theorem 16.55 Jacobi's necessary condition

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:1619 · p. 649

Rests on

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 corollary 7.36 cor:ana-mvt-consequences ¶
      - 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: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 definition 16.15 Variation; the first variation ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
      - proves proof ch:14-calculus-of-variations@proof-8 ¶
    - 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 proposition 16.16 Stationarity is necessary ¶
      - proves proof ch:14-calculus-of-variations@proof-11 ¶
    - 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 equation 16.20 eq:calcvar-euler-lagrange ¶ ↺
- depends_on equation 16.45 eq:calcvar-second-variation ¶
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 ¶

Supports

depends_on proposition 23.53 A caustic point is a conjugate point ¶
depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Weierstrass–Erdmann corner conditions	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1624
`depends_on`	→	Jacobi accessory equation; conjugate point	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1624
`depends_on`	→	Legendre's necessary condition	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1624
`depends_on`	←	A caustic point is a conjugate point	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:1801
`depends_on`	←	Other end conditions, and what buckling is variationally	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2013
`proves`	←	ch:14-calculus-of-variations@proof-27	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1627