definition 5.135 The symplectic group

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:5888 · p. 165

ground object -- no derivation owed

Rests on

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 definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
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 ¶

Supports

depends_on definition 22.40 Symplectic matrix ¶
depends_on definition 24.4 Symplectic group ¶
- 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 ¶
depends_on lemma 22.20 The symplectic condition ¶
- depends_on corollary 22.24 Invariance of the phase-space volume ¶
  - depends_on remark 22.25 This is Liouville's theorem ¶
- depends_on proposition 22.27 Properties of the Lagrange bracket ¶
  - depends_on remark 22.28 Where the content of the fundamental brackets lies ¶
- depends_on proposition 22.30 Properties of the Poisson bracket ¶
  - depends_on remark 22.31 The bracket is a Lie algebra structure ¶
    - depends_on remark 22.39 Poisson's theorem does not manufacture new constants ¶
- depends_on remark 22.33 What the duality is good for ¶
- depends_on remark 22.28 Where the content of the fundamental brackets lies ¶ ↺
- depends_on theorem 22.22 Poincaré ¶
  - depends_on proposition 32.40 A Hamiltonian section preserves area ¶
  - depends_on remark 22.23 What the invariant does not depend on ¶
depends_on proposition 5.139 Dimension of the symplectic group ¶
- depends_on proposition 24.5 Properties of the symplectic group ¶ ↺
- depends_on remark 22.21 The count ¶
depends_on proposition 5.136 $\Sp(2n,\R)$ is a subgroup of $\GL(2n,\R)$ ¶
depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
- depends_on corollary 22.24 Invariance of the phase-space volume ¶ ↺
- depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
- depends_on proposition 24.5 Properties of the symplectic group ¶ ↺

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`	→	Non-degenerate form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5906
`depends_on`	→	eq:lin-bilinear-expansion	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5906
`depends_on`	→	Normal form of a non-degenerate antisymmetric form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5906
`depends_on`	←	Symplectic matrix	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:1357
`depends_on`	←	Symplectic group	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:155
`depends_on`	←	The symplectic condition	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:597
`depends_on`	←	Dimension of the symplectic group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6107
`depends_on`	←	$\Sp(2n,\R)$ is a subgroup of $\GL(2n,\R)$	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5915
`depends_on`	←	A symplectic transformation has determinant $+1$	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6036