proposition 24.30 Linear non-squeezing

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:991 · p. 860

Rests on

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 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - depends_on definition 13.98 $k$-form ¶
      - depends_on definition 13.83 Covector ¶ ↺
      - depends_on definition 13.84 Tensor ¶
  - 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 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 24.9 The canonical form on a cotangent bundle ¶
      - depends_on lemma 13.107 Poincaré lemma ¶
      - 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 definition 24.4 Symplectic group ¶
- depends_on definition 5.135 The symplectic group ¶
  - depends_on definition 5.113 Non-degenerate form ¶
    - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
  - 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 ¶

Supports

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 ¶

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`	→	Ball and cylinder	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:994
`depends_on`	→	Symplectic group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:994
`depends_on`	→	eq:sym-omega-euclid	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
`proves`	←	ch:07-symplectic-geometry@proof-11	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:997