June 2014
Intermediate to advanced
233 pages
5h 42m
English
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
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Constrained variational problems 35
For comparison purposes, the figure also shows a parabola with dashed lines,
representing an approximation of the catenary and obeying the same bound-
ary conditions.
2.4 Closed-loop integrals
As a final topic in this chapter, we briefly view variational problems posed in
terms of closed-loop integrals, such as
I =
f(x, y, y
)dx = extremum,
subject to the constraint of
J =
g(x, y, y
)dx.
Note that there are no boundary points of the path given since it is a closed
loop. The substitution of
x = acos(t),y= asin(t),
changes the problem to the conventional form of
I =
t
1
t
0
F (x, y, ˙x, ˙y)dt,
subject to
J =
t
1
t
0
G(x, y, ˙x, ˙y)dt. ...
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