Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.2 Special cases

to these variations, we require

dI

dα

∣

∣

∣

∣

α=0

= 0 for allη(x). (22.3)

Substituting (22.2) into (22.1) and expanding as a Taylor series inαwe obtain

I(y, α)=

∫b

a

F(y+αη, y′+αη′,x)dx

=

∫b

a

F(y, y′,x)dx+

∫b

a

(

∂F

∂y

αη+

∂F

∂y′

αη′

)

dx+O(α^2 ).

With this form forI(y, α) the condition (22.3) implies that for allη(x)werequire

δI=

∫b

a

(

∂F

∂y

η+

∂F

∂y′

η′

)

dx=0,

whereδIdenotes the first-order variation in the value ofIdue to the variation

(22.2) in the functiony(x). Integrating the second term by parts this becomes

[

η

∂F

∂y′

]b

a

+

∫b

a

[

∂F

∂y

−

d

dx

(

∂F

∂y′

)]

η(x)dx=0. (22.4)

In order to simplify the result we will assume, for the moment, that the end-points

are fixed, i.e. not onlyaandbare given but alsoy(a)andy(b). This restriction

means that we requireη(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 arbitrary

η(x), it is easy to see that we require

∂F

∂y

=

d

dx

(

∂F

∂y′

)

. (22.5)

This is known as theEuler–Lagrange(EL) equation, and is a differential equation

fory(x),since the functionFis known.

22.2 Special cases

In certain special cases a first integral of the EL equation can be obtained for a

general form ofF.

22.2.1Fdoes not containyexplicitly

In this case∂F/∂y= 0, and (22.5) can be integrated immediately giving

∂F

∂y′

= constant. (22.6)