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The cotangent lift of an action is Hamiltonian with the tautological moment map

Statement

Assume ${A C}_{ω}$ . Let a smooth left action of $G$ on a smooth manifold $Q$ be given and let $μ_{0} : G \times Q \to Q$ denote it. Define the cotangent-lifted action on $T^{*} Q$ by

$\hat{g} (q, p) : = (g \cdot q, (d (a_{g})_{q}^{- 1})^{*} p),$

where $a_{g} (q) = g \cdot q$ . Then the lifted action is a smooth left action preserving the canonical symplectic form $ω_{c a n} = - d λ$ , and the tautological moment map

$⟨ μ (q, p), ξ ⟩ : = - p (ξ_{Q} (q)), ξ \in g,$

satisfies the component moment equations $d μ^{ξ} = - ι_{ξ_{T^{*} Q}} ω_{c a n}$ for the lifted fundamental fields. Its coadjoint equivariance, which makes $μ$ an equivariant moment map, is verified in the companion lemma.

Facts & Assumptions

Given: ${A C}_{ω}$ , a smooth left action of $G$ on $Q$ and the induced cotangent-lifted action on $T^{*} Q$ .

[A1]

${A C}_{ω}$ is countable choice; it is used only through the fundamental-field and cotangent-bundle interfaces cited below.

[F1]

$λ_{(q, p)} (v) = p (d π v)$ and $ω_{c a n} = - d λ$ is the canonical symplectic form on $T^{*} Q$ . Tautological one-form on a cotangent bundle.

[F2]

For a diffeomorphism $f : Q \to Q^{'}$ the cotangent lift $\hat{f} (q, p) = (f (q), (d f_{q}^{- 1})^{*} p)$ satisfies ${\hat{f}}^{*} λ_{Q^{'}} = λ_{Q}$ and ${\hat{f}}^{*} ω_{Q^{'}} = ω_{Q}$ . Cotangent lifts are symplectomorphisms.

[F3]

If $Y$ is a vector field on $Q$ and $Y^{#}$ is the infinitesimal generator of the inverse-transpose cotangent lifts of the local flow of $Y$ , then in cotangent coordinates $Y^{#} = Y^{i} \partial_{q^{i}} - p_{j} (\partial_{q^{i}} Y^{j}) \partial_{p_{i}}$ and $Y^{#}$ is Hamiltonian for $p (Y_{q})$ . The cotangent lift of a vector field is Hamiltonian.

[F4]

The fundamental field of the lifted action is $ξ_{T^{*} Q} (q, p) = {\frac{d}{d t} ∣}_{0} \hat{a_{\exp_{G} (- t ξ)}} (q, p)$ , and it projects to $ξ_{Q} (q)$ because $π \circ \hat{a_{h}} = a_{h} \circ π$ . Fundamental vector fields for a left action, Smooth left actions of Lie groups.

[F5]

$\exp_{G} ({Ad}_{g} ζ) = g \exp_{G} (ζ) g^{- 1}$ , hence $({Ad}_{g} ζ)_{Q} (g \cdot q) = d (a_{g})_{q} ζ_{Q} (q)$ for the fundamental fields of the action on $Q$ . Adjoint intertwines the exponential map, Fundamental vector fields for a left action.

Proof

technique · direct

1.1

The lifted action is a smooth left action: ${\hat{a}}_{g}$ is the cotangent lift of the diffeomorphism $a_{g}$ , the formula is smooth in $(g, q, p)$ , and ${\hat{a}}_{g} \circ {\hat{a}}_{h} = {\hat{a}}_{g h}$ by the chain rule. Each ${\hat{a}}_{g}$ preserves $λ$ and $ω_{c a n}$ by [F2], so the action is symplectic.

F2given

1.2

The fundamental field of the lifted action is the infinitesimal generator of the inverse-transpose lifts of the flow of $ξ_{Q}$ : it projects to $ξ_{Q}$ by [F4], and differentiating the lift formula in cotangent coordinates gives $(ξ_{Q})^{#} = ξ_{Q}^{i} \partial_{q^{i}} - p_{j} (\partial_{q^{i}} ξ_{Q}^{j}) \partial_{p_{i}} = ξ_{T^{*} Q}$ .

F3F4given

2.1

By [F3] the field $(ξ_{Q})^{#} = ξ_{T^{*} Q}$ is Hamiltonian with Hamiltonian function $p (ξ_{Q} (q))$ , that is $ι_{ξ_{T^{*} Q}} ω_{c a n} = d p (ξ_{Q} (q))$ . Therefore the function $μ^{ξ} (q, p) : = - p (ξ_{Q} (q))$ satisfies $d μ^{ξ} = - ι_{ξ_{T^{*} Q}} ω_{c a n}$ , the component moment equation of the library convention.

step 1.2F1F3

3.1

Equivariance holds as well: for $g \in G$ and $ξ \in g$ , $⟨ μ (\hat{g} (q, p)), ξ ⟩ = - ((d (a_{g})_{q}^{- 1})^{*} p) (ξ_{Q} (g \cdot q)) = - p ((d (a_{g})_{q}^{- 1}) ξ_{Q} (g \cdot q)) = - p (({Ad}_{g^{- 1}} ξ)_{Q} (q)) = ⟨ μ (q, p), {Ad}_{g^{- 1}} ξ ⟩ = ⟨ g \cdot μ (q, p), ξ ⟩,$ using [F5] with $g$ replaced by $g^{- 1}$ . Hence $μ$ is coadjoint equivariant and, with step 2.1, is an equivariant moment map for the lifted action.

step 2.1F5A1∎

Depends on

Used by

Angular momentum as the moment map for rotations of a cotangent bundle Example
Cotangent reduction for a principal bundle at zero Example
Moment maps are unique without normalization False statement
The cotangent-lift moment map has a plus sign under the library fundamental-field convention False statement
The general reduced dimension is dim M minus two dim G False statement
The tautological cotangent moment map is equivariant Lemma

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)
Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)