
18 Applied calculus of variations for engineers
reach the terminal point of the curve at the same time. This is also counter-
intuitive, since clearly they have different geometric distances to cover; how-
ever, since they are acting under the gravity and the slope of the curve is
different at the two locations, the particle starting from a higher location
gathers much bigger speed than the particle starting at a lower location.
This so-called tautochrone behavior may be proven by calculation of the
time of the particles using the formula developed earlier. Evaluation of this
integral between points (x
0
,y
0
)and(x
1
,y
1
) as well as between (x
2
,y
2
)and
(x
1