proposition 23.53 A caustic point is a conjugate point

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:1793 · p. 844

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depends_on definition 16.53 Jacobi accessory equation; conjugate point ¶
- depends_on equation 16.20 eq:calcvar-euler-lagrange ¶
- depends_on equation 16.45 eq:calcvar-second-variation ¶
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 theorem 16.55 Jacobi's necessary condition ¶
- depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
  - depends_on corollary 16.23 du Bois-Reymond form ¶
    - depends_on equation 16.20 eq:calcvar-euler-lagrange ¶ ↺
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - proves proof ch:14-calculus-of-variations@proof-12 ¶
  - depends_on theorem 16.32 Transversality ¶
    - depends_on equation 16.20 eq:calcvar-euler-lagrange ¶ ↺
    - depends_on theorem 16.31 Natural boundary condition ¶
      - depends_on equation 16.15 eq:calcvar-first-variation ¶ ↺
      - depends_on lemma 16.18 Fundamental lemma ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶ ↺
      - proves proof ch:14-calculus-of-variations@proof-16 ¶
    - proves proof ch:14-calculus-of-variations@proof-17 ¶
  - proves proof ch:14-calculus-of-variations@proof-18 ¶
- depends_on definition 16.53 Jacobi accessory equation; conjugate point ¶ ↺
- depends_on theorem 16.51 Legendre's necessary condition ¶
  - depends_on equation 16.45 eq:calcvar-second-variation ¶ ↺
  - depends_on proposition 16.16 Stationarity is necessary ¶ ↺
  - proves proof ch:14-calculus-of-variations@proof-26 ¶
- proves proof ch:14-calculus-of-variations@proof-27 ¶
proves proof ch:06-hamilton-jacobi@proof-16 ¶

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type	direction	node	provenance	where
`depends_on`	→	Jacobi accessory equation; conjugate point	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1801
`depends_on`	→	Lagrangian family; caustic	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1801
`depends_on`	→	Jacobi's necessary condition	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1801
`proves`	←	ch:06-hamilton-jacobi@proof-16	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1804