equation 16.15 eq:calcvar-first-variation

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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 16.38 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 16.31 Natural boundary condition ¶ ↺