definition 23.28 Action variables

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:901 · p. 833

ground object -- no derivation owed

Rests on

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 ¶

Supports

depends_on example 23.24 Harmonic oscillator ¶
depends_on proposition 23.38 Bohr–Sommerfeld and Einstein–Brillouin–Keller quantization ¶
- depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
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 23.29 Convention ¶
depends_on remark 29.45 What integrability buys, and where it stops ¶
depends_on theorem 23.30 Motion in action–angle variables ¶
- 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 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 23.34 Adiabatic invariance of the action ¶ ↺

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`	→	Abbreviated action	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:914
`depends_on`	→	Additive separability	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:914
`depends_on`	←	Harmonic oscillator	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:774
`depends_on`	←	Bohr–Sommerfeld and Einstein–Brillouin–Keller quantization	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1186
`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`	←	Convention	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:924
`depends_on`	←	What integrability buys, and where it stops	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1712
`depends_on`	←	Motion in action–angle variables	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:940
`depends_on`	←	Adiabatic invariance of the action	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1059