theorem A.68 Darboux

open in the book · appendices/A-long-proofs.tex:4425 · p. 2826

Rests on

depends_on definition A.67 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 13.83 Covector ¶
      - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.102 Wedge product ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on equation 13.239 eq:mfd-dual-basis ¶
  - depends_on definition 13.98 $k$-form ¶ ↺
- depends_on definition 13.98 $k$-form ¶ ↺
- depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
depends_on equation 5.144 eq:lin-symplectic-normal-form ¶
depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric 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 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 ¶
proves proof app:A-long-proofs@proof-57 ¶

Supports

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 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 example 24.29 A volume-preserving squeeze ¶
    - depends_on proposition 32.40 A Hamiltonian section preserves area ¶
    - depends_on remark 22.25 This is Liouville's theorem ¶
    - depends_on remark 24.22 Liouville's theorem is the foundation of statistical mechanics ¶
    - depends_on theorem 23.31 Liouville–Arnold ¶
      - depends_on proposition 24.56 The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy ¶
      - depends_on remark 23.33 Integrability is exceptional ¶
      - depends_on remark 23.32 What that proof imports ¶
      - depends_on theorem 32.51 Kolmogorov–Arnold–Moser, quoted ¶
      - depends_on theorem 32.47 Poincaré's non-integrability, quoted ¶
    - depends_on theorem 25.57 The Moyal equation, and its classical limit ¶
      - depends_on proposition 25.59 Homodyne tomography inverts a Radon transform ¶
    - 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 theorem 24.24 Poincaré recurrence ¶ ↺
- depends_on remark 24.13 What Darboux's theorem forbids ¶ ↺

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`	→	Symplectic manifold	declared	appendices/A-long-proofs.tex:4436
`depends_on`	→	eq:lin-symplectic-normal-form	declared	appendices/A-long-proofs.tex:4436
`depends_on`	→	Normal form of a non-degenerate antisymmetric form	declared	appendices/A-long-proofs.tex:4436
`depends_on`	←	No local invariants	declared	appendices/A-long-proofs.tex:4446
`depends_on`	←	The phase space of one particle in space	declared	appendices/A-long-proofs.tex:5198
`depends_on`	←	Darboux	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:355
`proves`	←	app:A-long-proofs@proof-57	declared	appendices/A-long-proofs.tex:5007