Put another way, while their positions are one set of 3 points, their momenta are another set of 3 points, and there is no requirement that 6 points will always lay on the same plane.
I wonder what phantom forces would appear when the reference frame changes in some complicated fashion. We get centrifugal "force" when we reconstruct F=dP/dt in a rotating reference frame, what would the 3-body "force" look like?
Oh makes perfect sense, do you have thoughts about real life examples? I did some research using AI and it said there were examples of restricted 3 body problems like the trojan asteroids, but no examples in real life similar to what is in this web app
Why would the plane keep changing? If there are only these three objects, won't the vectors of their gravitational pull to each other all be on this plane too?
One of the necessary conditions here is that the three objects return to their exact initial position, and so does the centre of mass. Initial conditions with non-zero momentum must trivially be ruled out. But this doesn't stop them from having velocities perpendicular to the initial plane that cancel out perfectly, so this doesn't refute the assertion that the planes keep changing.
They presumably meant non-zero total momentum. If the total momentum were non-zero, then the center of mass will be moving in a straight line, and will not return to where it began, and therefore the orbit would not be periodic.
Big picture the center of mass momentum is concerned and it does not matter if the system as a whole is moving up or down or to the right or the left. Like the Earth is basically orbiting the sun in an ellipse [1] so far as the sun is concerned and from the viewpoint of the solar system not care so much that it is moving around the galaxy unless we are interested that orbit being perturbed by other stars that we pass near over millions and milions of years.
[1] ignoring the parameters of that ellipse changing slightly and slowly thanks to the other planets
Oh sorry, yeah - I wasn't fully awake yet when I wrote it. As a sibling comment mentioned, I was talking about the total momentum. Anyway, it was in response to GP's assertion that GGP may be correct if "we're centered on the CG", which doesn't make sense because the CG can't move.
We have no observed examples in nature of three body equilibrium. But then again, all places we have looked are either influenced by the chaotic orbits around them of the Solar System, our surrounding galaxy, or nearby galaxies in a cluster.
There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.
> There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.
This is of course a relative statement. Every object affects every other object, subject to the limitations of lightspeed propagation of gravity waves through expanding space.
But as you point out, we still haven't noticed any examples that are stable short-term.
That depends on what you mean by stable short-term.
The Sun, Jupiter, and any small asteroid at L4 of L5 is, by itself, linearly stable. Meaning that any small perturbation will only grow linearly.
But, of course, this system interacts with Saturn. The arrangement of those three objects is still remarkably stable. But the interaction with Saturn makes for a chaotic system again.
