proposition 22.3 Invertibility of the Legendre map

open in the book · parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:82 · p. 805

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 equation 21.43 eq:lag-generalized-momentum ¶
proves proof ch:05-hamiltonian-mechanics@proof-1 ¶

Supports

depends_on definition 26.3 Singular Lagrangian ¶
- depends_on proposition 26.5 Primary constraints ¶
  - depends_on proposition 26.6 The canonical Hamiltonian is defined on the constraint surface ¶
    - depends_on definition 26.8 Total Hamiltonian ¶
      - depends_on definition 26.16 Extended Hamiltonian ¶
      - depends_on theorem 26.21 Properties of the Dirac bracket ¶
      - depends_on theorem 26.14 First-class constraints generate gauge transformations ¶
  - depends_on proposition 26.32 Constraint structure of the free Maxwell field ¶
    - depends_on proposition 26.34 Proca constraints are second class ¶
      - depends_on proposition 113.3 Degrees of freedom before and after ¶
    - depends_on proposition 26.37 Constraint structure of Yang–Mills theory ¶
  - depends_on proposition 26.29 Reparametrization invariance and the mass-shell constraint ¶
depends_on remark 22.4 When the Legendre map is singular ¶

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`	→	Inverse function theorem	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:92
`depends_on`	→	eq:lag-generalized-momentum	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:92
`depends_on`	←	Singular Lagrangian	declared	parts/03-classical-mechanics/09-constrained-hamiltonian-dynamics.tex:113
`depends_on`	←	When the Legendre map is singular	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:125
`proves`	←	ch:05-hamiltonian-mechanics@proof-1	declared	parts/03-classical-mechanics/05-hamiltonian-mechanics.tex:95