Mathematical Methods for Physics and Engineering - Matematica.NET

CALCULUS OF VARIATIONS22.3.1 Several dependent variablesHere we have F = F(y 1 ,y 1 ′ ,y 2,y 2 ′ ,...,y n,y n,x)whereeachy ′ i = y i (x). The analysisin this case proceeds as before, leading to n separate but simultaneous equationsfor the y i (x),∂F∂y i= ddx( ∂F∂y ′ i), i =1, 2,...,n. (22.12)22.3.2 Several independent variablesWith n independent variables, we need to extremise multiple integrals of the form∫ ∫ ∫ ()∂y ∂y ∂yI = ··· F y, , ,..., ,x 1 ,x 2 ,...,x n dx 1 dx 2 ···dx n .∂x 1 ∂x 2 ∂x nUsing the same kind of analysis as before, we find that the extremising functiony = y(x 1 ,x 2 ,...,x n ) must satisfy∂Fn∑( )∂y = ∂ ∂F, (22.13)∂x i ∂y xiwhere y xi stands for ∂y/∂x i .i=122.3.3 Higher-order derivativesIf in (22.1) F = F(y, y ′ ,y ′′ ,...,y (n) ,x) then using the same method as beforeand performing repeated integration by parts, it can be shown that the requiredextremising function y(x) satisfies∂F∂y − ddx( ∂F∂y ′ )+ d2dx 2 ( ∂F∂y ′′ )− ···+(−1) n dndx n ( ∂F∂y (n) )=0, (22.14)provided that y = y ′ = ··· = y (n−1) = 0 at both end-points. If y, or any of itsderivatives, is not zero at the end-points then a corresponding contribution orcontributions will appear on the RHS of (22.14).22.3.4 Variable end-pointsWe now discuss the very important generalisation to variable end-points. Suppose,as before, we wish to find the function y(x) that extremises the integralI =∫ baF(y, y ′ ,x) dx,but this time we demand only that the lower end-point is fixed, while we allowy(b) to be arbitrary. Repeating the analysis of section 22.1, we find from (22.4)782