Top o' the Morning to You, Prof. SyngeSeptember 17, 2000
Philip J. Davis (from the dust jacket, The Education of a Mathematician).
Over the years, Philip Davis's reviews have attracted many loyal followers. Here, SIAM News offers a glimpse of Davis as author, with a selection from his latest book (The Education of a Mathematician (AK Peters, Natick, Massachusetts, 2000, 353 pages)).
As a kid I loved to spin tops. I mean the kind of top you wind up with a long string and throw to the ground while retaining the string in your fist. Tops were "in"; you could buy a top for a nickel. The string cost a penny. All my friends spun tops. When I was in top form (so to speak), I could hold two tops in my hand and spin them simultaneously.
As this skill began to diminish, I began to collect all kinds of tops; little tops that came free in a box of Cracker Jack, wooden whip tops, tops that worked on a spring, big metal tops that you spun by pushing down on a spiralled shaft, toy gyroscopes, yo-yos, diabolos; any toy that spun rapidly. My collection has been dispersed but its hum lingers on.
When I joined the Bureau of Standards [in 1952], there were many distinguished visitors to the mathematics division who stayed for longer or shorter periods. There was also a shortage of room space. For a while I had to share an office with several such visitors and the very first was Professor John Lighton Synge, (Synge and Griffith: Principles of Mechanics, 1942). Synge was a companionable sort, an easy roommate. We had our stretches of silence during which we worked, and we had stretches when we chatted. I heard much about his older relative John Millington Synge (1871-1909), the well-known Irish playwright. [Another familiar relative of J. L. Synge is his daughter, Cathleen Morawetz of New York University.] I had read J. M.'s Playboy of the Western World, so I was able to energize J. L.'s narratives by asking a few intelligent questions.
I was with Synge for about a month and then he went on to other engagements. During this time, Hadassah [PJD's wife] had brought home a little top of a sort that I had never seen, intending it for our children. The top was about an inch and a quarter in length and made of plastic. Its body was a hollow sphere, or more precisely, a sphere with its "north polar cap" missing. Built into the sphere and emerging from it was a long, narrow cylindrical shaft by which the top was spun between forefinger and thumb.
But this was by no means an ordinary top, for its spherical shape---as opposed to the classical conical shape---enabled it to do a great trick. When spun, the top would continue to spin for a while in the normal fashion. Then, very gradually, the cylindrical shaft would approach the ground, and then, miraculously, so it seemed to me, its spherical body would rise, and the top would spin upside down on its shaft and even "go to sleep" in that position. How to explain that!
Well, as I have said, I was sharing an office with one of the world authorities on mechanics, and I recalled that in Synge and Griffith there was a long chapter on the motion of a rigid body in space. Vector analysis, cross products, gyroscopic couples, that sort of thing. I would bring the top in, spin it on Synge's desk, and ask him if he could account for the strange and wonderful behavior of my top.
In the morning, I did just that. Synge observed the motion of the top carefully. Then, rather non-commitally, he said something like "Very nice,
very nice," and went back to the manuscript of the book he was then working on: The Hypercircle Method in Mathematical Physics (1957).
The next morning I found on my desk a folio-sized reprint of an article in the Transactions of the Royal Irish Academy, bearing an inscription to me. Author: J. L. Synge. An approximate title: "On the Motion of the Tippe Top."
Well, topped by the Master! I thanked him and promised to read his article through. I did this over the next few days and was able to have my "revenge."
"If I'm not mistaken, Prof. Synge, you've added a frictional term to the usual equations."
"Yes, a non-linear frictional term."
"And that explains why the top doesn't go to sleep in the normal position. The normal position is unstable."
"Correct. That explains it."
"But your paper doesn't explain why the top turns over and goes to sleep upside down?"
"You're right. I'm afraid it doesn't. That would be a very difficult problem. A global problem. Prediction in the short term is often easy. But in the long term . . . aahh . . . ."
Yes, aahh . . . I let it go at that. But now I will ask: what has happened to the theory of the tippe top since the time I shared a room with Synge? In the intervening years the theory of dynamical systems has grown tremendously. Tops are again popular as an example of a "completely integrable system." Can the complete motion of the tippe top be derived theoretically from the principles of dynamical systems?
I asked my colleague Prof. Constantine Dafermos [whose expertise was also recognized recently by SIAM at its annual meeting]. His reaction was, "The tippe top is probably constructed with a high center of gravity and this would make the lower position less stable than the upper one. I doubt whether the complete motion can be derived. But I will see what's around."