remark 30.54 Other end conditions, and what buckling is variationally

Rests on

• depends_on phenomenon 30.52 Euler buckling ¶
  • depends_on proposition 30.51 Moment and curvature ¶
    • depends_on definition 30.31 The engineering moduli ¶
      • depends_on definition 30.20 Pressure and deviatoric stress ¶
        • depends_on proposition 30.19 The stress tensor is symmetric ¶
        • depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
      • depends_on equation 30.21 eq:elast-hooke-isotropic ¶
    • depends_on equation 30.21 eq:elast-hooke-isotropic ¶ ↺
    • proves proof ch:13-continuum-elasticity@proof-27 ¶
  • 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 corollary 7.44 Substitution and integration by parts ¶
        • depends_on proposition 7.31 Chain rule ¶
        • depends_on proposition 7.30 Leibniz rule ¶
        • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
        • 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 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 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 ¶
        • 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 ¶
  • proves proof ch:13-continuum-elasticity@proof-28 ¶
• depends_on theorem 16.55 Jacobi's necessary condition ¶
  • 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 ¶ ↺
      • 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 theorem 16.22 Euler–Lagrange ¶ ↺
        • 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 ¶

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