theorem 24.16 The Hamiltonian flow preserves the symplectic form

open in the book · parts/03-classical-mechanics/07-symplectic-geometry.tex:492 · p. 854

Rests on

depends_on definition 24.14 Hamiltonian vector field ¶
- 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 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 ¶

Supports

depends_on proposition A.570 Existence and uniqueness of the reduced form ¶
- depends_on proposition A.571 Invariant Hamiltonians descend with their flows ¶
depends_on remark 24.35 What non-squeezing does and does not say about nature ¶
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 ¶

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`	→	Hamiltonian vector field	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:499
`depends_on`	→	Symplectic manifold	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:499
`depends_on`	→	Cartan's magic formula	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:499
`depends_on`	←	Existence and uniqueness of the reduced form	declared	appendices/A-long-proofs.tex:27353
`depends_on`	←	What non-squeezing does and does not say about nature	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1225
`depends_on`	←	Liouville	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:676
`proves`	←	ch:07-symplectic-geometry@proof-5	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:503