definition 16.15 Variation; the first variation

open in the book · parts/02-mathematical-methods/14-calculus-of-variations.tex:454 · p. 636

ground object -- no derivation owed

Rests on

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 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 ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
- 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 definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - 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 ¶

Supports

depends_on definition 16.83 One-parameter transformation group; variational symmetry ¶
depends_on lemma 16.18 Fundamental lemma ¶
- depends_on theorem A.688 The Kirchhoff plate equation ¶
  - depends_on remark A.690 Poisson's count, and why it cannot be right ¶
  - 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 16.36 Euler–Lagrange equations for several independent variables ¶
  - depends_on definition 44.6 Stress–energy tensor ¶
    - depends_on definition 44.36 Energy conditions ¶
      - depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶
      - depends_on proposition 44.37 Energy conditions for the perfect fluid ¶
      - depends_on proposition 44.39 SEC is the attraction condition ¶
    - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on proposition 44.38 The electromagnetic field satisfies NEC, WEC, SEC and DEC ¶ ↺
    - depends_on theorem 44.19 Covariant conservation of stress–energy ¶
      - depends_on proposition 44.20 Perfect-fluid equations of motion ¶
    - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
      - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
      - depends_on proposition 44.30 The cosmological term ¶
  - depends_on proposition 28.42 The wave equation of a stretched string ¶
- depends_on theorem 16.38 Euler–Poisson equation ¶
  - depends_on theorem 30.56 The beam equation ¶
    - depends_on example 30.58 A steel ruler ¶
    - depends_on proposition 30.57 Bending waves are dispersive ¶
      - depends_on example 30.58 A steel ruler ¶ ↺
      - depends_on remark 30.59 Where Euler–Bernoulli fails ¶
  - depends_on theorem 30.64 The Kirchhoff plate equation ¶
    - depends_on phenomenon 30.65 Chladni figures ¶
    - depends_on proposition 30.66 Chladni scaling ¶
      - depends_on phenomenon 30.65 Chladni figures ¶ ↺
    - depends_on remark 30.67 Germain, Kirchhoff, and the edge conditions ¶
- 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 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 25.19 Hamilton's equations from the first-order action ¶
- depends_on theorem 25.22 Faddeev–Jackiw equations and brackets ¶
  - depends_on example 25.24 A charged particle in a strong magnetic field ¶
depends_on proposition 16.16 Stationarity is necessary ¶
- 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 23.53 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 theorem 16.31 Natural boundary 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 ¶ ↺
depends_on theorem 16.43 Euler's rule for integral constraints ¶
- depends_on example 16.45 Two solved isoperimetric problems ¶
- depends_on proposition 16.4 The hanging chain is a catenary ¶
- depends_on proposition 16.6 The isoperimetric extremal is a circle ¶ ↺

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`	→	Admissible class; functional	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:470
`depends_on`	→	eq:ana-functional-derivative	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:470
`depends_on`	←	One-parameter transformation group; variational symmetry	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:2665
`depends_on`	←	Fundamental lemma	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:544
`depends_on`	←	Stationarity is necessary	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:478
`depends_on`	←	Euler's rule for integral constraints	declared	parts/02-mathematical-methods/14-calculus-of-variations.tex:1292