proposition 5.119 Normal form of a non-degenerate antisymmetric form

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:5087 · p. 156

Rests on

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 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
depends_on definition 5.113 Non-degenerate form ¶
- depends_on definition 5.110 Bilinear map ¶ ↺
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 5.135 The symplectic group ¶
- 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 ¶ ↺
depends_on definition 24.2 Symplectic vector space ¶
- depends_on definition 24.8 Symplectic manifold ¶
  - depends_on definition 24.9 The canonical form on a cotangent bundle ¶
    - depends_on proposition 24.10 The canonical form is symplectic and intrinsic ¶
      - depends_on remark 24.11 The SI dimension of every object in this chapter ¶
  - depends_on definition 24.14 Hamiltonian vector field ¶
    - depends_on definition 24.43 Momentum map ¶
      - depends_on definition A.561 Coadjoint action and equivariance ¶
      - depends_on example 24.44 Linear and angular momentum ¶
      - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶
      - depends_on theorem 24.45 Noether, symplectic form ¶
    - depends_on definition 24.50 Prequantum operator ¶
      - depends_on proposition 24.51 Prequantization is a Lie-algebra homomorphism ¶
      - depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶
    - depends_on proposition 25.4 Leibniz rule and derivations ¶
      - depends_on lemma A.616 The four identities ¶
    - depends_on proposition 24.15 Poisson bracket from the symplectic form ¶
      - depends_on definition 24.37 Casimir function ¶
      - depends_on theorem 24.45 Noether, symplectic form ¶ ↺
    - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
      - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
      - depends_on remark 24.35 What non-squeezing does and does not say about nature ¶ ↺
      - depends_on theorem 24.21 Liouville ¶
  - depends_on definition 24.20 Liouville volume ¶
    - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
      - depends_on proposition A.619 The Dirac bracket is a projected Hamiltonian flow ¶
      - depends_on remark 26.26 What the factor is doing ¶
    - depends_on theorem 24.21 Liouville ¶ ↺
    - depends_on theorem 24.24 Poincaré recurrence ¶
      - depends_on remark 24.26 Recurrence and irreversibility ¶
      - depends_on remark 24.25 What the recurrence proof assumes about volume ¶
  - depends_on definition 24.53 Real polarization and polarized sections ¶
    - depends_on proposition 24.54 The vertical polarization gives wave mechanics ¶ ↺
  - depends_on definition 24.49 Prequantum datum ¶
    - depends_on definition 24.53 Real polarization and polarized sections ¶ ↺
    - depends_on definition 24.50 Prequantum operator ¶ ↺
    - depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
      - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
  - depends_on definition 24.17 Symplectomorphism ¶
    - depends_on definition 25.6 Infinitesimal canonical transformation ¶
      - depends_on definition 25.11 Generators of the Galilei transformations ¶
    - depends_on definition 24.28 Ball and cylinder ¶
      - depends_on definition 24.33 Symplectic capacity ¶
      - 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 definition 24.43 Momentum map ¶ ↺
    - depends_on theorem 24.18 Symplectic form of the transformation condition ¶
      - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶ ↺
  - depends_on lemma 24.46 The level set of the momentum map is the symplectic orthogonal of the orbit ¶ ↺
  - depends_on theorem 24.12 Darboux ¶
    - depends_on definition 24.20 Liouville volume ¶ ↺
    - depends_on remark 24.13 What Darboux's theorem forbids ¶
  - depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶ ↺
  - depends_on theorem 24.38 Symplectic foliation, quoted ¶
  - depends_on theorem 24.47 Marsden–Weinstein reduction ¶
    - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
      - depends_on example A.573 The abelian case, and eliminating a cyclic coordinate ¶
- depends_on lemma A.564 Dimension and double orthogonal ¶
  - depends_on proposition A.565 Freeness makes every value regular ¶
    - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on proposition A.570 Existence and uniqueness of the reduced form ¶ ↺
  - depends_on theorem A.568 The radical is the isotropy orbit ¶ ↺
- depends_on proposition 24.3 Even dimension and the canonical basis ¶
  - depends_on definition 24.4 Symplectic group ¶ ↺
  - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶ ↺
depends_on proposition 26.19 Second-class constraints come in pairs ¶
- depends_on definition 26.20 Dirac bracket ¶
  - depends_on definition 26.23 Gauge conditions ¶
    - depends_on proposition 26.24 The Faddeev–Popov determinant ¶
      - depends_on remark 26.26 What the factor is doing ¶ ↺
  - depends_on example 26.22 Holonomic constraints ¶
  - depends_on theorem A.617 Jacobi identity for the Dirac bracket ¶
  - depends_on theorem 26.21 Properties of the Dirac bracket ¶
    - depends_on postulate 26.47 Dirac quantization ¶
      - depends_on definition 26.50 The BRST charge ¶
      - depends_on proposition 26.48 Consistency of the state conditions ¶
- depends_on proposition 26.24 The Faddeev–Popov determinant ¶ ↺
- depends_on proposition 26.25 The Liouville measure of a second-class surface ¶ ↺
- depends_on theorem 26.17 Counting the physical degrees of freedom ¶
  - depends_on corollary A.642 The constraints are first class ¶
  - depends_on definition 26.23 Gauge conditions ¶ ↺
  - depends_on proposition 26.32 Constraint structure of the free Maxwell field ¶
    - depends_on proposition 26.34 Proca constraints are second class ¶
      - depends_on proposition 113.3 Degrees of freedom before and after ¶
    - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶
  - depends_on proposition 26.34 Proca constraints are second class ¶ ↺
  - depends_on proposition 26.29 Reparametrization invariance and the mass-shell constraint ¶
  - depends_on theorem 26.42 The Hamiltonian structure of general relativity ¶
    - depends_on notation A.636 Densities and smearing ¶
      - depends_on lemma A.637 Closed form of $H_{\parallel}$, and what it generates ¶
    - depends_on proposition 26.43 The hypersurface-deformation algebra ¶
- depends_on theorem 26.21 Properties of the Dirac bracket ¶ ↺
depends_on proposition 24.3 Even dimension and the canonical basis ¶ ↺
depends_on theorem A.68 Darboux ¶
- depends_on corollary A.69 No local invariants ¶
  - depends_on remark 24.13 What Darboux's theorem forbids ¶ ↺
- depends_on example A.80 The phase space of one particle in space ¶
- depends_on theorem 24.12 Darboux ¶ ↺
depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Bilinear map	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5106
`depends_on`	→	Non-degenerate form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5106
`depends_on`	→	Symmetric and antisymmetric forms	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5106
`depends_on`	→	eq:lin-bilinear-congruence	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5106
`depends_on`	←	The symplectic group	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5906
`depends_on`	←	Symplectic vector space	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:95
`depends_on`	←	Second-class constraints come in pairs	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:563
`depends_on`	←	Even dimension and the canonical basis	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:112
`depends_on`	←	Darboux	declared	appendices/A-long-proofs.tex:4436
`depends_on`	←	A symplectic transformation has determinant $+1$	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6036
`proves`	←	ch:03-linear-algebra-representations@proof-54	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5109