remark 24.35 What non-squeezing does and does not say about nature

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1198 · p. 862

remark -- no derivation owed by its kind

Rests on

depends_on remark 24.11 The SI dimension of every object in this chapter ¶
- depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
  - depends_on definition 13.52 Differential; pushforward ¶
    - depends_on definition 13.49 Smooth map between manifolds ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶ ↺
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 24.9 The canonical form on a cotangent bundle ¶
    - depends_on definition 13.103 Exterior derivative ¶
      - depends_on definition 13.98 $k$-form ¶
      - depends_on definition 13.102 Wedge product ¶
    - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on definition 13.105 Closed form ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on definition 24.2 Symplectic vector space ¶
  - depends_on lemma 13.107 Poincaré lemma ¶
    - depends_on definition 13.105 Closed form ¶ ↺
    - depends_on definition 13.106 Exact form ¶
      - depends_on definition 13.103 Exterior derivative ¶ ↺
      - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on equation 13.245 eq:mfd-nilpotency ¶
    - proves proof ch:11-manifolds-tensors-curvature@proof-21 ¶
  - proves proof ch:07-symplectic-geometry@proof-3 ¶
- depends_on remark 22.5 Units ¶
  - depends_on equation 22.1 eq:ham-hamiltonian ¶
  - depends_on equation 21.43 eq:lag-generalized-momentum ¶
depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
- 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 definition 24.8 Symplectic manifold ¶ ↺
- depends_on proposition 13.128 Cartan's magic formula ¶
  - depends_on definition 13.103 Exterior derivative ¶ ↺
  - depends_on equation 13.246 eq:mfd-leibniz-forms ¶
  - depends_on equation 13.245 eq:mfd-nilpotency ¶ ↺
  - depends_on proposition 13.127 Component formulas ¶
    - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
      - depends_on definition 13.88 General coordinate transformation ¶ ↺
    - depends_on equation 13.267 eq:mfd-lie-def ¶
    - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-29 ¶
- proves proof ch:07-symplectic-geometry@proof-5 ¶
depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
- depends_on definition 24.28 Ball and cylinder ¶
  - depends_on definition 24.17 Symplectomorphism ¶
    - depends_on definition A.70 Pullback ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on definition 24.8 Symplectic manifold ¶ ↺
  - depends_on remark 24.27 Fixing a scale, so that a ``ball'' means something ¶
    - depends_on equation 22.56 eq:ham-symplectic-matrix ¶
    - depends_on remark 24.11 The SI dimension of every object in this chapter ¶ ↺
- depends_on proposition 24.30 Linear non-squeezing ¶
  - depends_on definition 24.28 Ball and cylinder ¶ ↺
  - depends_on definition 24.4 Symplectic group ¶
    - depends_on definition 5.135 The symplectic group ¶
      - depends_on definition 5.113 Non-degenerate form ¶
      - depends_on equation 5.138 eq:lin-bilinear-expansion ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    - depends_on proposition 24.3 Even dimension and the canonical basis ¶
      - depends_on definition 24.2 Symplectic vector space ¶ ↺
      - depends_on equation 22.56 eq:ham-symplectic-matrix ¶ ↺
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
      - proves proof ch:07-symplectic-geometry@proof-1 ¶
  - depends_on equation 24.19 eq:sym-omega-euclid ¶
  - proves proof ch:07-symplectic-geometry@proof-11 ¶

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type	direction	node	provenance	where
`depends_on`	→	The SI dimension of every object in this chapter	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1225
`depends_on`	→	The Hamiltonian flow preserves the symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1225
`depends_on`	→	Gromov's non-squeezing theorem	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1225