theorem 24.24 Poincaré recurrence

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:824 · p. 858

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 theorem 24.21 Liouville ¶
- depends_on definition 24.20 Liouville volume ¶ ↺
- 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 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 equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - 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 13.90 Mixed tensor ¶
      - 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 ¶
proves proof ch:07-symplectic-geometry@proof-10 ¶

Supports

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:829
`depends_on`	→	Liouville	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:829
`depends_on`	←	Recurrence and irreversibility	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:897
`depends_on`	←	What the recurrence proof assumes about volume	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:884
`proves`	←	ch:07-symplectic-geometry@proof-10	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:832