Galileo and the Brachistochrone Problem

One of the most famous problems in the history of mathematics was posed by the Swiss mathematician Johann Bernoulli in 1696 as a challenge "to the most acute mathematicians of the entire world''. The problem can be stated as follows:

Given two points on a plane at different heights, what is the shape of the wire down which a bead will slide (without friction) under the influence of gravity so as to pass from the upper point to the lower point in the shortest amount of time?''

This is the brachistochrone ("shortest time'') problem. This was a different kind of optimization problem, since instead of asking for the value of a variable, among all possible values, that will maximize or minimize something, it asks for the optimal function (or curve), among all possible curves. The solution to this problem played a very important role in the history of the calculus. For the moment, we would like to discuss Galileo's work relevant to this problem, which occurred in 1638, well before the brachistochrone problem had been explicitly stated.

Galileo's Formula for Time of Descent of an Object Falling Along an Inclined Plane

Galileo studied motion under gravity, showing that a body falling in space traverses a distance proportional to the square of the time of descent. Using this law, he was able to compute the time of descent of an object, starting from rest, falling along an inclined plane from point $A$ to point $B$, assuming no friction.

Galileo1.png

The time to travel from $A$ to $B$ is

(1)
\begin{align} \sqrt{\frac{2}{g}}\frac{d}{\sqrt{h}} \end{align}

where $d$ is the distance between points $A$ and $B$, $h$ is the vertical distance between $A$ and $B$, and $g$ is the acceleration due to gravity, approximately 980 $\mbox{cm}/\mbox{sec}^2$. So for example, the time of descent along the straight line path $(0,0)\to (5,-5)$ is

(2)
\begin{align} \sqrt{\frac{2}{g}}\frac{\sqrt{50}}{\sqrt{5}}=\frac17\approx 0.1428571428\] \end{align}

Computing Times of Descent Along Polygonal Paths

Suppose we wish the body to follow the polygonal path $A\to B\to C$:

Galileo2.png

Then we can use equation (1) to compute the time of descent along the straight path $A\to B$, but equation (1) no longer applies for the path $B\to C$ since as the body falls from $A$ to $B$ it picks up speed, so the initial velocity along the path $B\to C$ is no longer $0$.

Galileo appeared to believe that the answer to the brachistochrone problem was a circle. Thus, if we bend a wire into the shape of the quarter circle with center (5,0) and radius 5, and let a bead slide down along the wire starting at (0,0), he speculated that this path would take the shortest time among all paths. As it turns out, Galileo was wrong in his conjecture that the circle would give the fastest path.

Exercises

1. In this worksheet, you will investigate Galileo's conjecture that the path of fastest descent is a circle. You will need this Mathematica notebook.

2. The applet below shows the construction of an upside-down cycloid, a curve generated as the path of a point on the circumference of a circle as the circle rolls along a straight line without slipping. Drag the center of the circle to see how the curve is generated as the path of the red point on the circumference. If was eventually shown that this curve is the solution to the brachistochrone problem! To see this, use the slider $r$ to find a circle that generates a cycloid that goes through the second point (5,-5). Now enter each of your points that you computed in Exercise 1 into the input field of the applet to graph the points. They should lie approximately on the cycloid.[[/span]]

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