definition 16.96 Action; Hamilton's principle

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:3073 · p. 665

ground object -- no derivation owed

Rests on

depends_on definition 21.31 Lagrangian ¶
depends_on equation 16.11 eq:calcvar-basic-functional ¶

Supports

depends_on definition 16.99 Abbreviated action ¶
- 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 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 ¶
depends_on definition 21.45 Action ¶
- depends_on definition 22.15 Canonical transformation ¶
  - depends_on proposition 22.17 Transformation conditions ¶
    - depends_on remark 22.21 The count ¶
  - depends_on proposition 22.18 The mixed generating functions ¶
    - depends_on definition 23.2 Hamilton's principal function and the Hamilton–Jacobi equation ¶
      - depends_on remark 23.12 Two functions, two roles ¶
    - depends_on proposition A.556 The Jacobian of the actions is the lattice matrix ¶
      - depends_on example A.557 The check that fixes the convention ¶
    - depends_on remark 22.19 Which form exists ¶
    - depends_on theorem 23.30 Motion in action–angle variables ¶ ↺
  - depends_on remark 22.16 Why the class is larger than the point transformations ¶
depends_on definition 28.3 Simple harmonic oscillator ¶
depends_on proposition 23.7 The principal function is the action ¶
- depends_on remark 23.8 The action as a function, not a functional ¶

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
`cites` (history)	→	On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function	derived	parts/02-mathematical-methods/14-calculus-of-variations.tex:3086
`depends_on`	→	Lagrangian	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:3087
`depends_on`	→	eq:calcvar-basic-functional	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:3087
`depends_on`	←	Abbreviated action	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:3155
`depends_on`	←	Action	declared	parts/03-classical-mechanics/04-lagrangian-mechanics.tex:1238
`depends_on`	←	Simple harmonic oscillator	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:101
`depends_on`	←	The principal function is the action	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:262