definition 24.20 Liouville volume

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:639 · p. 856

ground object -- no derivation owed

Rests on

depends_on definition 13.102 Wedge product ¶
- depends_on definition 13.98 $k$-form ¶
  - depends_on definition 13.83 Covector ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.83 Covector ¶ ↺
    - depends_on definition 13.81 Vector on a manifold ¶ ↺
- depends_on equation 13.239 eq:mfd-dual-basis ¶
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 13.102 Wedge product ¶ ↺
  - 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 definition 5.110 Bilinear map ¶
      - depends_on definition 5.37 Linear transformation ¶
    - 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 ¶
depends_on theorem 24.12 Darboux ¶
- depends_on definition 24.8 Symplectic manifold ¶ ↺
- depends_on theorem A.68 Darboux ¶
  - depends_on definition A.67 Symplectic manifold ¶
    - depends_on definition 13.105 Closed form ¶ ↺
    - depends_on definition 13.98 $k$-form ¶ ↺
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on equation 5.144 eq:lin-symplectic-normal-form ¶
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
  - proves proof app:A-long-proofs@proof-57 ¶
- proves proof ch:07-symplectic-geometry@prooflink-1 ¶

Supports

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 24.57 The half-integer, and the honest status of the construction ¶
  - 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 phenomenon 32.58 Regular and chaotic motion coexist ¶
    - depends_on phenomenon 32.59 The solar system is chaotic ¶
    - depends_on remark 23.33 Integrability is exceptional ¶ ↺
  - 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 ¶ ↺

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`	→	Wedge product	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:653
`depends_on`	→	Symplectic manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:653
`depends_on`	→	Darboux	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:653
`depends_on`	←	The Liouville measure of a second-class surface	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:876
`depends_on`	←	Liouville	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:676
`depends_on`	←	Poincaré recurrence	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:829