theorem 23.31 Liouville–Arnold

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:982 · p. 834

Rests on

depends_on equation 22.41 eq:ham-poisson-bracket ¶
depends_on theorem 23.30 Motion in action–angle variables ¶
- depends_on definition 23.28 Action variables ¶
  - depends_on definition 16.99 Abbreviated action ¶
    - depends_on definition 16.96 Action; Hamilton's principle ¶
      - depends_on definition 21.31 Lagrangian ¶
      - depends_on equation 16.11 eq:calcvar-basic-functional ¶
    - depends_on equation 16.86 eq:calcvar-hamilton-action ¶
  - depends_on definition 23.13 Additive separability ¶
    - depends_on definition 23.5 Complete integral ¶
      - depends_on equation 23.2 eq:hj-equation ¶
    - depends_on equation 23.15 eq:hj-characteristic-equation ¶
- depends_on proposition 22.18 The mixed generating functions ¶
  - depends_on definition 22.15 Canonical transformation ¶
    - depends_on definition 22.14 Phase-space coordinate transformation ¶
      - depends_on definition 22.2 Hamiltonian ¶
    - depends_on definition 21.45 Action ¶
      - depends_on definition 16.96 Action; Hamilton's principle ¶ ↺
      - depends_on equation 21.37 eq:lag-lagrangian ¶
    - depends_on theorem 22.6 Hamilton's equations ¶
      - depends_on definition 22.2 Hamiltonian ¶ ↺
      - depends_on equation 21.39 eq:lag-el-conservative ¶
      - depends_on equation 21.43 eq:lag-generalized-momentum ¶
      - proves proof ch:05-hamiltonian-mechanics@proof-2 ¶
  - depends_on equation 22.18 eq:ham-generating ¶
  - proves proof ch:05-hamiltonian-mechanics@proof-6 ¶
- depends_on theorem 22.6 Hamilton's equations ¶ ↺
- proves proof ch:06-hamilton-jacobi@proof-8 ¶
depends_on theorem 24.21 Liouville ¶
- 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 13.84 Tensor ¶
    - 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 13.98 $k$-form ¶ ↺
    - depends_on definition 24.2 Symplectic vector space ¶
      - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
  - depends_on 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 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 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:06-hamilton-jacobi@prooflink-1 ¶

Supports

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 ¶

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`	→	eq:ham-poisson-bracket	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:994
`depends_on`	→	Motion in action–angle variables	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:994
`depends_on`	→	Liouville	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:994
`depends_on`	←	The Bohr–Sommerfeld condition is a triviality condition on the prequantum holonomy	declared	parts/03-classical-mechanics/07-symplectic-geometry.tex:1900
`depends_on`	←	Integrability is exceptional	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1036
`depends_on`	←	What that proof imports	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1021
`depends_on`	←	Kolmogorov–Arnold–Moser, quoted	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1310
`depends_on`	←	Poincaré's non-integrability, quoted	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1201
`proves`	←	ch:06-hamilton-jacobi@prooflink-1	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:996