What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

§9. A PROBLEM IN MECHANICS

This is the one place where arguably Courant and Robbins made a mistake, although by adding further conditions it is possible to save their argument. Paradoxically, the flaw in their proof is most easily detected if we employ the topological approach to dynamics that their argument was intended to advocate.

We repeat the statement of the problem. Suppose a train travels between two railway stations along a straight track. A rod is hinged to the floor of one of the carriages, able to move without friction either forward or backward until it touches the floor (Fig. 175, p. 320). If it does touch the floor, assume it stays there throughout the subsequent motion. Suppose we specify in advance how the train moves. The motion need not be uniform: the train can speed up, stop suddenly, even go into reverse for a time. It must start at one station and end at the other.

Courant and Robbins ask whether it is always possible to place the rod in such a position that it never hits the floor during the journey. Their solution is to note that the final position of the rod depends continuously on its initial position. There is a continuous range of starting angles, from 0° to 180°. Because the final position depends continuously on the initial position, Bolzano’s theorem (p. 312) implies that the range of final angles is also continuous. If we start with the rod lying down forwards at 0°, it stays there. If we start with it lying down backwards at 180°, it stays there. So the range of final angles includes all values between 0° and 180°. In particular, it includes 90°, so we can arrange for the rod to finish up vertical. Since it stays on the floor when it hits it, it cannot hit the floor at all.

The difficulty is that the continuity assumption made in the above discussion is arguably not justified. The problem is not the intricacies of Newton’s laws of motion, but those “absorbing boundary conditions”: if the rod hits the floor, then it stays there. In order to see why the boundary conditions cause trouble, we introduce a topological picture of the possible motions of the system. This approach, known as a phase portrait, goes back to Poincaré. The idea is to draw a kind of spacetime diagram of the motion, not just for a single initial position of the rod, but for many different positions—in principle, all of them. The position of the rod is an angle between 0° and 360°, and we can graph this in the horizontal direction (see Fig. 294). Let time run in the vertical direction. Note that the left and right hand edges of this picture should be identified because 0° = 360°: conceptually, the rectangle is rolled into a cylinder.

Now, the path in space and time of the angle that determines the position of the rod forms some curve that runs up the cylinder—what Albert Einstein called a “world-line.” Different initial angles lead to different curves. The laws of dynamics show that these curves vary continuously as the initial angle varies continuously—provided the boundary conditions are not enforced. Without those conditions the rod is free to turn a full 360°—there is no floor to prevent it turning all the way round. A possible history is shown in Fig. 294a, and here the final position does depend continuously upon the initial position.

However, when the absorbing boundary conditions are put back (Fig. 294b), the final position need not depend continuously on the initial one. Curves that just graze the left-hand boundary can swing all the way over to the right. Indeed, in this particular picture all initial positions end up on the floor: contrary to what Courant and Robbins claim, there is no choice that keeps the rod off the floor throughout the motion.

Fig. 294. Possible history of the moving rod for different initial conditions. (a) Without boundary conditions. (b) What happens when the boundary conditions are imposed.

This error in Courant and Robbins’s reasoning was first pointed out by Tim Poston in 1976, but it is still not widely known. The continuity assumption can be resuscitated by imposing extra contraints on the motion, for example a perfectly level track, no springs on the train, and so forth. But it seems more instructive, as an exercise in the application of topology to dynamics, to understand why the absorbing boundary conditions destroy continuity. This difficulty is important in advanced topological dynamics, where it has given rise to the concept of an “isolating block,” which is a region such that no dynamical trajectories are tangent to its boundary.