feat(PhyslibAlpha): Souriau's moment map on a symplectic vector space, cocycle and Noether's theorem

[KevorkianPhilippe]

AI disclosure

This file was generated with Claude (Claude Code: Fable 5.1 for the file, Opus 5.5 for opening
this PR) under my supervision, following AI-POLICY.md and AGENTS.md. I have read every
definition and lemma statement and I vouch that they state what the docstrings say. The statements
were frozen before any proof was written and proved first in a separate development; the proofs
of this file were then written by Claude in Physlib style. The bibliographic reference (Souriau
1970, chapter 11, pp. 104-117, and the equation numbers cited in the docstrings) was checked by me
on the printed pages, not by the AI.

Summary

Adds PhyslibAlpha/ClassicalMechanics/MomentMap.lean, one self-contained file (imports Mathlib
only) on Souriau's moment map in the setting of his example (11.2): a real finite-dimensional
symplectic vector space E on which a real Lie algebra g acts by affine infinitesimally
symplectic vector fields Z_E(x) = A Z x + b Z, with Souriau's printed bracket convention
(11.22 a). It proves (11.7)-(11.9) for the moment, Souriau's cocycle f(Z)(Z') = σ(b Z, b Z')
and its identities, Noether's theorem (11.12) for a Hamiltonian flow on E, and that for the
translations of the plane the cocycle is not a coboundary. No manifold is involved.

Declarations added (namespace ClassicalMechanics)

Section A:

• IsSymplecticForm: structure, an alternating non-degenerate bilinear form.

Section B (sub-namespace AffineSymplecticAction unless noted):

• AffineSymplecticAction (structure, in ClassicalMechanics): the action Z_E(x) = A Z x + b Z.
• coadjoint (in ClassicalMechanics): the coadjoint vector field (11.16), ν ↦ ν ∘ ad(Z).
• vectorField: Z_E (6.11); moment: μ.Z (x) = -½ σ(A Z x, x) - σ(b Z, x) (11.7);
  cocycle: f(Z)(Z') = σ(b Z, b Z') (11.17 c).
• sigma_A_left, sigma_A_right, sigma_A_symm: infinitesimal symplecticity in three forms.
• moment_add, moment_smul: the moment is linear in Z.

Section C:

• sigma_vectorField_vectorField: (11.17 d), σ(Z_E, Z'_E) = μ[Z, Z'] + f(Z)(Z').
• cocycle_antisymm (11.30), cocycle_cyclic (11.33), cocycle_lie ((11.22)/(11.24), f is a
  Lie algebra cocycle for the coadjoint representation).
• sigma_vectorField_vectorField_shift (11.18), equivariant_of_coboundary (11.27),
  cocycle_eq_zero_of_linear (11.21, linear actions have zero cocycle).
• IsSymplecticGradient (in ClassicalMechanics): the symplectic gradient (9.16) with the sign
  of (11.7).

Section D:

• sigmaA, sigmaA_apply, sigmaB, sigmaB_apply, hasFDerivAt_quad,
  hasFDerivAt_moment_of_symm, hasFDerivAt_negSigmaVectorField: derivative computations.
• hasFDerivAt_moment: (11.7), σ(Z_E(x)) = -∇[μ.Z].
• vectorField_isSymplecticGradient: (11.9).
• moment_unique: (11.8 a), two moments differ by a constant torsor.
• fderiv_moment_vectorField: infinitesimal equivariance up to the cocycle (11.17 d).
• sigma_vectorField_bracket: (11.8 c).

Section E:

• noether, noether_const: (11.12), μ.Z is conserved along a motion of a Hamiltonian
  invariant under Z_E.

Section F (the plane):

• instLieRingPlane, instLieAlgebraPlane: ℝ × ℝ as an abelian Lie algebra.
• planeForm, isSymplecticForm_planeForm: the standard symplectic form of the plane.
• planeTranslations, planeTranslations_cocycle: the translations, cocycle σ(Z, Z').
• planeTranslations_not_coboundary: that cocycle is not a coboundary (11.21).

Also: one line in PhyslibAlpha.lean, and the entry Souriau1970 in docs/references.bib.

Reviewer map

1. The module docstring, then sections A and B (the two structures and the three definitions).
2. hasFDerivAt_moment and vectorField_isSymplecticGradient (section D): the moment is right.
3. sigma_vectorField_vectorField and the cocycle identities of section C.
4. noether (section E) and section F.
5. The derivative computations of section D (technical).

Checks run

• lake build -KCI PhyslibAlpha on the branch merged with current master: success, no warning in
  this file.
• lake exe runPhyslibAlphaLinters, alphaFileImports.py, noAlphaImports.py,
  ./scripts/PhyslibAlpha/alphaPythonLinters.sh, ./scripts/lint-style.sh: pass.
• #print axioms on every declaration: [propext, Classical.choice, Quot.sound].

🤖 Generated with Claude Code

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[jstoobysmith]

Approved, will merge shortly.