alpha}_{\mu }(q)=\mathcal{A}{(q)}^{\ast}\mu \in {T}_q^{\ast }Q;\,\, \, \mathrm{i}.\mathrm{e}.,\, \left\langle {\alpha}_{\mu }(q),{v}_q\right\rangle \\ {}\quad =\left\langle \mu, \mathcal{A}(q)\left({v}_q\right)\right\rangle \end{array}} $$

for all v _q  ∈ T _q Q. Note that α _μ is a G _μ -invariant one-form. It takes values in J ⁻¹(μ) since for any ξ ∈ $\mathfrak{g}$, we have

$$ {\displaystyle \begin{array}{l}\;\left\langle \mathbf{J}\left({\alpha}_{\mu }(q)\right),\xi \right\rangle =\left\langle {\alpha}_{\mu }(q),{\xi}_Q\right\rangle =\left\langle \mu, \mathcal{A}(q)\left({\xi}_Q(q)\right)\right\rangle \\ {}\qquad =\left\langle \mu, \xi \right\rangle .\end{array}} $$

In the Riemannian context, Smale (1970) constructed α _μ by a minimization process. Let ${\alpha}_q^{\sharp}\in {T}_qQ$ be the tangent vector that corresponds to ${\alpha}_q\in {T}_q^{\ast }Q$ via the metric ≪, ≫ on Q.

Proposition 8

The 1-form ${\alpha}_{\mu }(q)=\mathcal{A}{(q)}^{\ast}\mu \in {T}_q^{\ast }Q$ associated with the mechanical connection $\mathcal{A}$ given by Eq. 27 is characterized by

$$ K\left({\alpha}_{\mu }(q)\right)=\inf \left\{K\left({\beta}_q\right)|{\beta}_q\in {\mathbf{J}}^{-1}\left(\mu \right)\cap {T}_q^{\ast }Q\right\}, $$

(28)

where $K\left({\beta}_q\right)=\frac{1}{2}{\left\Vert {\beta}_q^{\sharp}\right\Vert}^2$ is the kinetic energy function on T ^* Q. See Fig. 2.

Fig. 2

The extremal characterization of the mechanical connection

The proof is a direct verification. We do not give here it since this proposition will not be used later in this book. The original approach of Smale (1970) was to take Eq. 28 as the definition of α _μ . To prove from here that α _μ is a smooth one-form is a nontrivial fact; see the proof in Smale (1970) or of Proposition 4.4.5 in Abraham and Marsden (2008). Thus, one of the merits of the previous proposition is to show easily that this variational definition of α _μ does indeed yield a smooth one-form on Q with the desired properties. Note also that α _μ (q) lies in the orthogonal space to ${T}_q^{\ast }Q\cap {\mathbf{J}}^{-1}\left(\mu \right)$ in the fiber ${T}_q^{\ast }Q$ relative to the bundle metric on T ^* Q defined by the Riemannian metric on Q. It also follows that α _μ (q) is the unique critical point of the kinetic energy of the bundle metric on T ^* Q restricted to the fiber ${T}_q^{\ast }Q\cap {\mathbf{J}}^{-1}\left(\mu \right)$.

Curvature

The curvature ℬ of a connection $\mathcal{A}$ is defined as follows.

Definition 4

The curvature of a connection $\mathcal{A}$ is the Lie algebra valued two-form on Q defined by

$$ \mathrm{\mathcal{B}}(q)\left({u}_q,{v}_q\right)=\mathbf{d}\mathcal{A}\left({\mathrm{hor}}_q\left({u}_q\right),{\mathrm{hor}}_q\left({v}_q\right)\right), $$

(29)

where d is the exterior derivative.

When one replaces vectors in the exterior derivative with their horizontal projections, then the result is called the exterior covariant derivative and one writes the preceding formula for ℬ as

$$ \mathrm{\mathcal{B}}={\mathbf{d}}^{\mathcal{A}}\mathcal{A}. $$

For a general Lie algebra valued k-form α on Q, the exterior covariant derivative is the k + 1-form d ^A α defined on tangent vectors v ₀ , v ₁ , … , v _k  ∈ T _q Q by

$$ {\displaystyle \begin{array}{l}\operatorname{}{\mathbf{d}}^{\mathcal{A}}\alpha \left({v}_0,{v}_1,\dots, {v}_k\right)\\ {}=\mathbf{d}\alpha \left({\mathrm{hor}}_q\left({v}_0\right),{\mathrm{hor}}_q\left({v}_1\right),\dots, {\mathrm{hor}}_q\left({v}_k\right)\right).\end{array}} $$

(30)

Here, the symbol ${\mathbf{d}}^{\mathcal{A}}$ reminds us that it is like the exterior derivative but that it depends on the connection $\mathcal{A}$. Curvature measures the lack of integrability of the horizontal distribution in the following sense.

Proposition 9

On two vector fields u, v on Q one has

$$ \mathrm{\mathcal{B}}\left(u,v\right)=-\mathcal{A}\left(\left[\mathrm{hor}(u),\mathrm{hor}(v)\right]\right). $$

Given a general distribution $\mathcal{D}$ ⊂ TQ on a manifold Q one can also define its curvature in an analogous way directly in terms of its lack of integrability. Define vertical vectors at q ∈ Q to be the quotient space ${T}_qQ/{\mathcal{D}}_q$ and define the curvature acting on two horizontal vector fields u, v (that is, two vector fields that take their values in the distribution) to be the projection onto the quotient of their Jacobi-Lie bracket. One can check that this operation depends only on the point values of the vector fields, so indeed defines a two-form on horizontal vectors.

Cartan Structure Equations

We now derive an important formula for the curvature of a principal connection.

Theorem 8 (Cartan structure equations)

For any vector fields u, v on Q we have

$$ \mathrm{\mathcal{B}}\left(u,v\right)=\mathbf{d}\mathcal{A}\left(u,v\right)-\left[\mathcal{A}(u),\mathcal{A}(v)\right], $$

(31)

where the bracket on the right hand side is the Lie bracket in $\mathfrak{g}$. We write this equation for short as

$$ \mathrm{\mathcal{B}}=\mathbf{d}\mathcal{A}-\left[\mathcal{A},\mathcal{A}\right]. $$

If the G-action on Q is a right action, then the Cartan Structure Equations read ℬ = d $\mathcal{A}$ + [$\mathcal{A}$, $\mathcal{A}$].

The following Corollary shows how the Cartan Structure Equations yield a fundamental equivariance property of the curvature.

Corollary 3

For all g ∈ G we have ${\Phi}_g^{\ast}\mathrm{\mathcal{B}}={\mathrm{Ad}}_g\circ \mathrm{\mathcal{B}}$. If the G-action on Q is on the right, equivariance means ${\varPhi}_g^{\ast}\mathrm{\mathcal{B}}={\mathrm{Ad}}_{g^{-1}}\circ \mathrm{\mathcal{B}}$.

Bianchi Identity

The Bianchi Identity, which states that the exterior covariant derivative of the curvature is zero, is another important consequence of the Cartan Structure Equations.

Corollary 4

If ℬ = ${\mathbf{d}}^{\mathcal{A}}$ $\mathcal{A}$ ∈ Ω²(Q; $\mathfrak{g}$) is the curvature two-form of the connection $\mathcal{A}$, then the Bianchi Identity holds:

$$ {\mathbf{d}}^{\mathcal{A}}\mathrm{\mathcal{B}}=0. $$

This form of the Bianchi identity is implied by another version, namely

$$ \mathbf{d}\mathrm{\mathcal{B}}={\left[\mathrm{\mathcal{B}},\mathcal{A}\right]}^{\wedge }, $$

where the bracket on the right hand side is that of Lie algebra valued differential forms, a notion that we do not develop here; see the brief discussion at the end of §9.1 in Marsden and Ratiu (1994). The proof of the above form of the Bianchi identity can be found in, for example, Kobayashi and Nomizu (1963).

Curvature as a Two-Form on the Base

We now show how the curvature two-form drops to a two-form on the base with values in the adjoint bundle.

The associated bundle to the given left principal bundle π _Q, G  : Q→Q/G via the adjoint action is called the adjoint bundle. It is defined in the following way. Consider the free proper action (g, (q, ξ)) ∈ G × (Q ×$\mathfrak{g}$)↦(g · q, Ad _g ξ)∈ Q ×$\, \mathfrak{g}$ and form the quotient $\tilde{\mathfrak{g}}:= Q{\times}_G\mathfrak{g}:= \left(Q\times \mathfrak{g}\right)/G$ which is easily verified to be a vector bundle ${\pi}_{\tilde{\mathfrak{g}}}:\tilde{\mathfrak{g}}\to Q/G$, where ${\pi}_{\tilde{\mathfrak{g}}}\left(g,\xi \right):= {\pi}_{Q,G}(q)$. This vector bundle has an additional structure: it is a Lie algebra bundle; that is, a vector bundle whose fibers are Lie algebras. In this case the bracket is defined pointwise:

$$ \left[{\pi}_{\tilde{\mathfrak{g}}}\left(g,\xi \right),{\pi}_{\tilde{\mathfrak{g}}}\left(g,\eta \right)\right]:= {\pi}_{\tilde{\mathfrak{g}}}\left(g,\left[\xi, \eta \right]\right) $$

for all g ∈ G and ξ, η ∈ $\mathfrak{g}$. It is easy to check that this defines a Lie bracket on every fiber and that this operation is smooth as a function of π _Q,G (q).

The curvature two-form ℬ ∈ Ω²(Q; $\mathfrak{g}$) (the vector space of $\mathfrak{g}$-valued two-forms on Q) naturally induces a two-form $\overline{\mathrm{\mathcal{B}}}$ on the base Q/G with values in $\tilde{\mathfrak{g}}$ by

$$ {\displaystyle \begin{array}{l}\overline{\mathrm{\mathcal{B}}}\left({\pi}_{Q,G}(q)\right)\left({T}_q{\pi}_{Q,G}(u),{T}_q{\pi}_{Q,G}(v)\right)\\ {}\operatorname{}:= {\pi}_{\tilde{\mathfrak{g}}}\left(q,\mathrm{\mathcal{B}}\left(u,v\right)\right)\end{array}} $$

(32)

for all q ∈ Q and u, v ∈ T _q Q. One can check that $\overline{\mathrm{\mathcal{B}}}$ is well defined.

Since Eq. 32 can be equivalently written as ${\pi}_{Q,G}^{\ast}\overline{\mathrm{\mathcal{B}}}={\pi}_{\tilde{\mathfrak{g}}}\circ \left({\mathrm{id}}_Q\times \mathrm{\mathcal{B}}\right)$ and π _Q,G is a surjective submersion, it follows that $\overline{\mathrm{\mathcal{B}}}$ is indeed a smooth two-form on Q/G with values in $\tilde{\mathfrak{g}}$.

Associated Two-Forms

Since ℬ is a $\mathfrak{g}$-valued two-form, in analogy with Eq. 24, for every μ ∈ ${\mathfrak{g}}^{\ast }$ we can define the μ-component of ℬ, an ordinary two-form ℬ _μ ∈ Ω²(Q) on Q, by

$$ {\mathrm{\mathcal{B}}}_{\mu }(q)\left({u}_q,{v}_q\right):= \left\langle \mu, \mathrm{\mathcal{B}}(q)\left({u}_q,{v}_q\right)\right\rangle $$

(33)

for all q ∈ Q and u _q  , v _q  ∈ T _q Q.

The adjoint bundle valued curvature two-form $\overline{\mathrm{\mathcal{B}}}$ induces an ordinary two-form on the base Q/G. To obtain it, we consider the dual ${\tilde{\mathfrak{g}}}^{\ast }$ of the adjoint bundle. This is a vector bundle over Q/G which is the associated bundle relative to the coadjoint action of the structure group G of the principal (left) bundle π _Q , G  : Q→Q/G on ${\mathfrak{g}}^{\ast }$. This vector bundle has additional structure: each of its fibers is a Lie-Poisson space and the associated Poisson tensors on each fiber depend smoothly on the base, that is, ${\pi}_{{\tilde{\mathfrak{g}}}^{\ast }}:{\tilde{\mathfrak{g}}}^{\ast}\to Q/G$ is a Lie-Poisson bundle over Q/G.

Given μ ∈ ${\mathfrak{g}}^{\ast }$, define the ordinary two-form ${\overline{\mathrm{\mathcal{B}}}}_{\mu }$ on Q/G by

$$ {\displaystyle \begin{array}{l}{\overline{\mathrm{\mathcal{B}}}}_{\mu}\left({\pi}_{Q,G}(q)\right)\left({T}_q{\pi}_{Q,G}\left({u}_q\right),{T}_q{\pi}_{Q,G}\left({v}_q\right)\right)\\ {}:= \left\langle {\pi}_{{\tilde{\mathfrak{g}}}^{\ast }}\left(q,\mu \right),\overline{\mathrm{\mathcal{B}}}\left({\pi}_{Q,G}(q)\right)\left({T}_q{\pi}_{Q,G}\left({u}_q\right),{T}_q{\pi}_{Q,G}\left({v}_q\right)\right)\right\rangle \\ {}=\left\langle \mu, \mathrm{\mathcal{B}}(q)\left({u}_q,{v}_q\right)\right\rangle ={\mathrm{\mathcal{B}}}_{\mu }(q)\left({u}_q,{v}_q\right),\end{array}} $$

(34)

where q ∈ Q, u _q , v _q ∈ T _q Q, and in the second equality $\left\langle, \right\rangle :{\tilde{\mathfrak{g}}}^{\ast}\times \tilde{\mathfrak{g}}\to \mathbb{R}$ is the duality pairing between the coadjoint and adjoint bundles. Since $\overline{\mathrm{\mathcal{B}}}$ is well defined and smooth, so is ${\overline{\mathrm{\mathcal{B}}}}_{\mu }$.

Proposition 10

Let $\mathcal{A}$ ∈ Ω¹(Q; $\mathfrak{g}$) be a connection one-form on the (left) principal bundle π _Q , G  : Q→Q/G and ℬ ∈ Ω²(Q; $\mathfrak{g}$) its curvature two-form on Q. If μ ∈ ${\mathfrak{g}}^{\ast }$, the corresponding two-forms ℬ _μ ∈ Ω²(Q) and ${\overline{\mathrm{\mathcal{B}}}}_{\mu}\in {\Omega}^2\left(Q/G\right)$ defined by Eqs. 33 and 34, respectively, are related by ${\pi}_{Q,G}^{\ast }{\overline{\mathrm{\mathcal{B}}}}_{\mu }={\mathrm{\mathcal{B}}}_{\mu }$. In addition, ℬ _μ satisfies the following G-equivariance property:

${\Phi}_g^{\ast }{\mathrm{\mathcal{B}}}_{\mu }={\mathrm{\mathcal{B}}}_{{\mathrm{Ad}}_g^{\ast}\mu }.$

Thus, if G = G _μ then $\mathbf{d}{\alpha}_{\mu }={\mathrm{\mathcal{B}}}_{\mu }={\pi}_{Q,G}^{\ast }{\overline{\mathrm{\mathcal{B}}}}_{\mu }$, where α _μ (q) = $\mathcal{A}$(q)^*(μ).

Further relations between α _μ and the μ-component of the curvature will be studied in the next section when discussing the magnetic terms appearing in cotangent bundle reduction.

The Maurer-Cartan Equations

A consequence of the structure equations relates curvature to the process of left and right trivialization and hence to momentum maps.

Theorem 9 (Maurer-Cartan equations)

Let G be a Lie group and let θ ^R: TG → $\mathfrak{g}$ be the map (called the right Maurer-Cartan form) that right translates vectors to the identity:

$$ {\theta}^R\left({v}_g\right)={T}_g{R}_{g^{-1}}\left({v}_g\right). $$

Then

$$ \mathbf{d}{\theta}^R-\left[{\theta}^R,{\theta}^R\right]=0. $$

There is a similar result for the left trivialization θ^L, namely the identity

$$ \mathbf{d}{\theta}^L+\left[{\theta}^L,{\theta}^L\right]=0. $$

Of course there is much more to this subject, such as the link with classical connection theory, Riemannian geometry, etc. We refer to Marsden et al. (2007) for further basic information and references and to Bloch (2003) for applications to nonholonomic systems, and to Cendra et al. Ratiu (2001a) for applications to Lagrangian reduction.