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Why does the three-body problem have no solution? Nothing is wrong with not having found a solution yet, but I've heard that we've proven that there is no solution. I know that it is chaotic, but is it just that we have formulas to predict "perfect" environments but nothing realistic?

I'm only in calc AB (pre integrals), but if an explanation requires them I can figure them out.

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    $\begingroup$ What do you mean by "no solution"? $\endgroup$ – BioPhysicist Oct 27 '19 at 1:11
  • $\begingroup$ From what I understand, people say that there is no way to solve it, if that helps. Sorry, I should have done more research before this. $\endgroup$ – Mr Chikn Oct 27 '19 at 2:02
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    $\begingroup$ Numerical solutions are definitely possible. There just isn't a general solution your can write down that covers all possible scenarios. i.e. you can't write out $\mathbf r_i(t)$ for all bodies for a general scenario $\endgroup$ – BioPhysicist Oct 27 '19 at 2:21
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    $\begingroup$ What do you mean by "why"? Are you looking for some rule that automatically tells one that the differential equation system has no closed-form solution? Or are you looking for what it means to have "no solution". Have you looked at Wikipedia's articles on Three-body problem and Closed Form Solution? $\endgroup$ – Bill N Oct 27 '19 at 2:23
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    $\begingroup$ Solutions exist for certain cases, e.g. Lagrange points, but there is no known general solution $\endgroup$ – hdhondt Oct 27 '19 at 5:11
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This is based on two things: one, a less-clear idea of what constitutes a "solution" to a problem, two, poor phrasing in the term "no solution" (which might also be related to those using it having a poor idea of what "solution" means).

Mathematically, a solution of any equation is simply some setting of the variables that makes the equation evaluate to "true". For example, given the equation

$$x + 2 = 9$$

the assignment of the variable $x$ given by $x := 7$ is a solution, because when substituted, you get

$$7 + 2 = 9$$

which becomes

$$9 = 9$$

which becomes "True".

In the case of a physical equation, like the one describing motion, the relevant "variable" stands for a function giving the positions of the (here, three) particles at each point in time, i.e. a trajectory or history, and the "solution" is just an assignment of a given function to that function variable which, likewise, causes the equation to become true - that is, a trajectory representing a physically permitted movement under the circumstances described by those equations.

If such a function exists, a solution exists, otherwise, it doesn't.

The three-body problem, in this case, clearly does have a solution, because there is clearly a set of trajectories that three bodies subject to mutual central forces, like gravity, follow, and hence which must, as time-series functions, satisfy the equatiosn of motion. When it is said that it "has no solution", what is really meant is that there is no way to represent this solution in a particular form: namely that using a composition of some chosen set of commonly-available mathematical functions, which is often called a "closed" or "exact" form but which I believe these are poor choices of terminology - a better term I have suggested elsewhere as an answer is a "solution by special functions", where the "special functions" reference the chosen set. Typically, these special functions include the arithmetic operations, exponentials and logarithms, and trigonometry, and perhaps also some others, but do not include every possible function.

But they are actually only just one possible representation of many, and they may not necessarily be the best one. Unfortunately much of how maths is taught conveys the impression that they are somehow "better" than other ways of doing so, which is not necessarily the case.

There is no one "best" answer to how to represent the solution to an equation, or any mathematical object, and no representation should be thought of as being more or less "true" than any other. Instead, what we are really after is which representation is most informative to us about the problem. For example, in many problems involving periodic motion, a Fourier series can actually be considered a better representation, since it basically shows you how it deviates from an ideal, simple harmonic motion, but is not a solution by special functions in this sense since it involves an infinite sum (technically you could call infinite sums a function of "infinite arity", i.e. taking infinitely many arguments, but SBSF generally are thought of as including functions of finite arity [number of input arguments] only). And in this regard, we actually do have very good and informative "representations" of the solution of this problem which consist in comprehensive natural and mathematical-language descriptions of the various behaviors it exhibits in different areas, combined with algorithms to approximate the solution to arbitrary precision. In fact, in some cases, an approximate description can be considered more informative than an exact one by boiling things down to the "essential" features of the behavior only (and very often can be "converted" to an exact by simply specifying on top of it a correction procedure which converges to the exact solution).

Of course, this doesn't answer the core question yet, which is why specifically does a special-functions representation not exist. Well, the "best" reason is, generally, that representations by special functions are limited by the set of functions that you include in that allowable set of special functions: for example, if the set of special functions were only the arithmetic functions, so you could not write anything other than a combination of arithmetical operations like $\frac{x + 3}{2}$ and so forth, then even sine could not be represented, and hence the equation of motion for simple harmonics

$$m \ddot{r}_x + kr_x = (t \mapsto 0)$$

would have inexpressible solution (the history of the $x$-coordinate $r_x$) in the same way as the three-body problem. Conversely, you could annex the three-body problem as an explicit special function, but we'd consider that essentially trivial and not enlightening at all. The ideal for such solutions is to use a set of suitably general and simply-understood functions - and for the useful sets that we have, the three-body problem just doesn't enter into any of them. In a sense, reality has "too much freedom" for all its possible behaviors, across all circumstances, to be captured in such a simple way.

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  • $\begingroup$ One often sees language such as "no solution expressible in terms of elementary functions". Of course one always has purely numerical solutions. $\endgroup$ – ZeroTheHero Oct 27 '19 at 14:43
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    $\begingroup$ You might want to mention that there is an infinite polynomial series solution in powers of $t^{1/3}$ that was found by Sundman in 1912. The solutions converges way too slowly to be useful for calculations (needing $10^{8,000,000}$ terms for astronomical purposes by one estimate). $\endgroup$ – Mark H Oct 27 '19 at 23:38

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