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CU-Boulder ASEN 5022 - Derivation of E-L Equation

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Calculus of Variations and Euler-Lagrange EquationA theoretical derivation of the Euler-Lagrange equations can be carried out by utilizing the calculusof variations of a definite integral. To this end, let us now address how one obtains a stationaryvalue of a function via the calculus of variations.First, the concept of virtual displacement should not be confused with the concept of the variationof a function. For we have the virtual displacement at our disposal, but the variation of the functionis not. Suppose we are to obtain a stationary value of =baG(y, y , x)dx (1)with the boundary conditionsy(a) = α, y(b) = β(2)Assume that y = f (x) by hypothesis gives a stationary value to . One way to prove this to betrue is to evaluate the same integral for a slightly modified function ¯y and establish that the rate ofchange of  due to the change in y is zero (why?). We can thus write¯y = f (x) + g(x) = y(x) + g(x)(3)where g(x) is an arbitrary function that must be continuous and differentiable as y. Since g(x) isan arbitrary function, this difference is called the variation of the function y and is denoted by δy:δy =¯y − y = g(x)(4)This is a seemingly trivial expression but there is an important property:The variation of δy refers to an arbitrary infinitesimal change of the value of thedependent variable of y, at the point x. The independent variable, x, does not participatein the process of variation.A consequence of the above statement isδx = 0 (5)henceδy(a) = δy(b) = 0 (6)Before we proceed to minimize  in (1), we need to establish two additional properties of theδ-process. Since G(y, y , x) involves y , we need to know how to express δy . To this end, we notefrom (4)∂∂xδy =∂∂x( ¯y − y) =∂∂x(g(x)) = g (7)1which is the derivative of the variation δy. On the other hand, for the variation of the derivative,we haveδy =¯y − y = (y + g) − y = g (8)Equations (7) and (8) give∂∂xδy = δ∂y∂x(9)Hence, the derivative of the variation is the same as the variation of the derivative. Similarly, onecan showδbaG(y, y , x)dx =baδG(y, y , x)dx (10)In other words, variation and differentiationare commutative. Similarly, onecan showthat variationand integration are also commutative.We are now ready to minimize  in (1). First, we haveδG(y, y , x) = G(y + g, y + g , x) − G(y, y , x)(11)= (∂G∂yg +∂G∂y g )(12)Now, we have for the variation of the definite integral (2) from (10) and (12):δ = δbaG(y, y , x)dx =baδG(y, y , x)dx (13)= ba(∂G∂yg +∂G∂y g )dx (14)But, via the rule of integration by parts, we haveba∂G∂y g dx = [∂G∂y g]ba−baddx(∂G∂y )gdx (15)Since the variation of y at x = a and x = b is zero from (6), we have the following variationalquantities:{δy(a) = g(a) = 0,δy(b) = g(b) = 0}=> g(a) = g(b) = 0 (16)so that we have[∂G∂y g(b)] − [∂G∂y g(a)] = 0 (17)2Substituting (15) and (17) into (15), we obtainδ = ba(∂G∂y−ddx∂G∂y )gdx (18)As  is associated with an arbitrary variation of y, the stationary value of  isδ=ba(∂G∂y−ddx∂G∂y )g(x)dx = 0 (19)Since g(x) is an arbitrary function, designating the variation of y, we must for the stationary valueof  have∂G∂y−ddx∂G∂y = 0 (20)This is the celebrated Euler-Lagrange equation in mechanics when G(y, y , x) is replaced by theLagrangian function, L , with substitutions of y by q and x by t:G(y, y , x) = L = T(˙q, q) − V(q)(21)so that we have from (20) the final equations of motionddt∂T∂˙q−∂T∂q+∂V∂q= 0 (22)When G is of the formG = G(y, y , y , x)(23)the resulting governing equation is given by∂G∂y−ddx∂G∂y +d2dx2∂G∂y = 0 (24)with the boundary conditions given by[(∂G∂y −ddx∂G∂y )δy +∂G∂y δy ]|ba= 0 (25)The preceeding equations are applicable for a beam under gravity load for which G becomesG = EI(∂2w(x)∂x2)2−


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