phenomenon 32.58 Regular and chaotic motion coexist

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1508 · p. 1098

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

Rests on

depends_on equation 32.27 eq:chaos-henon-heiles ¶
depends_on theorem 32.51 Kolmogorov–Arnold–Moser, quoted ¶
- 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 23.13 Additive separability ¶
    - depends_on proposition 22.18 The mixed generating functions ¶
      - depends_on definition 22.15 Canonical transformation ¶
      - depends_on equation 22.18 eq:ham-generating ¶ ↺
      - proves proof ch:05-hamiltonian-mechanics@proof-6 ¶
    - 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 ¶
    - 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 24.8 Symplectic manifold ¶
      - depends_on theorem 24.12 Darboux ¶
    - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - 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 proposition 13.128 Cartan's magic formula ¶
      - proves proof ch:07-symplectic-geometry@proof-5 ¶
    - proves proof ch:07-symplectic-geometry@proof-8 ¶
  - proves proof ch:06-hamilton-jacobi@prooflink-1 ¶
proves proof ch:15-nonlinear-dynamics-chaos@proof-19 ¶

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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:1519
`cites`	→	The Applicability of the Third Integral of Motion: Some Numerical Experiments	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1516
`cites`	→	The Applicability of the Third Integral of Motion: Some Numerical Experiments	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1524
`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:1519
`cites`	→	On Invariant Curves of Area-Preserving Mappings of an Annulus	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1519
`depends_on`	→	eq:chaos-henon-heiles	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1525
`depends_on`	→	Kolmogorov–Arnold–Moser, quoted	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1525
`proves`	←	ch:15-nonlinear-dynamics-chaos@proof-19	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1528