ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Diagonal action and addition of angular momenta

Example

Assume ${A C}_{ω}$ . Let $S O (3)$ act diagonally on $T^{*} R^{3} \times T^{*} R^{3}$ by the cotangent lifts of the rotations of each factor, with the product symplectic form. Then the moment map is the sum of the individual angular momenta:

$μ (q_{1}, p_{1}, q_{2}, p_{2}) = q_{1} \times p_{1} + q_{2} \times p_{2} \in R^{3},$

under the identification $s o (3)^{*} ≃ R^{3}$ . This is the classical addition of angular momenta for a two-particle system in $R^{3}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , the diagonal $S O (3)$ -action on the product of two cotangent bundles with the product form.

[A1]

Countable choice is The Axiom of Countable Choice ( ${A C}_{ω}$ ), inherited through both supplied Hamiltonian constructions; the finite addition uses no further choice.

[F1]

On each factor the tautological moment map of the rotation action is $μ_{i} (q_{i}, p_{i}) = q_{i} \times p_{i}$ under the identification $s o (3)^{*} ≃ R^{3}$ . Angular momentum as the moment map for rotations of a cotangent bundle.

[F2]

On a product with the diagonal action and the product form, the moment maps add: $μ = μ_{1} \circ {pr}_{1} + μ_{2} \circ {pr}_{2}$ , and the sum is equivariant. Products and opposites of symplectic moment maps.

[F3]

The tautological cotangent moment map is coadjoint equivariant (The tautological cotangent moment map is equivariant).

Verification

technique · direct

1.1

By [F1] each factor contributes the angular momentum $q_{i} \times p_{i}$ , computed from the tautological moment map with the library's negative fundamental-field convention.

F1given

1.2

These are specifically the tautological cotangent maps by [F1], so [F3] gives $μ_{i} (R \cdot (q_{i}, p_{i})) = {Ad}_{R}^{*} μ_{i} (q_{i}, p_{i})$ for every $R \in S O (3)$ . Thus each factor meets the equivariance hypothesis of [F2], independently of any covering-group example.

F1F3algebra

2.1

By [F2] the product moment map is the pointwise sum $μ = μ_{1} + μ_{2}$ , which under the identification of $s o (3)^{*}$ with $R^{3}$ is the vector sum $q_{1} \times p_{1} + q_{2} \times p_{2}$ . Its equivariance follows from step 1.2 and [F2]. This is the addition law for angular momenta in this model. It includes vanishing individual terms and cancellation of the two terms, since no division or general-position condition occurs. The inherited assumption is [A1].

step 1.1step 1.2F2A1∎

Depends on

Used by

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Sources

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