Mathematical Methods for Physicists: A concise introduction - Site Map

THE CALCULUS OF VARIATIONS
This is often called the second form of the Euler±Lagrange equation. If f does not
involve x explicitly, it can be integrated to yield
f y 0 @f
@y 0 ˆ c;
…8:8b†
where c is an integration constant.
The Euler±Lagrange equation can be extended to the case in which f is a
functional of several dependent variables:
f ˆ f y 1 …x†; y1…x†; 0 y 2 …x†; y2…x†; 0 ...;x :
Then, in analogy with Eq. (8.2), we now have
y i …"; x† ˆy i …0; x†‡" i …x†; i ˆ 1; 2; ...; n:
The development proceeds in an exactly analogous manner, with the result
Z
@I
x2
@" ˆ @f
d
@f
@y i dx @y
0
i …x†dx:
i
x 1
Since the individual variations, that is, the i …x†, are all independent, the vanishing
of the above equation when evaluated at " ˆ 0 requires the separate vanishing
of each expression in the brackets:
d @f
dx @yi
0 @f ˆ 0; i ˆ 1; 2; ...; n: …8:9†
@y i
Example 8.1
The brachistochrone problem: Historically, the brachistochrone problem was the
®rst to be treated by the method of the calculus of variations (®rst solved by
Johann Bernoulli in 1696). As shown in Fig. 8.2, a particle is constrained to
move in a gravitational ®eld starting at rest from some point P 1 to some lower
Figure 8.2
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