remark 24.11 The SI dimension of every object in this chapter

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:324 · p. 852

remark -- no derivation owed by its kind

Rests on

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 13.43 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 13.81 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 13.84 Tensor ¶
    - 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 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 ¶

Supports

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 proposition 24.34 Capacities exist if and only if non-squeezing holds ¶ ↺
      - depends_on remark 24.35 What non-squeezing does and does not say about nature ¶ ↺
  - 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`	→	The canonical form is symplectic and intrinsic	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:346
`depends_on`	→	Units	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:346
`depends_on`	←	What non-squeezing does and does not say about nature	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1225
`depends_on`	←	Liouville's theorem is the foundation of statistical mechanics	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:733
`depends_on`	←	Fixing a scale, so that a ``ball'' means something	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:942