Mathematical Methods for Physicists: A concise introduction - Site Map

THE CALCULUS OF VARIATIONS
p
Letting 1=c ˆ a and solving for y
0
gives
y 0 ˆ dy
dx ˆ
and solving for dx and integrating we obtain
Z
Z
dx ˆ
r
a y
;
y
r
y
dy:
a y
We then let
y ˆ a sin 2 ˆ a …1 cos 2†
2
which leads to
Z
x ˆ 2a
Z
sin 2 d ˆ a
…1 cos 2†d ˆ a …2 sin 2†‡k:
2
Thus the parametric equation of the path is given by
x ˆ b…1 cos †;
y ˆ b… sin †‡k;
where b ˆ a=2;ˆ 2. The path passes through the origin so we have k ˆ 0 and
x ˆ b…1 cos †;
y ˆ b… sin †:
The constant b is determined from the condition that the particle passes through
P 2 …x 2 ; y 2 †:
The required path is a cycloid and is the path of a ®xed point P 0 on a circle of
radius b as it rolls along the x-axis (Fig. 8.3).
A line that represents the shortest path between any two points on some surface
is called a geodesic. On a ¯at surface, the geodesic is a straight line. It is easy to
show that, on a sphere, the geodesic is a great circle; we leave this as an exercise
for the reader (Problem 8.3).
Figure 8.3.
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