remark 23.42 Phase speed and particle speed

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:1406 · p. 839

remark -- no derivation owed by its kind

Rests on

depends_on equation 23.14 eq:hj-separate-time ¶
depends_on example 23.22 Free particle ¶
- depends_on proposition 23.14 Cyclic coordinates ¶
  - 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 definition 21.35 Cyclic coordinate ¶
  - depends_on theorem 21.36 Conservation of the conjugate momentum ¶
    - depends_on definition 21.35 Cyclic coordinate ¶ ↺
    - depends_on equation 21.39 eq:lag-el-conservative ¶
    - depends_on equation 21.43 eq:lag-generalized-momentum ¶
    - proves proof ch:04-lagrangian-mechanics@proof-8 ¶
  - proves proof ch:06-hamilton-jacobi@proof-5 ¶
- depends_on theorem 23.6 Jacobi ¶
  - depends_on definition 23.5 Complete integral ¶ ↺
  - depends_on theorem A.286 Implicit function theorem ¶
    - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on equation 6.12 eq:top-euclidean-metric ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on definition 7.102 Differential of a function ¶
      - proves proof ch:05-real-analysis@proof-63 ¶
    - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-62 ¶
    - proves proof app:A-long-proofs@proof-185 ¶
  - depends_on theorem 22.6 Hamilton's equations ¶
    - depends_on definition 22.2 Hamiltonian ¶
      - depends_on definition 21.34 Generalized momentum ¶
      - depends_on definition 21.31 Lagrangian ¶
    - 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 ¶
  - proves proof ch:06-hamilton-jacobi@proof-1 ¶

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type	direction	node	provenance	where
`cites` (history)	→	Recherches sur la théorie des Quanta	derived	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1416
`depends_on`	→	eq:hj-separate-time	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1417
`depends_on`	→	Free particle	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1417