definition 16.99 Abbreviated action

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:3145 · p. 666

ground object -- no derivation owed

Rests on

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 ¶

Supports

depends_on definition 23.28 Action variables ¶
- 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 ¶ ↺
depends_on proposition 23.10 The characteristic function is the abbreviated action ¶
- depends_on remark 23.11 Whose principle, and where it fails ¶

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; Hamilton's principle	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:3155
`depends_on`	→	eq:calcvar-hamilton-action	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:3155
`depends_on`	←	Action variables	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:914
`depends_on`	←	The characteristic function is the abbreviated action	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:354