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In a paragraph written by a person emphasizing that rigour is not everything in mathematics (I wish I had written down the details), it was stated that the moon landings would have been impossible without the use of nonrigorous algorithms (I believe he was referring to the calculations of the trajectories of the spacecraft). What was the nonrigorous algorithm that was used? And why were rigorous algorithms insufficient? Was it because they were too slow on 60's hardware or was it because rigorous methods could not provide the answers required? And if such computations were to be done today, could they be done rigorously?

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This reminds me of the Turing-Wittgenstein conversation re: whether or not bridges would suddenly collapse if mathematics were found to contain contradictions: turing.org.uk/philosophy/ex4.html –  R Hahn Jan 19 '11 at 9:38
Perhaps the person who wrote that paragraph didn't understand the difference between rigour and exact solutions. If the numerical calculations that Arenstorf had done weren't rigorous, then we would have lost a few astronauts before realising what had happened. –  David Roberts Jan 19 '11 at 22:51
@David Roberts: "If the numerical calculations that Arenstorf had done weren't rigorous, then we would have lost a few astronauts..."? What?! You seem to be equating "nonrigorous" with "inaccurate", "unstable" or something similar...just because we can't prove something, it doesn't mean it's not true! –  Zen Harper Feb 25 '11 at 6:12
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up vote 27 down vote accepted

The existence of the Arenstorf Orbits was discovered in 1963 on the basis of numerical computations. The Arenstorf orbits appear as periodic solutions to the equations for the plane restricted three body problem (see the original paper by R. Arenstorf):

$$x''+2ix'-x=-\frac{(1-\mu)(x+\mu)}{|x+\mu|^3}-\frac{\mu(x+\mu-1)}{|x+\mu-1|^3},$$ where $x=x_1+ix_2$ is the complex position vector of the infinitesimal body referred to a co-system rotating with angular velocity $1$ about the center of gravity of the two attracting bodies of masses $1-\mu$ and $\mu$ ($0\leq\mu\leq 1$) as origin.

When $\mu=0$, the solutions describe the classical Keplerian motion. There are two types of the closed orbits which can be obtained by perturbation methods from $\mu=0$. The closed orbits of the first kind were discovered by Poincaré. They are close to circular Keplerian orbits. The closed orbits of the second kind (i.e. the Arenstorf 8-shaped orbits) are close to elliptic Keplerian orbits of arbitrary eccentricity.

The estimates showing that the Arenstorf orbits can pass very near the Moon and the Earth probably would not have been found without extensive numerical computations.

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Deleted my older comment, since I now think I see why it's non-rigorous: we cannot prove rigorously that the computation does converge to some limit function as $dt \to 0$, even though numerical evidence suggests that it does. But, if you assume the limit function does exist, doesn't it automatically have to be the solution, subject to mild conditions? So there shouldn't be anything wrong with using the computed solution, if we assume only that the limit as $dt \to 0$ really does exist. –  Zen Harper Feb 27 '11 at 12:27
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