Sunday, March 11, 2012

Two is an Ellipse, Three Chaos

A while back I finished reading one of my stellar Christmas gifts, Mathematics: The Loss of Certainty by Morris Kline. It has the singular distinction among books I’ve read that about a month after I’d finished it, I began to read it for the second time again. Yes, I do; I read books over and over; the good ones; but usually each reading years apart. One of the intriguing discoveries I made in this one, early, is that there is no straight-forward way of calculating with precision the movement of three bodies linked by gravitational attraction, what is known as the three-body problem. Newton tackled this problem after he had elegantly solved the two-body problem—thus, for example, the  movement of the earth around the sun. In such calculations, one assumes that one body is standing still and then measures the other, and the upshot is that the moving body’s orbit is elliptical. But when minimally three bodies are involved, precise prediction of their interacting movements can never be produced simply by the applications of algorithms; only approximations are possible. Imagine that. We can observe, we can measure, but we can’t predict precisely using math.

However splendid the Internet, mathematical subjects are almost never covered well for the amateur. Virtually all are written by mathematicians for other mathematicians. I make this post to note an exception. It is the explanation of the three-body problem by Ask a Mathematician / Ask a Physicist (link). Very nice explanation. The last comment on this article suggests that the problem has actually been solved satisfactorily. That comment is best ignored. The solution there requires that we succeed in summing up an infinite series; but who has time to do that?

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