This is a problem of calculus of variations with an equality constraint. This problem is similar to the problem of minimizing a function

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1 3-7 ISOPERIMETRIC PROBLEMS Isoperimetric implies constant perimeter. This is a problem of calculus of variations ith an equalit constraint in the integral form. ( V ) d ; W G( ) d This problem is similar to the problem of minimiing a function subject to an equalit constraint. The Lagrange multiplier method can be used to obtain the necessar conditions of optimalit. Let Γ' be a sub-class of admissible functions that also satisf W Constant condition. Use Lagrange multiplier method to rite the optimalit conditions and obtain the differential equation: V 1 ( λ) d ; λg λ Lagrange multiplier a constant d d d d Eample: The Catenar Problem: Given an inetensible string of length L ling in the gravitational field determine the shape of the curve that minimies the potential energ of the string. Solve the equation subject to the forced and natural boundar conditions. Lecture #17 66

2 3-8 AUXILIARY DIERENTIAL EQUATIONS Minimie the functional subject to auiliar differential equation V ( ) d G ( ) The admissible class Γ' is a subset of class Γ such that () satisfies the D.E. The first method is to solve the auiliar D.E. for (); choose the constants to satisf the.b.c. and Euler's equation - there ma not be enough freedom to satisf all these conditions. This case is similar to minimiation of a function of one variable ith one equalit constraint hich is not an interesting optimiation problem. A problem ith to unknon functions () and () ma be more interesting: V ( ) d ; G ( ) The solution of the D.E. ill have an arbitrar function ϕ() that can be required to satisf.b.c. and its on Euler's equation. The second approach is to use the Lagrange multiplier method; in this case λ ill be a function of. V 1 ( ( ) ( ) λ ( )) d ; λg Without designating the function λ() e ma give the functions and arbitrar variations that conform to the forced boundar Lecture #17 67

3 conditions. Thus Euler's equation for V and natural boundar conditions are obtained. Then e seek functions () and () that satisf the Euler equations the auiliar D.E. the forced boundar conditions and the natural boundar conditions. 3-9 IRST VARIATION O A DOUBLE INTEGRAL The potential energ of shells plates and membranes is given b a double integral. We must treat to independent variables and. Consider a finite region R in the () plane. R ma be multipl connected ring shaped or irregular region ith holes; R is considered to be closed. W() is the unknon function. The class Γ of admissible functions () is defined as follos: unctions in Γ are continuous ith 4 th order partial derivatives in R. unctions in Γ satisf linear forced boundar conditions of the tpe a b/n c; a b c are given point functions of R and /n is the normal derivative of on the boundar. Consider the integral: V ( ) dd R is a given function of eight variables ith continuous partial Lecture #17 68

4 derivatives to the third order ith respect to all real values of and for all values of () in R. Using the standard variational procedure using the Green's theorem to integrate b parts e get the Euler equation and the natural boundar conditions: 3-1 IRST VARIATION O A TRIPLE INTEGRAL Problems of equilibrium of elastic solids and problems of dnamics of plates shells and membranes lead to variations of triple integrals. The independent variables () ma be regarded as rectangular coordinates. Class Γ of admissible functions (): (i) continuous ith fourth order derivatives in R (ii) must satisf linear forced boundar condition a b/n c a b c are given point functions on the boundar of R and /n is the normal derivative of on the boundar. Consider the triple integral: ddd V R...) ( The standard variational process is used. The integration b parts process converts the volume integrals to the surface integrals b Lecture #17 69

5 use of the divergence theorem. These then provide the natural boundar conditions. Lecture #17 7