The two-body problem can be completely solved via two one-body problems, which only uses the four basic binary functions. However, the three-body problem cannot be solved with these functions and first-order integrals. So I am wondering, is there any finite numerical solution to the three-body problem?
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$\begingroup$ Israel's Technion researchers have found an effective solution to the famous age-old, three-body problem in physics. in a paper recently published in physical review with the current study by ginat and perets, the entire, multi-stage, three-body interaction is fully solved, statistically. scitechdaily.com/a-centuries-old-physics-mystery-solved $\endgroup$– AlexCommented Dec 10, 2021 at 2:04
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$\begingroup$ I just ran across this and the answer. I can't up vote either of you again, but I wonder if the answer is good enough to be accepted? $\endgroup$– uhohCommented Feb 3, 2022 at 10:15
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1$\begingroup$ @uhoh Yes, just waited a little too long to accept the answer and it got lost in my inbox. $\endgroup$– WarpPrimeCommented Feb 3, 2022 at 13:38
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$\begingroup$ @fasterthanlight I do that too, and in fact let many of my questions "age" for months or even longer sometimes. This one is so cool however that I'm really loving it! I added a comment under DH's answer btw. $\endgroup$– uhohCommented Feb 3, 2022 at 13:50
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1 Answer
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In a sense, even solving the two body problem as a function of time is unsolvable in terms of the elementary functions. The problem is that the solution involves the solving for the inverse of Kepler's problem, $M = E - e \sin E$. This inverse function is transcendental and cannot be expressed in terms of the elementary functions. That said, it's fairly easy to solve for $E$ to an arbitrary degree of precision.
Regarding the three body problem, there are two special cases that are trivially solvable. These are the triangular Lagrange points (L4 and L5) for the restricted circular three body problem. The three linear Lagrange points are solutions to fifth order polynomials, and the solutions cannot be expressed using the elementary functions. But once again, it's fairly easy to solve for the locations of the three three linear Lagrange points using approximation techniques.
Those restricted three body problem Lagrange points are special cases. The generic case of the three body problem is notoriously unsolvable in terms of the elementary functions. There is an infinite series solution, but nobody uses it. Over a century ago there was a prize for solving the three body problem. Karl Frithiof Sundman was awarded that prize in 1912 for showing that a solution exists in the form of an infinite series in which the terms are ever increasing powers of the cube root of time. The reason no one uses this solution is that the number of required terms can be huge. Eight million terms is not near enough. It's more on the order of $10^{8000000}$ terms.
answered Dec 10, 2021 at 0:15
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$\begingroup$ only slightly related is this Math SE answer to How do we know that is not possible to invert 𝑥=𝑡+cos𝑡 analytically? where it talks about "The space of mathematical expressions can be represented as a syntax tree. As well, algebraic equivalences can be represented as modifications on that syntax tree." $\endgroup$– uhohCommented Feb 3, 2022 at 13:50