John Pasta, a recently arrived physicist, assisted us in the task of flow diagramming, programming, and running the problem on the MANIAC. Fermi had decided to try to learn how to code the machine by himself. In those days it was more difficult than now, when there are set rules, ready-made programs, and the procedure itself is automated. One had to learn many little tricks in those early days. Fermi learned them very quickly and taught me some, even though I already knew enough to be able to estimate what kind of problems could be done, their duration in number of steps, and the principles of how they should be executed.
Our problem turned out to have been felicitously chosen. The results were entirely different qualitatively from what even Fermi, with his great knowledge of wave motions, had expected. The original objective had been to see at what rate the energy of the string, initially put into a single sine wave (the note was struck as one tone), would gradually develop higher tones with the harmonics, and how the shape would finally become "a mess" both in the form of the string and in the way the energy was distributed among higher and higher modes. Nothing of the sort happened. To our surprise the string started playing a game of musical chairs, only between several low notes, and perhaps even more amazingly, after what would have been several hundred ordinary up and down vibrations, it came back almost exactly to its original sinusoidal shape.
I know that Fermi considered this to be, as he said, "a minor discovery." And when he was invited a year later to give the Gibbs Lecture (a great honorary event at the annual American Mathematical Society meeting), he intended to talk about this. He became ill before the meeting, and his lecture never took place. But the account of this work, with Fermi, Pasta and myself as authors, was published as a Los Alamos report.
I should explain that the motion of a continuous medium like a string is studied on a machine by imagining the string to be composed of a finite number of particles — in our case, sixty-four or one hundred twenty-eight. (It is better to take a power of two for the number of elements, which is more convenient to handle on the computer.) These particles are connected to each other by forces which are not only linear in terms of their distance but by additional small non-linear quadratic terms. Then the machine quickly computes in short time-steps the motion of each of these points. After having computed this, it goes to the next time-step, computes the new positions, and so on for many times. There is absolutely no way to perform this numerical work with pencil and paper; it would literally take thousands of years. An analytic closed-form solution using the mathematical techniques of classical analysis of the nineteenth and twentieth centuries is also completely unlikely.
The results were truly amazing. There were many attempts to find the reasons for this periodic and regular behavior, which was to be the starting point of what is now a large literature on non-linear vibrations. Martin Kruskal, a physicist in Princeton, and Norman Zabuski, a mathematician at Bell Labs, wrote papers about it. Later, Peter Lax contributed signally to the theory. They made interesting mathematical analyses of problems of this sort. A mathematician will know that the so-called Poincaré return-type of dynamical system containing that many particles is terrifically long — on an astronomical scale — and the fact that it came back so soon to its original form is what is so surprising.
Another Los Alamos physicist, Jim Tuck, was curious to see if after this near return to the original position, another period started again from this condition and what it would be after a second "period." With Pasta and Metropolis, he tried it again and, surprisingly, the thing came back, a percent or so less exactly. These continued and, after six or twelve such periods, it started improving again and a sort of superperiod appeared. Again this is most peculiar.
Other authors, among them several Russian mathematicians, have studied this problem and written papers about it. Last year I received a request from the Japanese Academy for permission to reprint the Fermi-Pasta-Ulam paper. I assented without hesitation and shortly thereafter a whole volume appeared containing studies of these questions by many authors.
I might say here that John Pasta was a very interesting person. A physicist by profession, he spent several years during the depression as a policeman on a beat in New York City. He joined my group in Los Alamos. On the whole very taciturn, he could occasionally make very caustic, humorous remarks. When Johnny became an AEC Commissioner, impressed by Pasta's common sense, ability, and knowledge of the Los Alamos scene, he invited him to join the AEC in Washington.
As for James Tuck, he was a British physicist who had come to Los Alamos with the British Mission during the war. He had returned to Oxford after the war, but then came back to join the laboratory again. We collaborated on a method for obtaining energy from fusion in a non-explosive way and during the war had written a joint report on this which may still be classified.
As a very young physicist Tuck was for a time assistant to Lindemann, who later became Lord Cherwell, Churchill's science advisor. He has a fund of interesting and amusing stories about this experience, and he still defends Cherwell vigorously against accusations or criticisms. He reminds me of the English eccentrics described by Jules Verne and by Karl May. Very tall, moving in an abrupt, somewhat uncoordinated way, by his awkwardness he causes many amusing incidents that are always the delight of his friends. For many years Tuck directed a Los Alamos program for the peaceful uses of fusion. The laboratory is still vigorously engaged in a large effort to find methods to extract energy "peacefully" from the fusion of deuterium.
There was another problem which Fermi wanted to study but which we somehow never came to formulate well or to work on. He said one day, "It would be interesting to do something purely kinematical. Imagine a chain consisting of very many links, rigid, but free to rotate around each other. It would be curious to see what shapes the chain would assume when it was thrown on a table, by studying purely the effects of the initial energy and the constraints, no forces."
During these years we and the von Neumanns started the practice of spending Christmas together. Claire, our daughter, was a small child, and it became a tradition that on Christmas Eve Johnny and Klari would help us assemble her toys. I remember a large cardboard dollhouse which took many hours; both Johnny and I, especially I, being inept at following the instructions which called for inserting tab A into slot B. To this day I am incapable of following written instructions, whether for filling out forms or assembling parts. Johnny, on the other hand, loved it. In Princeton he actively followed the smallest details of the construction of the Princeton MANIAC. According to Bigelow, its engineer, Johnny had learned all the electronic parts and supervised their assembly. When the machine neared completion, I remember how once he made fun of it at his own expense. He told me, "I don't know how really useful this will be. But at any rate it will be possible to get a lot of credit in Tibet by coding 'Om Mane Padme Hum' [Oh, thou flower of lotus] a hundred million times in an hour. It will far exceed anything prayer wheels can do."