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Unformatted text preview: P Introduction to Calculus of Variations The calculus of variations is a generalization of the minimum and maximum problem of ordinary calculus. It seeks to determine a function, ( 29 y f x = , that minimizes/maximizes a definite integral ( 29 , , , , . . . . . . 2 1 x x I F x y y y ′ ′′ = ∫ (1) which is called a functional (function of functions) and whose integrand contains y and its derivatives and the independent variable x. Although the calculus of variations is similar to the maximum and minimum problems of ordinary calculus, it does differ in one important aspect. In ordinary calculus, one obtains the actual value of a variable for which a given function has an extreme point. In the calculus of variations, one does not obtain a function that provides extreme value for a given integral (functional). Instead, one only obtains the governing differential equation that the function must satisfy to make the given functional have a stationary value. Hence, the calculus of variations is not a computational tool but it is only a devise for obtaining the governing differential equation of the physical stationary value problem. ∆ b x P P dy dx ds 1 The bifurcation buckling behavior of a bothendpinned column shown in the sketch may be examined in two different perspectives. Consider first that the static deformation prior to buckling has taken place and the examination is being conducted in the neighboring equilibrium position where the axial compressive load has reached the critical value and the column bifurcates (is disturbed) without any further increase of the load. The strain energy stored in the elastic body due to this flexural action is ( 29 ( 29 2 T v v 1 1 EIy EI U dv y y y dv y dx 2 2 I 2 σ ε ′′ ′′ ′′ = = = ∫ ∫ ∫ l % % (2) In calculating the strain energy, the contributions from the shear strains are generally neglected as they are very small compared to those from normal strains....
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 Summer '10
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 Calculus, Derivative, dx

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