example A.573 The abelian case, and eliminating a cyclic coordinate

Rests on

• depends_on remark 24.48 Reduction is what physicists do without saying so ¶
  • depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
    • depends_on definition 24.37 Casimir function ¶
      • depends_on definition 24.36 Poisson manifold ¶
        • depends_on definition 22.29 Poisson bracket ¶
        • depends_on equation 22.46 eq:ham-jacobi ¶
      • depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
        • depends_on definition 22.29 Poisson bracket ¶ ↺
        • depends_on definition 24.14 Hamiltonian vector field ¶
        • depends_on equation 22.46 eq:ham-jacobi ¶ ↺
        • proves proof ch:07-symplectic-geometry@proof-4 ¶
    • depends_on definition 24.40 Lie–Poisson bracket ¶
      • depends_on definition 5.127 Lie algebra ¶
        • depends_on definition 5.121 Algebra ¶
      • depends_on definition 24.36 Poisson manifold ¶ ↺
    • depends_on theorem 29.25 Euler's equations ¶
      • depends_on postulate 19.22 Newton's second law, rotational case ¶
        • depends_on definition 19.18 Angular momentum ¶
        • depends_on postulate 19.8 Newton's second law ¶
      • depends_on theorem 29.16 Principal axes ¶
        • depends_on definition 29.13 Inertia tensor, continuum form ¶
        • depends_on proposition 29.15 Positivity ¶
        • depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
        • proves proof ch:12-rigid-body-rotating-frames@proof-10 ¶
      • depends_on theorem 29.7 Transport theorem ¶
        • depends_on definition 29.4 Angular velocity ¶
        • depends_on proposition 29.5 The velocity field of a rigid body ¶
        • proves proof ch:12-rigid-body-rotating-frames@proof-5 ¶
      • proves proof ch:12-rigid-body-rotating-frames@proof-16 ¶
    • proves proof ch:07-symplectic-geometry@proof-13 ¶
  • depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    • depends_on definition 24.8 Symplectic manifold ¶
      • depends_on definition 13.105 Closed form ¶
        • depends_on definition 13.103 Exterior derivative ¶
        • depends_on definition 13.98 $k$-form ¶
      • depends_on definition 13.98 $k$-form ¶ ↺
      • depends_on definition 24.2 Symplectic vector space ¶
        • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    • depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      • depends_on definition 24.8 Symplectic manifold ¶ ↺
      • depends_on definition 24.43 Momentum map ¶
        • depends_on definition 14.2 Lie group ¶
        • depends_on definition 24.14 Hamiltonian vector field ¶ ↺
        • depends_on definition 24.17 Symplectomorphism ¶
      • depends_on theorem 13.59 Regular value theorem ¶
        • depends_on definition 13.52 Differential; pushforward ¶
        • depends_on definition 13.58 Hypersurface ¶
        • depends_on definition 13.49 Smooth map between manifolds ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-10 ¶
      • proves proof ch:07-symplectic-geometry@proof-15 ¶
    • depends_on theorem 13.59 Regular value theorem ¶ ↺
    • proves proof ch:07-symplectic-geometry@prooflink-2 ¶
• depends_on theorem A.563 Marsden–Weinstein reduction ¶
  • depends_on definition A.561 Coadjoint action and equivariance ¶
    • depends_on definition 4.21 Group ¶
      • depends_on definition 4.11 Inverse element ¶
        • depends_on definition 4.9 Neutral element ¶
      • depends_on definition 4.20 Monoid ¶
        • depends_on definition 4.9 Neutral element ¶ ↺
        • depends_on definition 4.18 Semigroup ¶
    • depends_on definition 14.2 Lie group ¶ ↺
    • depends_on definition 24.43 Momentum map ¶ ↺
  • depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶ ↺
  • depends_on theorem 13.67 Quotient manifold theorem ¶
    • depends_on definition 13.66 Smooth action; free; proper; orbit ¶
      • depends_on definition 13.48 Differentiable manifold ¶
        • depends_on definition 13.47 Differentiable structure ¶
      • depends_on definition 13.49 Smooth map between manifolds ¶ ↺
      • depends_on definition 6.9 Compact set ¶
        • depends_on definition 6.5 Open cover ¶
    • depends_on definition 13.54 Embedded submanifold ¶
      • depends_on definition 13.53 Immersion, submersion, embedding ¶
        • depends_on definition 13.52 Differential; pushforward ¶ ↺
        • depends_on definition 6.7 Homeomorphism ¶
      • depends_on definition 13.48 Differentiable manifold ¶ ↺
    • depends_on theorem 13.63 Constant rank theorem ¶
      • depends_on corollary A.292 Inverse function theorem ¶
        • depends_on proposition 7.104 Chain rule in several variables ¶
        • depends_on theorem A.286 Implicit function theorem ¶
        • proves proof app:A-long-proofs@proof-186 ¶
      • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.97 Partial derivative; gradient ¶
      • depends_on theorem A.286 Implicit function theorem ¶ ↺
      • proves proof ch:11-manifolds-tensors-curvature@proof-12 ¶
    • proves proof app:A-long-proofs@proof-329 ¶
    • proves proof ch:11-manifolds-tensors-curvature@prooflink-2 ¶
  • proves proof app:A-long-proofs@proof-346 ¶

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