Yes just to add to the other answers. They may have switched places in the past, in the early solar system, when the solar system was still forming, in the Nice model. (Named after the Nice observatory)
Nice model
But forward simulations of the solar system for billions of years show that the orbits of the outer planets are now stable. Page on arxiv.org
The orbit of Pluto is chaotic, in one sense. It's impossible to predict where it will be in its orbit over tens of millions of years,
Though the actual orbital resonance 3:2 of Pluto with Neptune is stable over billions of years timescales.
So - that means that the entire outer solar system is not totally predictable because of effects of Pluto on the other planets.
So - in principle that means that the other planets could be perturbed also. But in simulations, then the entire outer solar system is stable.
The inner solar system is another matter. Over very long timescales of hundreds of millions of years, Mercury could be ejected from its orbit due to resonances with Jupiter - very similar rate of precession of its perihelion. So if the two fall into sync, then that can lead to increasing perturbations of Mercury. It could hit the Sun, Jupiter, Earth, Venus, or be ejected from the solar system. And the result could also destabilize the orbit of the lighter Mars in some of the scenarios.
Those various scenarios are unlikely, 98% - 99% probability that the inner solar system remains stable.
There is also a very tiny 1 in 2500 chance of Mars passing very close to Earth or hitting it, or being ejected from the solar system leading also to other possibilities such as Mercury or even Venus hitting Earth. Earth-Venus smash-up possible
In detail
And in any case no way that tiny Mercury, or even Mars, could significantly change the orbits of Neptune or Uranus - in these simulations their orbits remain stable.
The other possibility is a close fly by of our solar system by another star, close enough to perturb the outer planets. But the chance of that is tiny.
There are many stars that will come close to our solar system, and every hundred thousand years one will pass close enough to disturb the Oort cloud and send more comets towards the sun. But the solar system, right out to Neptune, is a tiny target compared to the interstellar distances. Stars Passing Close to the Sun - and for perhaps the next one to pass close to our solar system: Scholz's Star: A Close Flyby.
In one of the comments on the centauri dreams article Scholz's Star: A Close Flyby. someone says that Jacques Laskar has done an integration investigating effect of stability of the solar system taking account of stellar flybys and that there was no major disturbance of the arrangement of planets over 5 billion years - but doesn't give a cite
I have posted a comment asking if he knows more, haven't found the paper in a search. Also, if anyone here knows more about this do say in the comments.
See also: Stability of the Solar System (wikipedia) and more detail and more technical: Stability of the solar system (scholarpedia)
Nice model
But forward simulations of the solar system for billions of years show that the orbits of the outer planets are now stable. Page on arxiv.org
CHAOS OF PLUTO ORBIT - BUT OUTER SYSTEM STILL STABLE OVER BILLIONS OF YEARS
The orbit of Pluto is chaotic, in one sense. It's impossible to predict where it will be in its orbit over tens of millions of years,
Though the actual orbital resonance 3:2 of Pluto with Neptune is stable over billions of years timescales.
So - that means that the entire outer solar system is not totally predictable because of effects of Pluto on the other planets.
So - in principle that means that the other planets could be perturbed also. But in simulations, then the entire outer solar system is stable.
IS INNER SOLAR SYSTEM STABLE OVER BILLIONS OF YEARS?
The inner solar system is another matter. Over very long timescales of hundreds of millions of years, Mercury could be ejected from its orbit due to resonances with Jupiter - very similar rate of precession of its perihelion. So if the two fall into sync, then that can lead to increasing perturbations of Mercury. It could hit the Sun, Jupiter, Earth, Venus, or be ejected from the solar system. And the result could also destabilize the orbit of the lighter Mars in some of the scenarios.
Those various scenarios are unlikely, 98% - 99% probability that the inner solar system remains stable.
There is also a very tiny 1 in 2500 chance of Mars passing very close to Earth or hitting it, or being ejected from the solar system leading also to other possibilities such as Mercury or even Venus hitting Earth. Earth-Venus smash-up possible
In detail
"Nevertheless, as predicted by the secular equations, in 1% of the cases, the eccentricity of Mercury increases considerably. In many cases, this deformation of the orbit of Mercury then leads to a collision with Venus, or with the Sun in less than 5 Ga, while the orbit of the Earth remained little affected. However, for one of these orbits, the increase in the eccentricity of Mercury is followed by an increase in the eccentricity of Mars, and a complete internal destabilisation of the inner Solar System (Mercury, Venus, Earth, Mars) in about 3.4 Gyr. Out of 201 additional cases studied in the vicinity of this destabilisation at about 3.4 Gyr, 5 ended by an ejection of Mars out of the Solar System. Others lead to collisions between the planets, or between a planet and the Sun in less than 100 million years. One case resulted in a collision between Mercury and Earth, 29 cases in a collision between Mars and the Earth and 18 in a collision between Venus and the Earth (Laskar and Gastineau, 2009)."
That's from the scholarpedia entry maintained by Jacques Laskar one of the authors of several of these studies. Stability of the solar system (scholarpedia)
And in any case no way that tiny Mercury, or even Mars, could significantly change the orbits of Neptune or Uranus - in these simulations their orbits remain stable.
CLOSE FLYBY BY ANOTHER STAR
The other possibility is a close fly by of our solar system by another star, close enough to perturb the outer planets. But the chance of that is tiny.
There are many stars that will come close to our solar system, and every hundred thousand years one will pass close enough to disturb the Oort cloud and send more comets towards the sun. But the solar system, right out to Neptune, is a tiny target compared to the interstellar distances. Stars Passing Close to the Sun - and for perhaps the next one to pass close to our solar system: Scholz's Star: A Close Flyby.
In one of the comments on the centauri dreams article Scholz's Star: A Close Flyby. someone says that Jacques Laskar has done an integration investigating effect of stability of the solar system taking account of stellar flybys and that there was no major disturbance of the arrangement of planets over 5 billion years - but doesn't give a cite
I have posted a comment asking if he knows more, haven't found the paper in a search. Also, if anyone here knows more about this do say in the comments.
See also: Stability of the Solar System (wikipedia) and more detail and more technical: Stability of the solar system (scholarpedia)
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
4.8m answer views110.3k this month
Top Writer2017, 2016, and 2015
Published WriterHuffPost, Slate, and 4 moreHuffPost, Slate, Forbes, Newsweek, Business Insider, and Science 2.0