Mathematical Methods for Physics and Engineering : A Comprehensive Guide

CALCULUS OF VARIATIONS

22.3.1 Several dependent variables

Here we haveF=F(y 1 ,y′ 1 ,y 2 ,y 2 ′,...,yn,yn′,x)whereeachyi=yi(x). The analysis

in this case proceeds as before, leading tonseparate but simultaneous equations

for theyi(x),

∂F

∂yi

=

d

dx

(

∂F

∂yi′

)

,i=1, 2 ,...,n. (22.12)

22.3.2 Several independent variables

Withnindependent variables, we need to extremise multiple integrals of the form

I=

∫∫

···

∫

F

(

y,

∂y

∂x 1

,

∂y

∂x 2

,...,

∂y

∂xn

,x 1 ,x 2 ,...,xn

)

dx 1 dx 2 ···dxn.

Using the same kind of analysis as before, we find that the extremising function

y=y(x 1 ,x 2 ,...,xn) must satisfy

∂F

∂y

=

∑n

i=1

∂

∂xi

(

∂F

∂yxi

)

, (22.13)

whereyxistands for∂y/∂xi.

22.3.3 Higher-order derivatives

If in (22.1)F=F(y, y′,y′′,...,y(n),x) then using the same method as before

and performing repeated integration by parts, it can be shown that the required

extremising functiony(x) satisfies

∂F

∂y

−

d

dx

(

∂F

∂y′

)

+

d^2

dx^2

(

∂F

∂y′′

)

−···+(−1)n

dn

dxn

(

∂F

∂y(n)

)

=0, (22.14)

provided thaty=y′=···=y(n−1)= 0 at both end-points. Ify, or any of its

derivatives, is not zero at the end-points then a corresponding contribution or

contributions will appear on the RHS of (22.14).

22.3.4 Variable end-points

We now discuss the very important generalisation to variable end-points. Suppose,

as before, we wish to find the functiony(x) that extremises the integral

I=

∫b

a

F(y, y′,x)dx,

but this time we demand only that the lower end-point is fixed, while we allow

y(b) to be arbitrary. Repeating the analysis of section 22.1, we find from (22.4)