theorem 23.6 Jacobi

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:161 · p. 824

Rests on

depends_on definition 23.5 Complete integral ¶
- depends_on equation 23.2 eq:hj-equation ¶
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 7.2 Absolute value ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - 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 7.9 Real function ¶ ↺
  - depends_on definition 7.99 Differentiability at a point ¶ ↺
  - 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 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 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - 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 definition 7.11 Sum of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-13 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-19 ¶
  - 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 ¶

Supports

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 23.23 Uniform gravity, and the parabola ¶
depends_on proposition 23.7 The principal function is the action ¶
- depends_on remark 23.8 The action as a function, not a functional ¶

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:173
`depends_on`	→	Implicit function theorem	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:173
`depends_on`	→	Hamilton's equations	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:173
`depends_on`	←	Central force in plane polar coordinates	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:804
`depends_on`	←	Free particle	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:724
`depends_on`	←	Uniform gravity, and the parabola	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:752
`depends_on`	←	The principal function is the action	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:262
`proves`	←	ch:06-hamilton-jacobi@proof-1	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:177