remark 24.48 Reduction is what physicists do without saying so

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1642 · p. 867

remark -- no derivation owed by its kind

Rests on

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 definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 22.14 Phase-space coordinate transformation ¶
    - 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 definition 24.8 Symplectic manifold ¶
      - depends_on equation 22.4 eq:ham-pdot ¶
      - depends_on equation 22.3 eq:ham-qdot ¶
    - 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 4.4 Internal binary operation; magma ¶
      - depends_on definition 4.33 Vector space ¶
  - 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 definition 18.3 Particle ¶
      - depends_on definition 19.5 Linear momentum ¶
    - depends_on postulate 19.8 Newton's second law ¶
      - depends_on definition 19.5 Linear momentum ¶ ↺
      - depends_on postulate 19.6 Newton's first law ¶
  - depends_on theorem 29.16 Principal axes ¶
    - depends_on definition 29.13 Inertia tensor, continuum form ¶
      - depends_on definition 7.125 Multiple integral ¶
      - depends_on definition 13.5 Tensor under orthogonal transformations ¶
      - depends_on definition 19.42 Inertia tensor ¶
    - depends_on proposition 29.15 Positivity ¶
      - depends_on definition 29.13 Inertia tensor, continuum form ¶ ↺
      - proves proof ch:12-rigid-body-rotating-frames@proof-9 ¶
    - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
      - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on definition 5.41 Adjoint ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-31 ¶
    - 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 definition 14.26 The rotation group ¶
      - depends_on lemma 14.62 The Levi-Civita symbol is invariant ¶
      - depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
    - depends_on proposition 29.5 The velocity field of a rigid body ¶
      - depends_on definition 29.4 Angular velocity ¶ ↺
      - depends_on equation 29.1 eq:rigid-constraint ¶
      - proves proof ch:12-rigid-body-rotating-frames@proof-3 ¶
    - 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 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 definition A.70 Pullback ¶
      - depends_on definition 24.8 Symplectic manifold ¶ ↺
  - depends_on theorem 13.59 Regular value theorem ¶
    - depends_on definition 13.52 Differential; pushforward ¶
      - depends_on definition 13.49 Smooth map between manifolds ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.58 Hypersurface ¶
      - depends_on definition 13.54 Embedded submanifold ¶
    - 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 ¶

Supports

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

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The free rigid body is a Lie–Poisson system	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1655
`depends_on`	→	Marsden–Weinstein reduction	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1655
`depends_on`	←	The abelian case, and eliminating a cyclic coordinate	declared	appendices/A-long-proofs.tex:27524