Saari and Xia’s “Off to infinity in finite time” (1995) is another terrific survey on this topic.

@article{Saari:1995aa, Author = {Saari, Donald G. and Xia, Zhihong}, Journal = {Notices Amer. Math. Soc.}, Number = {5}, Pages = {538--546}, Title = {Off to infinity in finite time}, Volume = {42}, Year = {1995}}

my understanding is that there is some major link of this problem (3 body problem) to the new field of chaotic dynamics with findings of instability & which also relate to the solar system stability. there the lorentz equation is the prototypical case study. re galactic algorithms, an expanding subj esp in the modern age with mass TCS research scale. it seems one could write an entire book on the subj.

but lets face it the concept of “closed form solution” is both expansive and a bit primitive. even simply finding roots of polynomials barely adhere to that right? also there is no “formula” for primes, except there are many, many algorithms. whats the difference? the distinction blurs an awful lot in the 21st century & probably will continue to blur.

I don’t think it is quite true to say that Kepler solved the 2-body problem before Newton. Kepler’s “solution” was observational, that is, obeyed by the sun and a planet to a relatively high degree of accuracy (based on Tycho Brahe’s data). Newton’s job was to find the equation which led to this solution, as he famously did.

I must be missing something about the first “impossibity result”. Stated as it is, wouldn’t it suggest that even $2$-body problem is hopeless, as for $N=2$ you get 12 variables but only 10 integral equations?

Yes I understand — with WordPress, it’s so annoying.

To Dick and Ken: Would you please be able to install a “preview” button so that a poster can preview, before clicking on “Post Comment”? This would be ideal.

Saari and Xia’s “Off to infinity in finite time” (1995) is another terrific survey on this topic.

`@article{Saari:1995aa, Author = {Saari, Donald G. and Xia, Zhihong}, Journal = {Notices Amer. Math. Soc.}, Number = {5}, Pages = {538--546}, Title = {Off to infinity in finite time}, Volume = {42}, Year = {1995}}`

my understanding is that there is some major link of this problem (3 body problem) to the new field of chaotic dynamics with findings of instability & which also relate to the solar system stability. there the lorentz equation is the prototypical case study. re galactic algorithms, an expanding subj esp in the modern age with mass TCS research scale. it seems one could write an entire book on the subj.

but lets face it the concept of “closed form solution” is both expansive and a bit primitive. even simply finding roots of polynomials barely adhere to that right? also there is no “formula” for primes, except there are many, many algorithms. whats the difference? the distinction blurs an awful lot in the 21st century & probably will continue to blur.

I don’t think it is quite true to say that Kepler solved the 2-body problem before Newton. Kepler’s “solution” was observational, that is, obeyed by the sun and a planet to a relatively high degree of accuracy (based on Tycho Brahe’s data). Newton’s job was to find the equation which led to this solution, as he famously did.

I must be missing something about the first “impossibity result”. Stated as it is, wouldn’t it suggest that even $2$-body problem is hopeless, as for $N=2$ you get 12 variables but only 10 integral equations?

My apologies for the two typos (“impossibility”, and “even the -body”)…

Yes I understand — with WordPress, it’s so annoying.

To Dick and Ken: Would you please be able to install a “preview” button so that a poster can preview, before clicking on “Post Comment”? This would be ideal.