theorem 32.51 Kolmogorov–Arnold–Moser, quoted

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1297 · p. 1096

Rests on

depends_on definition 32.50 Diophantine frequency vector ¶
- depends_on proposition 32.49 The small-denominator obstruction ¶
  - depends_on equation 22.18 eq:ham-generating ¶
  - depends_on equation 23.52 eq:hj-angle-linear ¶
  - proves proof ch:15-nonlinear-dynamics-chaos@proof-17 ¶
depends_on theorem 23.31 Liouville–Arnold ¶
- 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 equation 16.86 eq:calcvar-hamilton-action ¶
    - depends_on definition 23.13 Additive separability ¶
      - depends_on definition 23.5 Complete integral ¶
      - depends_on equation 23.15 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 21.45 Action ¶
      - depends_on theorem 22.6 Hamilton's equations ¶
    - 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 equation 13.239 eq:mfd-dual-basis ¶
    - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on definition 13.105 Closed form ¶
      - depends_on definition 13.98 $k$-form ¶ ↺
      - depends_on definition 24.2 Symplectic vector space ¶
    - depends_on theorem 24.12 Darboux ¶
      - depends_on definition 24.8 Symplectic manifold ¶ ↺
      - depends_on theorem A.68 Darboux ¶
      - 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 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 theorem 7.34 Rolle ¶
      - 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 ¶
      - 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 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 ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`cites`	→	Proof of a Theorem of A. N. Kolmogorov on the Preservation of Conditionally Periodic Motions under a Small Perturbation of the Hamiltonian	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1309
`cites`	→	On Conservation of Conditionally Periodic Motions for a Small Change in Hamilton's Function	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1309
`cites`	→	On Invariant Curves of Area-Preserving Mappings of an Annulus	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1309
`depends_on`	→	Diophantine frequency vector	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1310
`depends_on`	→	Liouville–Arnold	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1310
`depends_on`	←	Regular and chaotic motion coexist	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1525
`depends_on`	←	The solar system is chaotic	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1604
`depends_on`	←	Integrability is exceptional	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1036