remark 22.19 Which form exists

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:549 · p. 811

remark -- no derivation owed by its kind

Rests on

depends_on corollary A.292 Inverse function theorem ¶
- depends_on proposition 7.104 Chain rule in several variables ¶
  - depends_on definition 7.13 Composite function ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on definition 7.99 Differentiability at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on equation 6.12 eq:top-euclidean-metric ¶
  - depends_on definition 7.102 Differential of a function ¶
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - proves proof ch:05-real-analysis@proof-63 ¶
- depends_on theorem A.286 Implicit function theorem ¶
  - depends_on definition 7.99 Differentiability at a point ¶ ↺
  - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
  - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶ ↺
    - depends_on definition 7.99 Differentiability at a point ¶ ↺
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-62 ¶
  - proves proof app:A-long-proofs@proof-185 ¶
- proves proof app:A-long-proofs@proof-186 ¶
depends_on proposition 22.18 The mixed generating functions ¶
- depends_on definition 22.15 Canonical transformation ¶
  - depends_on definition 22.14 Phase-space coordinate transformation ¶
    - depends_on definition 22.2 Hamiltonian ¶
      - depends_on definition 21.34 Generalized momentum ¶
      - depends_on definition 21.31 Lagrangian ¶
  - depends_on definition 21.45 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 21.37 eq:lag-lagrangian ¶
  - depends_on theorem 22.6 Hamilton's equations ¶
    - depends_on definition 22.2 Hamiltonian ¶ ↺
    - 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 ¶
- depends_on equation 22.18 eq:ham-generating ¶
- proves proof ch:05-hamiltonian-mechanics@proof-6 ¶

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type	direction	node	provenance	where
`depends_on`	→	Inverse function theorem	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:564
`depends_on`	→	The mixed generating functions	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:564