definition 23.52 Lagrangian family; caustic

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:1774 · p. 843

ground object -- no derivation owed

Rests on

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 6.1 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 proposition 7.31 Chain rule ¶
      - depends_on proposition 7.30 Leibniz rule ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-26 ¶
    - depends_on lemma 16.19 du Bois-Reymond ¶
      - depends_on equation 16.18 eq:calcvar-dbr-hyp ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:14-calculus-of-variations@proof-9 ¶
    - 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 16.11 Admissible class; functional ¶
      - depends_on equation 7.177 eq:ana-functional-derivative ¶
    - depends_on definition 16.13 Weak and strong extrema ¶
      - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - 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 ¶

Supports

depends_on proposition 23.54 The geometrical amplitude diverges ¶
- 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 ¶
depends_on proposition 23.53 A caustic point is a conjugate point ¶

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`	→	Field of extremals; slope function	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1790
`depends_on`	→	Complete integral	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1790
`depends_on`	←	The geometrical amplitude diverges	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1840
`depends_on`	←	A caustic point is a conjugate point	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1801