theorem 23.30 Motion in action–angle variables

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:927 · p. 834

Rests on

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 21.34 Generalized momentum ¶
      - depends_on definition 21.31 Lagrangian ¶ ↺
  - 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 ¶

Supports

depends_on example 23.24 Harmonic oscillator ¶
depends_on theorem 23.34 Adiabatic invariance of the action ¶
- depends_on example 23.35 The slowly stiffened oscillator ¶
- depends_on example 23.27 Charged particle in a uniform magnetic field ¶
- depends_on remark 23.36 How good the invariance really is ¶
- depends_on remark 23.37 Why this matters observationally ¶
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 ¶

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`	→	Action variables	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:940
`depends_on`	→	The mixed generating functions	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:940
`depends_on`	→	Hamilton's equations	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:940
`depends_on`	←	Harmonic oscillator	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:774
`depends_on`	←	Adiabatic invariance of the action	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1059
`depends_on`	←	Liouville–Arnold	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:994
`proves`	←	ch:06-hamilton-jacobi@proof-8	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:944