Galileo and the brachistochrone problemA controversy crossing more than a century

Andrea Baracco – graduate at Politecnico di Torino

Figure 1 - Inclined plane - Museo Galileo - Firenze

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 1andFigure 2).

Galileo proposed in 1638 the problem 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.

Figure 2 - Brachistochronous fall - Museo Galileo - Firenze

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).

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 “ActaEruditorum” 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 SolidWorksMotion 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 togetherbut this is false and the SolidWorks Motion simulation clearly shows that (as you can see in the model simulation or in the enclosed video).

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