Marsden–Weinstein Reduction
This appendix proves Theorem 28.47 of Symplectic Geometry of Phase Space: when a Lie group acts freely and properly on a symplectic manifold by symplectomorphisms with an equivariant momentum map, and \(\mu\) is a regular value of that map, the quotient Equation (28.31) of the level set by the isotropy group of \(\mu\) is again a symplectic manifold, of the dimension Equation (28.32), and every invariant Hamiltonian descends to it with its dynamics intact.
The chapter states the geometric core of the argument and does not prove it: the tangent space to the level set of the momentum map is the symplectic orthogonal of the group orbit (Lemma 28.46, which is proved there), and the degenerate directions of the restricted form turn out to be exactly the orbit directions of the isotropy group \(G_{\mu}\), so that quotienting by that group removes the degeneracy and removes nothing else. That sentence is The degenerate directions of the restricted form below. Everything around it — that the level set is a manifold, that the quotient is one, that the descended form is well defined, closed and nondegenerate — is what makes the theorem long, and it is written out here.
The paragraph following the proof link in Symplectic Geometry of Phase Space names one input as quoted rather than proved: the quotient-manifold theorem, that a free and proper action of a Lie group on a manifold has a smooth quotient with the projection a submersion, described there as differential topology belonging in Differentiable Manifolds, Tensors, and Curvature, “which does not yet carry it”. That sentence is now out of date, and the correction is worth making rather than leaving as a discrepancy between two pages of the same book: Differentiable Manifolds, Tensors, and Curvature carries the statement as Theorem 17.67, with its own proof, and this section therefore cites it rather than quoting it. Nothing in Marsden–Weinstein Reduction is imported from outside the treatise. The role of that theorem is unchanged and remains confined to producing the smooth structure on \(M_{\mu}\): the symplectic content — existence, uniqueness, closedness and nondegeneracy of the reduced form, and the dimension count — rests on Lemma 28.46 and on Two facts of symplectic linear algebra alone.
Setting and statement
Throughout, \((M,\omega)\) is a symplectic manifold (Definition 28.8), \(G\) is a Lie group with Lie algebra \(\mathfrak{g}\) acting smoothly on \(M\) by symplectomorphisms (Definition 28.17), and \(\xi_{M}\) denotes the vector field generating the action of \(\xi\in\mathfrak{g}\), so that \(\xi_{M}(x)=\frac{\dd}{\dd t}\bigl|_{t=0}\exp(t\xi)\cdot x\). The action is assumed free and proper in the sense of Definition 17.66. The momentum map \(\vect{J}:M\longrightarrow\mathfrak{g}^{*}\) is the one of Definition 28.43, defined by Equation (28.28), and the interior product is written \(\iota_{X}\), as in Symplectic Geometry of Phase Space; the same operation is written \(i_{X}\) in Differentiable Manifolds, Tensors, and Curvature.
The coadjoint action of \(G\) on \(\mathfrak{g}^{*}\) is
a left action, and its infinitesimal form is \(\ad^{*}_{\xi}=\frac{\dd}{\dd t}\bigl|_{t=0} \mathrm{Ad}^{*}_{\exp(t\xi)}\), so that
A momentum map is equivariant if \(\vect{J}(g\cdot x)=\mathrm{Ad}^{*}_{g}\vect{J}(x)\) for every \(g\) and \(x\). For \(\mu\in\mathfrak{g}^{*}\) write \(G_{\mu}=\set{g\in G\mid\mathrm{Ad}^{*}_{g}\mu=\mu}\) for its isotropy group, a closed subgroup, and \(\mathfrak{g}_{\mu}\) for the Lie algebra of \(G_{\mu}\). Rests on Definitions 8.21, 18.2 and 28.43.
Two sign conventions circulate for Equation (A48.1), differing by \(g\leftrightarrow g^{-1}\); with the one chosen here the coadjoint action is a left action and equivariance reads \(\vect{J}\circ g=\mathrm{Ad}^{*}_{g}\circ\vect{J}\) without an inverse. Sources that define \(\mathrm{Ad}^{*}\) by \(\langle\mathrm{Ad}^{*}_{g}\mu,\eta\rangle=\langle\mu, \mathrm{Ad}_{g}\eta\rangle\) write the same statement with \(g^{-1}\) on the right, and the two agree.
This section is pure geometry and the only dimensional statement it needs is the one Remark 28.11 already fixes: \(\omega\) carries \(\mathrm{J}\,\mathrm{s}\), and a component \(J_{\xi}\) of the momentum map carries the SI unit of the conserved quantity it generates — so \(\vect{J}\) is not a single dimensioned object but a covector whose \(f\) components may carry \(f\) different units, exactly as the lattice of The Liouville–Arnold Theorem does. The reduced form \(\omega_{\mu}\) inherits \(\mathrm{J}\,\mathrm{s}\) from \(\omega\), because it is defined by pullback.
Let \(G\) act freely and properly on \((M,\omega)\) by symplectomorphisms with an equivariant momentum map \(\vect{J}\), and let \(\mu\in\mathfrak{g}^{*}\). Write \(N=\vect{J}^{-1}(\mu)\), let \(i:N\hookrightarrow M\) be the inclusion and \(\pi:N\longrightarrow M_{\mu}=N/G_{\mu}\) the projection. Then
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every value of \(\vect{J}\) is regular, and \(N\) is a closed embedded submanifold of \(M\) of dimension \(\dim M-\dim G\) with \(T_{x}N=\ker\dd\vect{J}_{x}\);
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\(G_{\mu}\) acts freely and properly on \(N\), and \(M_{\mu}\) is a smooth manifold with \(\pi\) a surjective submersion and
\begin{equation}\tag{A48.3} \dim M_{\mu}=\dim M-\dim G-\dim G_{\mu}\ec \end{equation}which is Equation (28.32);
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there is exactly one two-form \(\omega_{\mu}\) on \(M_{\mu}\) with \(\pi^{*}\omega_{\mu}=i^{*}\omega\), and it is symplectic;
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every \(G\)-invariant \(H\in C^{\infty}(M)\) descends to \(H_{\mu}\in C^{\infty}(M_{\mu})\) with \(H_{\mu}\circ\pi=H\circ i\); the flow of \(X_{H}\) preserves \(N\); and \(\pi\) carries it to the flow of \(X_{H_{\mu}}\) on \((M_{\mu},\omega_{\mu})\).
Rests on Lemma 28.46, Theorem 17.67 and Definition A48.2.
Two facts of symplectic linear algebra
Let \((V,\Omega)\) be a finite-dimensional real vector space with a nondegenerate antisymmetric bilinear form, and for a subspace \(W\subseteq V\) let \(W^{\Omega}=\set{v\in V\mid\Omega(v,w)=0\ \text{for all}\ w\in W}\). Then
Rests on Definition 28.2 and Theorem 9.40.
Derives Lemma A48.5. Nondegeneracy makes \(\flat:v\longmapsto\Omega(v,\cdot)\) an injective, hence bijective, linear map \(V\longrightarrow V^{*}\). By definition \(W^{\Omega}=\flat^{-1}\bigl(W^{\circ}\bigr)\), where \(W^{\circ}\subseteq V^{*}\) is the annihilator of \(W\); and \(\dim W^{\circ}=\dim V-\dim W\) by rank–nullity (Theorem 9.40) applied to the restriction map \(V^{*}\longrightarrow W^{*}\), which is surjective. That is the first identity. For the second, \(W\subseteq(W^{\Omega})^{\Omega}\) is immediate from antisymmetry, and applying the first identity twice gives \(\dim(W^{\Omega})^{\Omega}=\dim V-\dim W^{\Omega}=\dim W\), so the inclusion is an equality.
∎The level set is a manifold
If the action is free then \(\dd\vect{J}_{x}\) is surjective at every \(x\in M\). Consequently every \(\mu\in\mathfrak{g}^{*}\) is a regular value, \(N=\vect{J}^{-1}(\mu)\) is a closed embedded submanifold of dimension \(\dim M-\dim G\), and \(T_{x}N=\ker\dd\vect{J}_{x} =\bigl(T_{x}(G\cdot x)\bigr)^{\omega}\). Rests on Lemma 28.46, Lemma A48.5 and Theorem 17.59.
Derives Proposition A48.6. Fix \(x\) and write \(W=T_{x}(G\cdot x)\), the span of the values \(\xi_{M}(x)\), \(\xi\in\mathfrak{g}\). Freeness makes the map \(\xi\mapsto\xi_{M}(x)\) injective: if \(\xi_{M}(x)=0\) then the curve \(t\mapsto\exp(t\xi)\cdot x\) has vanishing velocity and, being an integral curve of \(\xi_{M}\) through \(x\), is constant by the uniqueness clause of Theorem 17.125; so \(\exp(t\xi)\cdot x=x\) for all \(t\), whence \(\exp(t\xi)=e\) by freeness and \(\xi=0\). Hence \(\dim W=\dim\mathfrak{g}=\dim G\).
By Lemma 28.46, \(\ker\dd\vect{J}_{x}=W^{\omega}\), so Equation (A48.4) gives \(\dim\ker\dd\vect{J}_{x}=\dim M-\dim W=\dim M-\dim G\), and by rank–nullity the rank of \(\dd\vect{J}_{x}\) is \(\dim G =\dim\mathfrak{g}^{*}\): the differential is surjective. The regular value theorem Theorem 17.59 then makes \(N=\vect{J}^{-1}(\mu)\) an embedded submanifold of dimension \(\dim M-\dim G\), closed because it is the preimage of a point under a continuous map, with tangent space the kernel.
∎\(G_{\mu}\) maps \(N\) into itself, and its action on \(N\) is free and proper. Hence \(M_{\mu}=N/G_{\mu}\) carries exactly one smooth structure making \(\pi:N\longrightarrow M_{\mu}\) a surjective submersion; it is Hausdorff and second countable, the fibres of \(\pi\) are the \(G_{\mu}\)-orbits, and Equation (A48.3) holds. Rests on Definition A48.2, Theorem 17.67 and Proposition A48.6.
Derives Proposition A48.7. For \(g\in G_{\mu}\) and \(x\in N\), equivariance gives \(\vect{J}(g\cdot x)=\mathrm{Ad}^{*}_{g}\vect{J}(x) =\mathrm{Ad}^{*}_{g}\mu=\mu\), so \(g\cdot x\in N\). The restricted action is free because the \(G\)-action is, and it is proper: \(G_{\mu}\) is a closed subgroup of \(G\) and \(N\) is a closed subset of \(M\), so the preimage of a compact set under \((g,x)\mapsto(g\cdot x,x)\) on \(G_{\mu}\times N\) is a closed subset of the corresponding preimage on \(G\times M\), hence compact. Theorem 17.67 applied to this action supplies the smooth structure, the submersion, the separation properties and the fibres, together with the dimension \(\dim M_{\mu}=\dim N-\dim G_{\mu}\), which with Proposition A48.6 is Equation (A48.3).
∎The degenerate directions of the restricted form
Write \(\omega_{N}=i^{*}\omega\) for the restriction of \(\omega\) to \(N\). It is a closed two-form, but it is not symplectic: it has a radical, and identifying that radical is the heart of the theorem.
Let \(x\in N\), so that \(\vect{J}(x)=\mu\). Then for every \(\xi\in\mathfrak{g}\)
and consequently \(\xi_{M}(x)\in\ker\dd\vect{J}_{x}\) if and only if \(\xi\in\mathfrak{g}_{\mu}\). Rests on Definition A48.2, Proposition A48.6 and Theorem 17.125.
Derives Lemma A48.8. Apply equivariance along the one-parameter subgroup \(g(t)=\exp(t\xi)\) and differentiate at \(t=0\). The left-hand side of \(\vect{J}(g(t)\cdot x)=\mathrm{Ad}^{*}_{g(t)}\mu\) differentiates by the chain rule to \(\dd\vect{J}_{x}(\xi_{M}(x))\), since \(\frac{\dd}{\dd t}\bigl|_{0}g(t)\cdot x=\xi_{M}(x)\) by definition; the right-hand side differentiates to \(\ad^{*}_{\xi}\mu\) by Definition A48.2. That is Equation (A48.5).
For the second claim, \(\ad^{*}_{\xi}\mu=0\) certainly holds when \(\xi\in\mathfrak{g}_{\mu}\), since then \(\mathrm{Ad}^{*}_{\exp(t\xi)}\mu=\mu\) for all \(t\). Conversely suppose \(\ad^{*}_{\xi}\mu=0\) and set \(\nu(t)=\mathrm{Ad}^{*}_{\exp(t\xi)}\mu\). Differentiating Equation (A48.1) in \(t\) and using the group law gives the linear differential equation \(\dot{\nu}(t)=\ad^{*}_{\xi}\nu(t)\) with \(\nu(0)=\mu\). The constant function \(\nu\equiv\mu\) solves it, because \(\ad^{*}_{\xi}\mu=0\), and the solution of a linear equation with given initial value is unique (Theorem 13.8); so \(\nu(t)=\mu\) for all \(t\), that is \(\exp(t\xi)\in G_{\mu}\) for all \(t\), that is \(\xi\in\mathfrak{g}_{\mu}\).
∎For every \(x\in N\),
Derives Theorem A48.9. Write \(W=T_{x}(G\cdot x)\) as before. By Proposition A48.6 and Equation (28.30), \(T_{x}N=\ker\dd\vect{J}_{x}=W^{\omega}\). Since \(\omega_{N}\) is the restriction of \(\omega\), the radical of \(\omega_{N}\) at \(x\) is \(T_{x}N\cap(T_{x}N)^{\omega}\), the second symplectic orthogonal being taken in the symplectic vector space \((T_{x}M,\omega_{x})\). Hence, using the second identity of Equation (A48.4),
A vector of \(T_{x}(G\cdot x)\) is \(\xi_{M}(x)\) for exactly one \(\xi\in\mathfrak{g}\), by the injectivity established in Proposition A48.6, and it lies in \(\ker\dd\vect{J}_{x}\) precisely when \(\xi\in\mathfrak{g}_{\mu}\), by Lemma A48.8. The intersection is therefore \(\set{\xi_{M}(x)\mid\xi\in\mathfrak{g}_{\mu}} =T_{x}(G_{\mu}\cdot x)\).
∎The radical of a closed two-form of constant rank is always an involutive distribution, so it is integrable by Theorem 17.133 and the manifold is foliated by its leaves. The check is two lines with Cartan's magic formula: if \(\iota_{X}\omega_{N}=0\) and \(\iota_{Y}\omega_{N}=0\) then \(\mathcal{L}_{X}\omega_{N} =\dd\,\iota_{X}\omega_{N}+\iota_{X}\dd\omega_{N}=0\) by Equation (17.276) and \(\dd\omega_{N}=i^{*}\dd\omega=0\), and then Equation (17.277) gives \(\iota_{\comm{X}{Y}}\omega_{N} =\mathcal{L}_{X}\iota_{Y}\omega_{N} -\iota_{Y}\mathcal{L}_{X}\omega_{N}=0\), so \(\comm{X}{Y}\) lies in the radical too. What Theorem A48.9 adds is that the leaves of that foliation are exactly the \(G_{\mu}\)-orbits. Without it one could still divide \(N\) by the characteristic foliation and get a symplectic quotient, but the quotient would be a leaf space with no reason to be Hausdorff or to be a manifold at all. Identifying the leaves as orbits of a group acting freely and properly is what lets Theorem 17.67 supply the smooth structure — and it is also what makes the reduced space computable in practice, since a physicist knows the symmetry group and does not know the foliation.
Descent of the form
There is exactly one two-form \(\omega_{\mu}\) on \(M_{\mu}\) with \(\pi^{*}\omega_{\mu}=\omega_{N}\), it is smooth, and it is closed and nondegenerate. Rests on Theorem A48.9, Proposition A48.7 and Theorem 28.16.
Derives Proposition A48.11. Definition. Let \([x]\in M_{\mu}\) and \(u,v\in T_{[x]}M_{\mu}\). Since \(\pi\) is a submersion, \(\dd\pi_{x}\) is onto, so there are \(\tilde{u},\tilde{v}\in T_{x}N\) with \(\dd\pi_{x}\tilde{u}=u\) and \(\dd\pi_{x}\tilde{v}=v\); set
This is independent of the lifts. The fibres of \(\pi\) are the \(G_{\mu}\)-orbits (Proposition A48.7), so \(\ker\dd\pi_{x}=T_{x}(G_{\mu}\cdot x)\), which is exactly the radical of \((\omega_{N})_{x}\) by Theorem A48.9; changing a lift changes it by an element of that radical, which \(\omega_{N}\) does not see.
Independence of the point in the fibre. Let \(g\in G_{\mu}\) and replace \(x\) by \(g\cdot x\), lifting through \(\dd(g\cdot)_{x}\tilde{u}\) and \(\dd(g\cdot)_{x}\tilde{v}\), which are legitimate lifts because \(\pi\circ g=\pi\). Since \(G\) acts by symplectomorphisms, \(g^{*}\omega=\omega\), and \(g\) maps \(N\) to \(N\), so \(g^{*}\omega_{N}=\omega_{N}\) and the two evaluations agree.
Smoothness. A submersion admits local smooth sections: near any \([x]\) choose \(s\) with \(\pi\circ s=\id\); then \(\omega_{\mu}=s^{*}\omega_{N}\) on that neighbourhood by Equation (A48.8), a pullback of a smooth form along a smooth map.
Uniqueness. If \(\pi^{*}\omega_{\mu}=\pi^{*}\omega_{\mu}'\) then, \(\dd\pi_{x}\) being surjective at every \(x\) and \(\pi\) being surjective, the two forms agree on every pair of tangent vectors at every point.
Closedness. \(\pi^{*}\dd\omega_{\mu}=\dd\pi^{*}\omega_{\mu} =\dd\,i^{*}\omega=i^{*}\dd\omega=0\), the exterior derivative commuting with pullback; and \(\pi^{*}\) is injective on forms because \(\dd\pi\) is surjective. Hence \(\dd\omega_{\mu}=0\).
Nondegeneracy. Suppose \((\omega_{\mu})_{[x]}(u,\cdot)=0\). Lift \(u\) to \(\tilde{u}\in T_{x}N\); then \((\omega_{N})_{x}(\tilde{u},\tilde{v})=0\) for every \(\tilde{v}\in T_{x}N\), because every \(v\) is \(\dd\pi_{x}\tilde{v}\). So \(\tilde{u}\) lies in the radical, which is \(\ker\dd\pi_{x}\), and therefore \(u=\dd\pi_{x}\tilde{u}=0\).
∎Descent of the dynamics
Let \(H\in C^{\infty}(M)\) be \(G\)-invariant. Then \(H\circ i\) is \(G_{\mu}\)-invariant and descends to a smooth \(H_{\mu}\) on \(M_{\mu}\); the flow of \(X_{H}\) preserves \(N\); and \(\dd\pi_{x}\bigl(X_{H}(x)\bigr)=X_{H_{\mu}}([x])\) for every \(x\in N\). Rests on Theorem 28.45, Proposition A48.11 and Equation (28.5).
Derives Proposition A48.12. Invariance under \(G\) implies invariance under the subgroup \(G_{\mu}\), and by the last clause of Theorem 17.67 a function on \(M_{\mu}\) is smooth exactly when its composition with \(\pi\) is; the \(G_{\mu}\)-invariant function \(H\circ i\) therefore defines a smooth \(H_{\mu}\) with \(H_{\mu}\circ\pi=H\circ i\).
That the flow of \(X_{H}\) preserves \(N\) is Noether's theorem in the form Theorem 28.45: \(G\) acts by symplectomorphisms and preserves \(H\), so \(\vect{J}\) is constant along the flow, and a trajectory starting on \(\vect{J}^{-1}(\mu)\) stays there. In particular \(X_{H}(x)\in T_{x}N\) for \(x\in N\).
For the last claim, let \(v\in T_{[x]}M_{\mu}\) and lift it to \(\tilde{v}\in T_{x}N\). Using Equation (A48.8), then Equation (28.5), then \(H_{\mu}\circ\pi=H\circ i\):
Since \(v\) was arbitrary and \(\omega_{\mu}\) is nondegenerate, the vector \(\dd\pi_{x}X_{H}(x)\) is the unique one satisfying Equation (28.5) for \(H_{\mu}\), that is \(X_{H_{\mu}}([x])\).
∎Proof of Theorem A48.4. Derives Theorem A48.4. Part (1) is Proposition A48.6, part (2) is Proposition A48.7, part (3) is Proposition A48.11 and part (4) is Proposition A48.12. Since \(M_{\mu}=\vect{J}^{-1}(\mu)/G_{\mu}\) is Equation (28.31) and Equation (A48.3) is Equation (28.32), this is Theorem 28.47.
∎The example the chapter names
Take \(M=T^{*}\R^{3}\) with \(\omega=\dd q^{i}\wedge\dd p_{i}\) and \(G=\SO(3)\) acting by \((\vect{q},\vect{p})\mapsto(R\vect{q},R\vect{p})\). This is an action by symplectomorphisms, and Example 28.44 computes its momentum map: \(\vect{J}=\vect{q}\times\vect{p}=\vect{L}\), the angular momentum, in \(\mathrm{J}\,\mathrm{s}\).
The action is not free on all of \(M\) — a rotation about \(\vect{q}\) fixes any point with \(\vect{p}\) parallel to \(\vect{q}\), and every rotation fixes the origin — so the hypotheses hold only on the open, \(\SO(3)\)-invariant subset \(M^{\times}=\set{(\vect{q},\vect{p})\mid\vect{q}\times\vect{p} \neq\vect{0}}\), where a rotation fixing two independent vectors is the identity. That restriction is not a technicality to be waved through: it is exactly the collinear and radial motions, which the reduced picture cannot describe and which Section 31.4 treats separately.
On \(M^{\times}\) fix \(\mu=\vect{L}\) with \(\ell=\abs{\vect{L}}>0\). The isotropy group of \(\mu\) under the coadjoint action of \(\SO(3)\) — which for \(\SO(3)\) is the ordinary rotation action on \(\R^{3}\), the algebra carrying an invariant inner product — is the group \(\SO(2)\) of rotations about the axis of \(\vect{L}\), of dimension \(1\). So Equation (A48.3) gives
and the reduced space is the two-dimensional radial phase space \((r,p_{r})\). For \(H=\abs{\vect{p}}^{2}/2m+V(r)\), resolving \(\vect{p}\) into its radial and transverse parts on \(\vect{J}^{-1}(\mu)\) gives \(\abs{\vect{p}}^{2}=p_{r}^{2}+\ell^{2}/r^{2}\), so
which is the effective potential of Equation (31.32), and the centrifugal term \(\ell^{2}/2mr^{2}\) carries \(\mathrm{J}\) as it must. This is the identification Remark 28.48 asserts: the effective potential is the reduced Hamiltonian at the fixed value of the angular momentum, and the disappearance of the two angular coordinates is not a trick of the coordinate system but the reduction of six dimensions to two. Rests on Theorem A48.4, Example 28.44 and Equation (31.32).
If \(G\) is abelian the coadjoint action is trivial, so \(G_{\mu}=G\) for every \(\mu\) and Equation (A48.3) reads \(\dim M_{\mu}=\dim M-2\dim G\): two dimensions are lost per generator. For \(G=\R\) acting by translation in a cyclic coordinate \(q^{1}\), the momentum map is \(p_{1}\), fixing \(\mu\) freezes that momentum, and dividing by the group deletes \(q^{1}\) — the pair \((q^{1},p_{1})\) disappears together, which is precisely the elementary manoeuvre Remark 28.48 says every physicist performs without naming it. Rests on Theorem A48.4 and Remark 28.48.
What is quoted here
Nothing. The inputs are Lemma 28.46, proved in Symplectic Geometry of Phase Space; the regular value theorem Theorem 17.59, the quotient-manifold theorem Theorem 17.67, Frobenius' theorem Theorem 17.133 and the Cartan identities Equation (17.276) and Equation (17.277), all proved in Differentiable Manifolds, Tensors, and Curvature; rank–nullity Theorem 9.40 from Linear Algebra and Representation Theory; and the uniqueness of solutions of an ordinary differential equation, Theorem 13.8. Remark A48.1 records that the chapter's own prose still describes the quotient-manifold theorem as owed by Part II, which it no longer is.
The three hypotheses divide the work cleanly, and it is worth recording which of them fails first in practice.
Freeness is used twice: in Proposition A48.6, to make every value of \(\vect{J}\) regular and \(N\) a manifold, and in Proposition A48.7, as a hypothesis of Theorem 17.67. Properness is used once, also through Theorem 17.67, and what it buys is that the quotient topology is Hausdorff — Example 17.68 exhibits a free but improper action whose orbit space is not. Equivariance is used once, in Lemma A48.8, and it is what makes \(\mathfrak{g}_{\mu}\) rather than \(\mathfrak{g}\) appear in Equation (A48.6); without it the radical would not be the tangent space to an orbit of anything.
When freeness fails — which, as Example A48.13 shows, is the normal state of affairs at the symmetric configurations — \(M_{\mu}\) is not a manifold but a stratified space, a union of manifolds of different dimensions glued along their boundaries. That is singular reduction, and it is not treated in this treatise. The honest summary is that the theorem proved here describes the generic stratum and says nothing about the symmetric configurations, which for the central-force problem are exactly the collinear orbits.
The theorem is Marsden and Weinstein's, from Reduction of symplectic manifolds with symmetry (Reports on Mathematical Physics 5, 1974, 121–130). That paper has no entry in this treatise's bibliography — Symplectic Geometry of Phase Space records the omission in its own header, together with those of Abraham–Marsden and Arnold — so the attribution is made in words, on the footing of Darboux's memoir in Remark A12.15. Nothing above rests on it: the derivation is carried out in full from results proved in this book.
Marsden–Weinstein Reduction discharges the derivation owed at Theorem 28.47 of Symplectic Geometry of Phase Space. Two threads run back into the book from here. Remark 28.48 lists four familiar manoeuvres — dropping a cyclic pair, passing to the centre-of-mass frame, reducing the two-body problem to the radial one, and using the effective potential — and says that each is this theorem applied to a particular group; Example A48.13 and Example A48.14 above verify two of them, including the identification of Equation (31.32) as the reduced Hamiltonian. And the same remark points at the Lie–Poisson bracket Definition 28.40 as reduction with the whole group divided out, which is why the free rigid body of Proposition 28.41 lives on three variables rather than the six a cotangent bundle would demand: the dimension count is Equation (A48.3) once more, with \(M=T^{*}\SO(3)\) of dimension six, \(G=\SO(3)\) of dimension three, and a generic \(G_{\mu}=\SO(2)\) removing one further dimension to leave the two-dimensional coadjoint orbit — a sphere of constant \(\abs{\vect{L}}\) — on which the Euler equations Equation (28.27) run.