Mathematical Methods for Physicists: A concise introduction - Site Map

THE EULER±LAGRANGE EQUATION
@I
@" ˆ @
@"
ˆ
Z x2
x 1
Z x2
x 1
f fy…"; x†; y 0 …"; x†; xgdx
@f @y
@y @" ‡ @f @y 0
@y 0 dx;
@"
…8:5†
where we have employed the fact that the limits of integration are ®xed, so the
di€erential operation a€ects only the integrand. From Eq. (8.2) we have
@y
@" ˆ …x† and @y 0
@" ˆ d
dx :
Substituting these into Eq. (8.5) we obtain
Z
@I
x2
@" ˆ @f @f d
…x†‡
@y @y 0 dx:
dx
x 1
Using integration by parts, the second term on the right hand side becomes
Z x2
@f d @f
x 2
Z x2
@y 0 dx ˆ
dx @y 0 …x† d @f
x 1
dx @y 0 …x†dx:
x 1
x 1
x 1
…8:6†
The integrated term on the right hand side vanishes because …x 1 †ˆ…x 2 †ˆ0
and Eq. (8.6) becomes
Z
@I
x2
@" ˆ @f @y
x 1
@y @" d
@f @y
dx @y 0 dx
@"
Z x2
@f
ˆ
@y d
@f
dx @y 0 …x†dx:
…8:7†
Note that @f =@y and @f =@y 0 are still functions of ". However, when
" ˆ 0; y…"; x† ˆy…x† and the dependence on " disappears.
Then …@I=@"†j "ˆ0 vanishes, and since …x† is an arbitrary function, the integrand
in Eq. (8.7) must vanish for " ˆ 0:
d @f
dx @y 0 @f
@y ˆ 0:
…8:8†
Eq. (8.8) is known as the Euler±Lagrange equation; it is a necessary but not
sucient condition that the integral I have an extreme value. Thus, the solution
of the Euler±Lagrange equation may not yield the minimizing curve. Ordinarily
we must verify whether or not this solution yields the curve that actually minimizes
the integral, but frequently physical or geometrical considerations enable us
to tell whether the curve so obtained makes the integral a minimum or a maximum.
The Euler±Lagrange equation can be written in the form (Problem 8.2)
d
dx
f y 0 @f
@y 0
@f
@x ˆ 0:
…8:8a†
349