theorem 24.21 Liouville

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:668 · p. 856

Rests on

depends_on definition 24.20 Liouville volume ¶
- 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 13.84 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 5.113 Non-degenerate form ¶
      - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
      - 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 ¶
depends_on proposition 7.105 Clairaut–Schwarz ¶
- depends_on definition 7.20 Continuity at a point ¶
  - depends_on definition 7.16 Limit ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on proposition 7.29 Linearity ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:05-real-analysis@proof-64 ¶
depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
- depends_on definition 24.14 Hamiltonian vector field ¶
  - depends_on definition 24.8 Symplectic manifold ¶ ↺
  - depends_on equation 22.4 eq:ham-pdot ¶
  - depends_on equation 22.3 eq:ham-qdot ¶
- depends_on definition 24.8 Symplectic manifold ¶ ↺
- depends_on proposition 13.128 Cartan's magic formula ¶
  - depends_on definition 13.103 Exterior derivative ¶ ↺
  - depends_on equation 13.246 eq:mfd-leibniz-forms ¶
  - depends_on equation 13.245 eq:mfd-nilpotency ¶
  - depends_on proposition 13.127 Component formulas ¶
    - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.90 Mixed tensor ¶
      - depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
      - depends_on definition 13.88 General coordinate transformation ¶ ↺
    - depends_on equation 13.267 eq:mfd-lie-def ¶
    - proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-29 ¶
- proves proof ch:07-symplectic-geometry@proof-5 ¶
proves proof ch:07-symplectic-geometry@proof-8 ¶

Supports

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 ¶

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`	→	Liouville volume	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:676
`depends_on`	→	Clairaut–Schwarz	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:676
`depends_on`	→	The Hamiltonian flow preserves the symplectic form	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:676
`depends_on`	←	A volume-preserving squeeze	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:988
`depends_on`	←	A Hamiltonian section preserves area	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1042
`depends_on`	←	This is Liouville's theorem	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:843
`depends_on`	←	Liouville's theorem is the foundation of statistical mechanics	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:733
`depends_on`	←	Liouville–Arnold	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:994
`depends_on`	←	The Moyal equation, and its classical limit	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:2063
`depends_on`	←	Poincaré recurrence	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:829
`proves`	←	ch:07-symplectic-geometry@proof-8	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:680