Mathematical Methods for Physics and Engineering - Matematica.NET

22.2 SPECIAL CASESto these variations, we requiredIdα∣ = 0α=0for all η(x). (22.3)Substituting (22.2) into (22.1) and expanding as a Taylor series in α we obtainI(y, α) ==∫ ba∫ baF(y + αη, y ′ + αη ′ ,x) dxF(y, y ′ ,x) dx +∫ ba( )∂F ∂Fαη +∂y ∂y ′ αη′ dx +O(α 2 ).With this form for I(y, α) the condition (22.3) implies that for all η(x) werequire∫ b( ∂FδI =a ∂y η + ∂F )∂y ′ η′ dx =0,where δI denotes the first-order variation in the value of I due to the variation(22.2) in the function y(x). Integrating the second term by parts this becomes[η ∂F ] b ∫ b[ ∂F∂y ′ +a a ∂y − d ( )] ∂Fdx ∂y ′ η(x) dx =0. (22.4)In order to simplify the result we will assume, for the moment, that the end-pointsare fixed, i.e. not only a and b are given but also y(a) andy(b). This restrictionmeans that we require η(a) =η(b) = 0, in which case the first term on the LHS of(22.4) equals zero at both end-points. Since (22.4) must be satisfied for arbitraryη(x), it is easy to see that we require∂F∂y = ddx( ∂F∂y ′ ). (22.5)This is known as the Euler–Lagrange (EL) equation, and is a differential equationfor y(x), since the function F is known.22.2 Special casesIn certain special cases a first integral of the EL equation can be obtained for ageneral form of F.22.2.1 F does not contain y explicitlyIn this case ∂F/∂y = 0, and (22.5) can be integrated immediately giving∂F∂y ′ = constant. (22.6)777