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    Re: The Three-body Problem
    From: Alexandre Eremenko
    Date: 2021 Oct 25, 08:46 -0700

    Yes, 3 body problem is significant for navigation. Or at least was significant, historically. The short story is this.
    In 17 century the problem of determining longitude by astronomical methods became important. Why this is so, most members of this community know very well, and I will not repeat it.

    In principle, two methods were proposed:
    1. Make a good clock which could keep time when transported.
    2. Use pure astronomical methods. 

    When Newton was asked which method is more likely to lead to success, he said: "astronomical methods", he did not believe that good enough clock could be made. Of the pure astronomical methods, 2 were proposed:
    a) Based on eclipses of Jupiter satellites,
    b) Based on Lunar distances.

    The Jupiter satellite methods was the best available for use on land, but not practical on ship. To observe eclipses of Juliter satellites, one needs to stabilize a large telescope on a ship, and this was found to be impossible.

    For Lunar distances, one needs tables of position of the Moon of high accuracy for t least few years in advance. The motion of the Moon is quite irregular, and such tables could not be made without a good theory of motion of the Moon. Such a theory is possible based on the Gravity law of Newton, and it is he 3 body problem. The 3 bodies are earth, Moon and Sun. The problem is to solve the equation of motion of three bodies under gravity.

    Newton himself could solve 2 body problem, and thus derive Kepler Laws. Most planets obey Kepler Laws with very high accuracy (the 2 bodies are Sun and the planet). Jupiter satellites also obey Kepler Laws (the two bodies are Jupiter and a satellite, the influence of the Sun is negligible. But the case of the Moon is a genuine 3 body case: we cannot neglect either of the three bodies.

    The first attempt to solve  this problem was made by Newton himself, but his results were not precise enough for navigation. Then several 18 century mathematicians contributed, especially Clairaut and Euler. Based on Euler's theory, Tobias Mayer finally created tables which were sufficient for navigation purposes.  Approximately at the same time, chronometer was invented, so Mayer obtained only a small part of the Longitude prize. Still smaller part, 500 pounds went to Euler.

    Solution of Euler and Mayer was partially analytic and partially numerical. It was sufficient for practical application of Lunar distances, but mathematicians continued to study the problem until the end of 19 century, hoping to obtain some neat analytic solution. In the end of 19 century it was shown by Poincare that this is impossible.

    Alex.

       
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