proposition 16.16 Stationarity is necessary

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:473 · p. 636

Rests on

depends_on definition 16.15 Variation; the first variation ¶
- depends_on definition 16.11 Admissible class; functional ¶
  - 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 theorem 7.40 Continuous functions are integrable ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
    - 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 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 definition 16.11 Admissible class; functional ¶ ↺
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
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 ¶
  - depends_on definition 7.16 Limit ¶ ↺
  - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
  - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
  - proves proof ch:05-real-analysis@proof-6 ¶
- proves proof ch:05-real-analysis@proof-17 ¶
proves proof ch:14-calculus-of-variations@proof-6 ¶

Supports

depends_on theorem 16.22 Euler–Lagrange ¶
- depends_on definition 16.62 Field of extremals; slope function ¶
  - depends_on definition 23.52 Lagrangian family; caustic ¶
    - depends_on proposition 23.54 The geometrical amplitude diverges ¶
      - depends_on proposition 23.55 The fold, and the quarter-power law ¶
      - depends_on remark 23.59 What is observed ¶
    - depends_on proposition 23.53 A caustic point is a conjugate point ¶
  - depends_on remark 23.8 The action as a function, not a functional ¶
  - depends_on theorem 16.63 Hilbert's invariant integral ¶
    - depends_on remark 23.8 The action as a function, not a functional ¶ ↺
    - depends_on theorem 16.64 Weierstrass' sufficient condition ¶
  - depends_on theorem 16.64 Weierstrass' sufficient condition ¶ ↺
- depends_on phenomenon 30.52 Euler buckling ¶
  - depends_on example 30.53 A steel rod buckles at a twentieth of its crushing load ¶
  - depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶
  - depends_on remark 30.55 Beyond the critical load: the elastica ¶
  - depends_on remark 30.68 Shells carry load in a different way ¶
- depends_on proposition 29.39 Steady precession ¶
- depends_on theorem 16.35 System of Euler–Lagrange equations ¶
  - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶
  - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
    - depends_on proposition 21.56 The two prescriptions and their difference ¶
- depends_on theorem 16.31 Natural boundary condition ¶
  - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
  - depends_on theorem A.691 Kirchhoff's edge conditions and the corner force ¶
    - depends_on remark A.692 The corner force is real, and you can feel it ¶
  - depends_on theorem 16.32 Transversality ¶
    - depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
      - depends_on theorem 16.55 Jacobi's necessary condition ¶
depends_on theorem 16.51 Legendre's necessary condition ¶
- depends_on corollary 16.65 Sufficiency for a weak minimum ¶
- depends_on theorem 16.55 Jacobi's necessary condition ¶ ↺

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`	→	Variation; the first variation	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:478
`depends_on`	→	Weak and strong extrema	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:478
`depends_on`	→	Fermat: interior extremum	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:478
`depends_on`	←	Euler–Lagrange	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:656
`depends_on`	←	Legendre's necessary condition	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1528
`proves`	←	ch:14-calculus-of-variations@proof-6	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:481