So, the 3-body problem has no bearing on the solar system?
The "3-body problem" involves the finding of an analytic solution to the orbits a 3-body system operating under the Laws of Motion and Gravitation.
The sun and the each of the inner planets, Mercury, Venus, Earth, Mars, Jupiter and Saturn can be very closely approximated as six separate 2-body systems.
This is because the mass of the sun, being 99.8% of the whole solar system's mass, dominates the solar system's operation.
This explains why Keplers Laws, being only for separate 2-bodied systems, worked as well as they did.
The Sun-earth-moon would be better analysed as a 3-bodied problem but luckily again the earth-moon system can be quite closely approximated as a "double-planet" orbiting the far more massive sun.
So, the 3-body problem has little bearing on the solar system.
The Solar has the sun, 8 planets, earth's huge moon, quite a number of dwarf planets and numerous asteroids so a solution to the 3-body problem would not be a great help.
But people still work on it because that's what they do
!
Real simulations are done numerically using either Newtonian Laws or General Relativity and can predict behaviour years into the future.
The accuracy of that prediction falls of the further into the future they go. That is why the best that can be claimed is that, for example, an asteroid has a 1 in XXXX cnance of hitting the earth in yy years time.
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