Mathematical Methods for Physics and Engineering - Matematica.NET

22.3 SOME EXTENSIONSy(x)+η(x)∆yy(x)h(x, y) =0∆xbFigure 22.5 Variation of the end-point b along the curve h(x, y) =0.that we require[η ∂F ] b∂y ′ +a∫ ba[ ∂F∂y − d ( )] ∂Fdx ∂y ′ η(x) dx =0. (22.15)Obviously the EL equation (22.5) must still hold for the second term on the LHSto vanish. Also, since the lower end-point is fixed, i.e. η(a) = 0, the first term onthe LHS automatically vanishes at the lower limit. However, in order that it alsovanishes at the upper limit, we require in addition that∣∂F ∣∣∣x=b∂y ′ =0. (22.16)Clearly if both end-points may vary then ∂F/∂y ′ must vanish at both ends.An interesting and more general case is where the lower end-point is againfixed at x = a, but the upper end-point is free to lie anywhere on the curveh(x, y) = 0. Now in this case, the variation in the value of I due to the arbitraryvariation (22.2) is given to first order by[ ] ∂F b ∫ b( ∂FδI =∂y ′ η +a a ∂y − d )∂Fdx ∂y ′ ηdx+ F(b)∆x, (22.17)where ∆x is the displacement in the x-direction of the upper end-point, asindicated in figure 22.5, and F(b) is the value of F at x = b. In order for (22.17)to be valid, we of course require the displacement ∆x to be small.From the figure we see that ∆y = η(b) +y ′ (b)∆x. Since the upper end-pointmust lie on h(x, y) = 0 we also require that, at x = b,∂h ∂h∆x + ∆y =0,∂x ∂ywhich on substituting our expression for ∆y and rearranging becomes( )∂h ∂h+ y′ ∆x + ∂h η =0. (22.18)∂x ∂y ∂y783