proposition 23.54 The geometrical amplitude diverges

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:1829 · p. 844

Rests on

depends_on definition 23.52 Lagrangian family; caustic ¶
- depends_on definition 16.62 Field of extremals; slope function ¶
  - depends_on definition 6.17 Simply connected space ¶
    - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 6.15 Path-connected space ¶
      - depends_on definition 6.6 Continuous map ¶ ↺
  - depends_on theorem 16.22 Euler–Lagrange ¶
    - depends_on equation 16.15 eq:calcvar-first-variation ¶
    - depends_on lemma 16.20 Mixed form ¶
      - depends_on corollary 7.44 Substitution and integration by parts ¶
      - depends_on lemma 16.19 du Bois-Reymond ¶
      - proves proof ch:14-calculus-of-variations@proof-10 ¶
    - depends_on proposition 16.16 Stationarity is necessary ¶
      - depends_on definition 16.15 Variation; the first variation ¶
      - depends_on definition 16.13 Weak and strong extrema ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - proves proof ch:14-calculus-of-variations@proof-6 ¶
    - proves proof ch:14-calculus-of-variations@proof-11 ¶
- depends_on definition 23.5 Complete integral ¶
  - depends_on equation 23.2 eq:hj-equation ¶
depends_on equation 23.66 eq:hj-transport ¶
depends_on theorem 7.133 Gauss ¶
- depends_on definition 7.127 Simple regions ¶
  - 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 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶ ↺
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on remark 7.128 What the derivations below take as given ¶
  - depends_on definition 7.125 Multiple integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶ ↺
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
      - proves proof ch:05-real-analysis@proof-10 ¶
    - proves proof ch:05-real-analysis@proof-23 ¶
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
- depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
  - depends_on definition 7.41 Antiderivative ¶
    - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
      - depends_on theorem 7.35 Mean value theorem ¶
      - proves proof ch:05-real-analysis@proof-20 ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
  - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
    - depends_on theorem 7.24 Extreme value theorem ¶ ↺
    - proves proof ch:05-real-analysis@proof-24 ¶
  - proves proof ch:05-real-analysis@proof-25 ¶
- proves proof ch:05-real-analysis@proof-79 ¶
proves proof ch:06-hamilton-jacobi@proof-17 ¶

Supports

depends_on proposition 23.55 The fold, and the quarter-power law ¶
- depends_on proposition 23.56 The phase jump at a fold ¶
- depends_on remark 23.58 The caustics that are not turning surfaces ¶
depends_on remark 23.59 What is observed ¶

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`	→	Lagrangian family; caustic	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1840
`depends_on`	→	eq:hj-transport	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1840
`depends_on`	→	Gauss	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1840
`depends_on`	←	The fold, and the quarter-power law	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1891
`depends_on`	←	What is observed	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:2096
`proves`	←	ch:06-hamilton-jacobi@proof-17	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1843