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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 '14 at 20:11

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:

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

If you want to see what the current state of the art is you could do worse than watch Sean Raymond's presentation at "Protostars and Planets VI" from 2013. You can find the actual write up here. Or from the same conference there is Melvyn Davies' review of the long-term dynamics of planetary systems. The talk can be seen here. I think this review does contain the sort of information you are looking for. Certainly it discusses the past and future evolution of our solar system, as well as planetary systems in general.

A brief summary would be that the solar system is probably stable for the remaining lifetime of the Sun. However, there is the intriguing possibility that Mercury could fall into the Sun or collide with Venus in the next billion years or that Mars could be ejected from the solar system on a similar timescale (e.g. Battygin & Laughlin 2008).

These questions cannot really be settled using old analytical arguments - the approach of Lagrange and Laplace predicts complete stability but ignored the idea of resonances between the planets. The reason that our solar system appears to have been stable on timescales of billions of years is that none of the planets are in orbits that resonate with each other - and that would lead to chaotic, unpredictable behaviour. Now predictions are made from the results of extensive suites of N-body simulations that simply integrate forward in time starting from initial conditions that sample the (very small) range of uncertainty in the present state of the solar system.

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The classical work on the stability of the solar system was of course done by Laplace, see e.g.

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
What exactly is meant by stable here anyways? And what is a long-time scale? Long compared to what? – TROLLKILLER 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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