definition 23.13 Additive separability

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:405 · p. 827

ground object -- no derivation owed

Rests on

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 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 example 23.26 The Kepler problem, and the Bohr formula ¶
depends_on proposition 23.14 Cyclic coordinates ¶
- depends_on example 23.25 Central force in plane polar coordinates ¶
- depends_on example 23.22 Free particle ¶
  - depends_on remark 23.42 Phase speed and particle speed ¶
- depends_on example 23.27 Charged particle in a uniform magnetic field ¶ ↺
- depends_on example 23.23 Uniform gravity, and the parabola ¶
depends_on proposition 23.20 Separability implies integrability ¶
- depends_on remark 23.21 The tensorial statement, and what Part II owes it ¶
- depends_on remark 23.32 What that proof imports ¶ ↺
depends_on theorem 23.17 Stäckel's conditions are sufficient ¶
- depends_on remark 23.18 The converse, and what this treatise does not prove ¶
- depends_on remark 23.19 The same conditions govern the Helmholtz equation ¶

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`	→	Complete integral	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:413
`depends_on`	→	eq:hj-characteristic-equation	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:413
`depends_on`	←	Action variables	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:914
`depends_on`	←	The Kepler problem, and the Bohr formula	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:856
`depends_on`	←	Cyclic coordinates	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:431
`depends_on`	←	Separability implies integrability	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:637
`depends_on`	←	Stäckel's conditions are sufficient	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:513