Maybe there’s some astro-buff who knows how is it that:
The three-bodied problem allegedly drove Newton insane. Yet, we CAN predict where a planet will be in its orbit. This seems paradoxical to me. Maybe someone can explain how we predict the unpredictable. …Maybe the Bad Astronomer will stop by…
Thanks,
There is no analytical solution to the general 3-body problem. That is, you can’t write down an equation that describes the orbit. However, there are various tricks you can use. For example, if you have 3 bodies but one is much less massive than the other two, you can approximate the third body as a massless body and calculate the orbit. This is good enough when, for example, calculating the path of an itnerstellar probe or an asteroid.
You can also find numerical solutions (as opposed to analytical), which means you do a brute-force calculation and find the paths of the planets. This generally involves a computer. The result is not a nice equation, but a massive table of data telling you the positions of each planet at each time. This way you can simulate/predict the behaviour of whole galaxies or even groups of galaxies. You’re only limited by the computing power available. One of the fastest supercomputers in the world, the GRAPE-6, was designed specifically for this task.
I had to figure out where the Earth was once. It took me several weeks. Literally.
I used something called VSOP, which is an acronym for a bunch of French words. It is a vast set of numbers which basically describe a mathematical equation. The equation is a fit, an approximation, using the known positions of the planets over a long period of time. If you make the equation complicated enough, you can get a pretty good fit to the positions.
We were able to get the Earth’s position relative to the Sun accurately enough for our calculations, which was good, because it seemed ridiculous that it was so hard. 
Also, it’s impossible to make long-term predictions about the orbits of some planets, IIRC.
The allegation that the 3-body problem drove Newton insane is itself part of the lunatic fringe. This story is probably from what H.L. Menken called “The American Credo,” one part of which is that scientific geniuses all have a screw loose and are impractical dreamers. This might be part of an additional credo that those who are exceptional in one area are dumb as dirt in others, sometimes called the “it all averages out” theory.
What is the evidence that Newton was insane?
As others have said, there is no closed form (analytical) solution for the 3-body problem. However, there are several series or numerical methods, using present known positions as inputs, for getting precise answers for a considerable time into the future.
Someone please educate me here.
I am pretty well schooled in mathematics, and even had a basic Astro class in college, but why can’t you get an analytical solution for the 3-body problem? What is it about it that makes it unsolvable?
Thanks.
It was right under your nose the whole time.
For two body systems, the equations come out second order polynomials which can be solved using the classic quadratic equation. Note that fifth and higher polynomials have no analytic solutions. Three body problems, when you reduce to single variables, come out as such high degree polynomials, ergo no analytic solution.
Note that because of the ancient history of polynomials and trig functions, we are somewhat “pre-programmed” to think in those terms. There is no reason why analytic solutions can’t exist that just don’t use them. Perhaps someday a new class of functions will be discovered that allow reasonable to express solutions.
Somebody did find a solution to the general nth-degree polynomial, but the functions used are complicated at best. IIRC, it was done at the beginning of the 20th century. I’ll poke around mathworld and see if I can find a reference, but I’m pretty sure on this one, just cause it struck me as remarkable.
All right, so my claim was a bit too strong. Any polynomial that can be reduced to x[sup]p[/sup] + bx[sup]q[/sup] + c can be solved in terms of hypergeometric functions. I found it over here, down towards the end of the page.
But is the solution in a closed form?
IANAM but such functions are defined in terms of hypergeometric series which is an infinite series. So for numerical solutions we are back to numerical methods. Or do we?
For practical purposes, yes, numerical methods are where it’s at.
Let me point out that many hypergeometric series are useful functions (which, of course, we have to evaluate numerically, I suppose). Example: exponentials are hypergeometric, if memory serves. Many more hypergeomtric series (and multi-variate extensions) are not simple functions, of course.
I do have some questions, though, for ftg. What quadratic equation are you talking about for the two body solution? I don’t recall ever having to solve one to completely solve the two-body problem. Similarly, I’m not sure what higher order polynomial one has to solve for the three body problem. How do you even separate the differential equations? It’s trivial to get separable (albeit nonlinear) DE’s for the two-body problem, but I’m not aware of any way to do this for the three-body problem. I’m genuinely curious.
One other question… how many terms in a perturbation expansion would we need to get accurate answers for the planetary orbits? I mean, they knew about the precession of mercury’s orbit easily 100 years ago, they could calculate the part due to the effects of other planets, and I’m pretty sure they didn’t use computers to do this… 