theorem 16.22 Euler–Lagrange

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:646 · p. 638

Rests on

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 definition 7.13 Composite function ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-11 ¶
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-15 ¶
  - depends_on proposition 7.30 Leibniz rule ¶
    - depends_on definition 7.12 Product of functions ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - proves proof ch:05-real-analysis@proof-14 ¶
  - 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: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 definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
  - 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 3.38 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 definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶ ↺
    - 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 ¶
proves proof ch:14-calculus-of-variations@proof-11 ¶

Supports

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

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`	→	eq:calcvar-first-variation	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:656
`depends_on`	→	Mixed form	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:656
`depends_on`	→	Stationarity is necessary	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:656
`depends_on`	←	Field of extremals; slope function	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1866
`depends_on`	←	Euler buckling	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1927
`depends_on`	←	Steady precession	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:1430
`depends_on`	←	System of Euler–Lagrange equations	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:993
`depends_on`	←	Natural boundary condition	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:861
`proves`	←	ch:14-calculus-of-variations@proof-11	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:660