definition 24.4 Symplectic group

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:145 · p. 850

ground object -- no derivation owed

Rests on

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 definition 4.33 Vector space ¶
- depends_on equation 5.138 eq:lin-bilinear-expansion ¶
- 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 definition 5.110 Bilinear map ¶ ↺
    - depends_on equation 5.137 eq:lin-bilinear-matrix ¶
  - depends_on equation 5.139 eq:lin-bilinear-congruence ¶
  - proves proof ch:03-linear-algebra-representations@proof-54 ¶
depends_on proposition 24.3 Even dimension and the canonical basis ¶
- depends_on definition 24.2 Symplectic vector space ¶
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
- 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 ¶

Supports

depends_on proposition 24.5 Properties of the symplectic group ¶
- depends_on remark 24.6 Connectedness, and what it rests on ¶
- depends_on remark 24.7 Contrast with the orthogonal group ¶
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 ¶

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 symplectic group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:155
`depends_on`	→	Even dimension and the canonical basis	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:155
`depends_on`	←	Properties of the symplectic group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:165
`depends_on`	←	Linear non-squeezing	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:994