proposition 24.10 The canonical form is symplectic and intrinsic

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:272 · p. 851

Rests on

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 13.43 Coordinate system ¶
  - depends_on definition 13.45 Differentiable map on a topological space ¶
    - depends_on definition 13.44 Atlas ¶ ↺
    - depends_on definition 13.43 Coordinate system ¶ ↺
  - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.47 Differentiable structure ¶
      - depends_on definition 13.44 Atlas ¶ ↺
- depends_on definition 13.81 Vector on a manifold ¶
  - depends_on definition 13.71 Differentiable curve ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - 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 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶ ↺
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - depends_on definition 13.102 Wedge product ¶
    - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on equation 13.239 eq:mfd-dual-basis ¶
- 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 definition 5.110 Bilinear map ¶
      - depends_on definition 5.113 Non-degenerate form ¶
      - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
      - proves proof ch:03-linear-algebra-representations@proof-54 ¶
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 ¶

Supports

depends_on remark 24.11 The SI dimension of every object in this chapter ¶
- depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
- depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
- depends_on remark 24.27 Fixing a scale, so that a ``ball'' means something ¶
  - depends_on definition 24.28 Ball and cylinder ¶
    - depends_on definition 24.33 Symplectic capacity ¶
      - depends_on proposition 24.34 Capacities exist if and only if non-squeezing holds ¶
    - depends_on example 24.29 A volume-preserving squeeze ¶
    - depends_on proposition 24.30 Linear non-squeezing ¶
      - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶
    - depends_on theorem 24.31 Gromov's non-squeezing theorem ¶ ↺
  - depends_on definition 24.33 Symplectic capacity ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Differential; pushforward	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:280
`depends_on`	→	The canonical form on a cotangent bundle	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:280
`depends_on`	→	Poincaré lemma	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:280
`depends_on`	←	The SI dimension of every object in this chapter	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:346
`proves`	←	ch:07-symplectic-geometry@proof-3	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:283