Lectur21 - Lecture XXI Applications of Calculus of...

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Lecture XXI Applications of Calculus of Variations II I. Special Cases A. In some cases, the solution of the differential equation implied by the Euler equation can be simplified by the structure of the problem. Particularly, f(t,x,x’) if t, x, or x’ are zero, then the differential equation implied by Euler is simplified. B. Case 1: f depends on t and x’ only (f=f(t,x’)). 1. Following the calculus of variations f df dt x x == 0 If d f x’ /d t=0 then f x’ is constant. 2. For example [] (29 3 32 23 3 2 2 0 0 01 0 2 xt txt d t ft x t c tx t c xt c t c t t t x ’( ) - =- = -= - = - = C. Case 2: F depends on x and x’ only ff x f x x f x xx x x x x x x =+ -- = ’’ 0 Multiplying by x’ xf f x f x fx xf x xf x fx xf x xf x dxf x x x x x x x x = -+ = +⇐ 0 0 Thus, df xf f c x x - =⇒ - = 0 0 D. Case 3: f depends on x’ only. The Euler equation is f x’x’ x’’=0. Thus, along the extremal case either: 1. f x’x’ =0 {in Kamien and Schwartz f x’x’ (x’)=0 because f x’x’ is a function of x’}, or 2. x’’(t)=0 which is identical to x’ = constant which yields an extremal of the form x(t)=c 1 t+c 2 . 3. If f is linear in x’-f x’x’ =0, the Euler equation is an identity and any x satisfies it.
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4. If f depends solely on x’, but is not linear, the the graphs of extremals are straight lines. (29 xt e d t st x t x x t x xt t t ’( ) () ,() 3 00 11 0 1 == fx t e f t ex t e x x =⇒ = =+ 3 23 0 3 The solution in this case must be of the form x(t)=c 1 t+c 2 . D. Case 4: f depends on t and x only. The Euler equation becomes f x (t,x)=0 which is the static problem. E. Case 5: f is linear in x’ of the form f=A(t,x)+B(t,x)x’. 1. The Euler equation is: AB x BB x fA B x fB t x xx tx xxx x += +⇐ = = ’’ (, ) 2. This is not a differential equation. If the solution x(t) of this equation satisfies the boundary conditions, it may be the optimal solution. 3. Alternatively, A x =B t may be an identity A x (t,x)=B t (t,x), satisfied by any function x. Then according to the integrability theorem of differential equations, there is a function P(t,x) such that P t =A, P x =B, P tx =A x =B t and ftxx A B x P Px dP dt ( ,,’ ) =+ = + = Thus, the integrand is the total derivative of P and xd t d t Pt xt Pt xt t t t t = - ∫∫ ’( , ( ) ) ( , ( ) ) 0 1 0 1 The value of the integral depends only on the endpoints; the path joining them is irrelevant. Any feasible path is optimal. This is
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This note was uploaded on 11/08/2011 for the course AEB 6533 taught by Professor Moss during the Fall '08 term at University of Florida.

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Lectur21 - Lecture XXI Applications of Calculus of...

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