remark 22.4 When the Legendre map is singular

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:109 · p. 806

remark -- no derivation owed by its kind

Rests on

depends_on definition 21.66 Kinetic metric ¶
depends_on proposition 22.3 Invertibility of the Legendre map ¶
- 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 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 definition 7.102 Differential of a function ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - 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 7.99 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 ¶
  - proves proof app:A-long-proofs@proof-186 ¶
- depends_on equation 21.43 eq:lag-generalized-momentum ¶
- proves proof ch:05-hamiltonian-mechanics@proof-1 ¶

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type	direction	node	provenance	where
`depends_on`	→	Kinetic metric	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:125
`depends_on`	→	Invertibility of the Legendre map	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:125