GALILEO AND THE BRACHISTOCHRONE PROBLEM
A CONTROVERSY CROSSING MORE THAN A CENTURY
ANDREA BARACCO – graduate at Politecnico di Torino
According to tradition, Galileo Galilei
(1564 - 1642) - father of modern physics
– is said to have dropped balls from
the leaning tower of Pisa to
demonstrate that their time of descent
was independent of their mass, which
was contrary to what Aristotle had
taught. Although this is probably a
legend, he was anyhow the first one to
derive the correct kinematic motion of
a mass under uniform acceleration
and to prove it using inclined planes
(Figure 1 and Figure 2).
Galileo proposed in 1638 the problem Figure 1 - Inclined plane - Museo Galileo - Firenze
of determining what was the curve
between two points A and B along which a point starting in A, without friction and
in the gravitational field, reaches B in the shortest time.
Ordinary thinking would suggest that
the quickest way to reach two points is
a straight line (third inclined plane in
the enclosed model and in the Figure
3). Actually, a straight line could be
faster or slower than a broken line or a
curve (fourth and fifth inclined plane in
the model are an example of two
slower paths).
Figure 2 - Brachistochronous fall - Museo Galileo Firenze
Galileo suggested that the arc of a
circle was the solution, but he could
not prove his assumption, which in fact
was incorrect (second inclined plane
in the model).
60 years later Johann Bernoulli (1667 - 1748) posed again the problem in his “Acta
Eruditorum” and in 1697 five mathematicians (among them Newton, Leibnitz and
his brother Jakob) argued that the correct path of the quickest descent from A to
B was the arc of a cycloid (first inclined plane in the model).
A propriety of the cycloid is to be a brachistochrone curve (from the Greek
brachistos = the shortest, chronos = time), which is the curve between two points
through which a body moves, under the force of gravity and assuming no friction,
in the least time.
However the equations were not exhaustively solved by that time and the method
had to be refined by Euler and Lagrange by the “calculus of variation” and
completely solved only later in 1766.
Figure 3 - Rendering of my model with PhotoView360
Today, by using a SolidWorks Motion simulation, it is evident that neither the straight
line, nor the Galileo’s arc of a circle or broken/interpolated lines are the fastest
curve to solve the problem; the quickest path is the arc of a cycloid. The cycloid is
actually very close to the arc of a circle; the two balls seem to arrive almost
together but this is false and the SolidWorks Motion simulation clearly shows that (as
you can see in the model simulation or in the enclosed video).
-SITOGRAPHY:
http://en.wikipedia.org/wiki/Brachistochrone_curve
http://www.museogalileo.it/en/index.html
http://www-history.mcs.st-and.ac.uk/HistTopics/Brachistochrone.html