From: John Clements
Date: 2021 Oct 24, 21:59 -0700
Well, if you were trolling for compulsive math over-explainers, you hooked me[*].
The three-body problem is indeed deeply significant in navigation, and points to a common misunderstanding about physics and math in general. After students have spent a few years studying how to differentiate and (especially) how to integrate, it's a good idea to ask them this: "what fraction of the differentiation and integration problems can we now solve?" The answer, of course, is zero percent. The fraction of problems that we can solve essentially rounds to nothing. It does turn out that we have nice "closed-form" solutions for some very important problems, like what happens to a rocket ship when we assume that the earth is a large mass that is traveling along a fixed path, but once you have (as the phrase goes) three bodies in your differential equations, there simply are no nice simple answers for any but a small set of near-trivial conditions (e.g., all three bodies are smashed together and stationary. I can solve that one for you.)
Instead, we typically fall back to what are called "numeric" solutions. That is, we use a number of well-developed approximation algorithms that allow us to simulate the motion of the bodies by (essentially) considering a single point in time, and the forces in that fixed frame, and accelerating each of the bodies *very slightly* in the directions indicated by the forces. The hard part is knowing how small you have to make your time increments in order to bound the total error, and many smart people know lots more about that than me. Uh... Navier-Stokes?
This, as I understand it, is the general technique used by our governments in order to anticipate the movements of the planets in the future, which does indeed affect navigation.
Now, are you going to tell me you already knew all of that? I hope not, but if so, I apologize for telling you again.
Oh, I also apologize in advance for anything else I got wrong. Thanks for letting me blather.
[*] I'm of course using the word "trolling" in its original internet sense, as popularized on alt.folklore.urban, back before usenet was ruined in 1995.