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One of the pertinent questions about many body systems that causes me much wonder is why the solar system is so stable for billions of years. I came across the idea of "resonance" and albeit an useful concept, it hardly explains the long stability of the solar system. Normally an N body problem with inverse square mutual interaction is an example of a chaotic system. Is there any real progress about this stability issue?

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It is important to note that chaotic systems can still have bounded orbits; chaos does not preclude stability. –  Chris White May 15 at 20:11

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Jack Wisdom at MIT has extensively studied the question of the stability of the solar system. He has a list of papers with links to freely-readable PDF files on his website:

http://groups.csail.mit.edu/mac/users/wisdom/

A good starting point might be "Is the Solar System Stable? and Can We Use Chaos to Make Measurements?" (PDF) (in Chaos, proceedings of the ''Joint Soviet-American Chaos Conference'' held at Woods Hole, June, 1989).

The abstract begins:

This talk addresses two separate questions: "Is the solar system stable?" and "Can we use chaos to make better measurements?" In the first part, a review is presented of the numerical experiments which indicate that the motion of Pluto, and indeed the whole solar system, is chaotic.

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A more recent analysis, extending through $5\cdot10^9$, years can be found at Existence of collisional trajectories of Mercury, Mars and Venus with the Earth. –  mmc Apr 21 '11 at 1:10

The classical work on the stability of the solar system was of course done by Laplace, see e.g. http://en.wikipedia.org/wiki/Pierre-Simon_Laplace#Stability_of_the_solar_system

Using perturbation theory he found an argument why the system Sun-Jupiter-Saturn is stable. However, nowadays people doubt the assumptions he made about the behavior of higher order terms; Instead high precision numerical simulations are used to predict the future of the solar system (links to this research have been posted in answers already).

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The sun contains about 99.9% of the mass of the solar system, so the motion of a planet is approximately that of a two body problem with $m<<M$, which is quite stable.

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Perturbations of one planet's orbit due to the others are small, but it's not at all clear that they wouldn't add up to produce instability over long time scales. As I recall, the long-time stability of the solar system was an active research question not all that long ago (the 80s and 90s?), but I don't know how or if it was eventually resolved. –  Ted Bunn Apr 18 '11 at 15:41
    
Experimentally it's stable for around 5Bn yr - the future is less certain! –  Martin Beckett Apr 19 '11 at 20:09
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What exactly is meant by stable here anyways? And what is a long-time scale? Long compared to what? –  user1708 Apr 19 '11 at 20:21
    
@Martin: the experiment is skewed by the boundary condition that we are here to observe it. Maybe the stability scale is only 100mn years, and we got lucky. –  Ron Maimon Oct 8 '11 at 21:57
    
@Ron well presumably it's been stable enough for 5Bn years to have the earth remain intact. And stable enough for life for 3.5Bn –  Martin Beckett Oct 8 '11 at 22:24

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