proposition 23.7 The principal function is the action

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:252 · p. 826

Rests on

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 22.1 eq:ham-hamiltonian ¶
depends_on theorem 23.6 Jacobi ¶
- 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 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - 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 ¶
    - 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 7.97 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 ¶
- 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 ¶
proves proof ch:06-hamilton-jacobi@proof-2 ¶

Supports

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`	→	Action; Hamilton's principle	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:262
`depends_on`	→	eq:ham-hamiltonian	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:262
`depends_on`	→	Jacobi	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:262
`depends_on`	←	The action as a function, not a functional	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:296
`proves`	←	ch:06-hamilton-jacobi@proof-2	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:265