definition 24.28 Ball and cylinder

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:945 · p. 859

ground object -- no derivation owed

Rests on

depends_on definition 24.17 Symplectomorphism ¶
- depends_on definition A.70 Pullback ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
  - depends_on definition 13.83 Covector ¶
    - 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 13.98 $k$-form ¶
    - depends_on definition 13.83 Covector ¶ ↺
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶ ↺
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
- 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 13.102 Wedge product ¶
    - 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 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 24.10 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 13.81 Vector on a manifold ¶ ↺
    - depends_on definition 24.9 The canonical form on a cotangent bundle ¶
      - depends_on definition 13.103 Exterior derivative ¶ ↺
      - depends_on definition 24.8 Symplectic manifold ¶ ↺
    - depends_on lemma 13.107 Poincaré lemma ¶
      - depends_on definition 13.105 Closed form ¶ ↺
      - depends_on definition 13.106 Exact 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 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 ¶ ↺

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`	→	Symplectomorphism	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:960
`depends_on`	→	Fixing a scale, so that a ``ball'' means something	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:960
`depends_on`	←	Symplectic capacity	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1117
`depends_on`	←	A volume-preserving squeeze	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:988
`depends_on`	←	Linear non-squeezing	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:994
`depends_on`	←	Gromov's non-squeezing theorem	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1068