Mathematical Methods for Physicists: A concise introduction - Site Map

THE CALCULUS OF VARIATIONS
is to ®nd the function u…x; y; z† for which I is stationary with respect to small
changes in the functional form u…x; y; z†.
Generalizing Eq. (8.2), we now let
u…x; y; z;"†ˆu…x; y; z; 0†‡"…x; y; z†;
…8:40†
where …x; y; z† is an arbitrary well-behaved (that is, di€erentiable) function which
vanishes at the boundary S. Then we have, from Eq. (8.40),
and similar expressions for u y ; u z ;and
ZZZ
@I
@"
"ˆ0ˆ
V
u x …x; y; z;"†ˆu x …x; y; z; 0†‡" x ;
@f
@u ‡ @f
@u x ‡ @f
x @u y ‡ @f
y @u z
z
dxdydz ˆ 0:
We next integrate each of the terms …@f =@u i † i using `integration by parts' and the
integrated terms vanish at the boundary as required. After some simpli®cations,
we ®nally obtain
ZZZ
V
@f
@u @ @f
@ @f
@ @f
@x @u x @y @u y @z @u z
…x; y; z†dxdydz ˆ 0:
Again, since …x; y; z† is arbitrary, the term in the braces may be set equal to zero,
and we obtain the Euler±Lagrange equation:
@f
@u @ @f
@ @f
@ @f
ˆ 0:
@x @u x @y @u y @z @u z
…8:41†
Note that in Eq. (8.41) @=@x is a partial derivative, in that y and z are constant.
But @=@x is also a total derivative in that it acts on implicit x dependence and on
explicit x dependence:
@ @f
ˆ
@2 f
‡
@2 f
u
@x @u x @x@u x @u@u x ‡ @2 f
‡ @2 f
u
x @u y @u xy ‡
@2 f
u
x @u z @u xz :
x
@u 2 x
…8:42†
Example 8.9
The SchroÈ dinger wave equation. The equations of motion of classical mechanics
are the Euler±Lagrange di€erential equations of Hamilton's principle. Similarly,
the SchroÈ dinger equation, the basic equation of quantum mechanics, is also a
Euler±Lagrange di€erential equation of a variational principle the form of which
is, in the case of a system of N particles, the following
Z
Ld ˆ 0;
…8:43†
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