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Recently in a documentary I heard that Newton's law of gravitation very well explains motion of two bodies such as sun and earth. And when applied to three bodies the answer is chaotic and not stable.

Then how do physicists explain the motion of higher number of bodies such as the solar system or just the three bodies of sun, earth and moon?

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    $\begingroup$ It does explain it, only that it's very difficult and we have to use numerical solutions. In fact, a simple computer program can give a good approximation. See en.wikipedia.org/wiki/N-body_problem $\endgroup$ – jinawee Dec 28 '13 at 0:34
  • $\begingroup$ @jinawee Could you please recommend a book to read about this? thanks $\endgroup$ – triomphe Dec 28 '13 at 1:04
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Here is an example of a (relatively) recent paper to deal with the many-body gravitational problem in regards to the orbit of Mercury. The upshot is that big masses are only slightly perturbed by smaller masses, but small masses can be strongly influenced by the behavior of larger masses. So think of it this way: the details of Jupiter's orbit are pretty messy because of the other planets in the solar system, but it can be approximated very well without worrying about higher-order perturbations (like the pull of Venus - as one example). On the other hand, Mercury's orbit is much messier because the other planets (Venus and Jupiter especially) have a big affect on its orbit.

Poincare famously showed that even in the limit of two bodies undergoing mutual gravitation without any perturbation, a test particle introduced into the system would exhibit (what would later become known as) chaotic motion.

So the answer is this: Newton's Law of Gravitation works fine for three or more bodies, but the system will generally have exponential sensitivity to its initial conditions (i.e. be chaotic). This is strongly evident in the orbit of Mercury, but less obvious in the orbit of, say, Jupiter.

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