definition 24.33 Symplectic capacity

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:1100 · p. 861

ground object -- no derivation owed

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 remark 24.27 Fixing a scale, so that a ``ball'' means something ¶ ↺

Supports

depends_on proposition 24.34 Capacities exist if and only if non-squeezing holds ¶

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:1117
`depends_on`	→	Fixing a scale, so that a ``ball'' means something	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1117
`depends_on`	←	Capacities exist if and only if non-squeezing holds	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1146