I recently finished watching the 3 Body Problem on Netflix so this question came to mind. Can anyone explain (in simple terms) why the 3 body problem was deemed unsolvable even by the advanced alien race in the series? Even better, can anyone here simplify what the 3 Body Problem actually is in Physics? It really got my curiosity. Thanks! 🙂
In: Physics
A three body system is an example of a **chaotic** system. This means that while we technically know how the system behaves and can simulate its future states (in this case we know all the equations of gravity), a very very small change in any input variable can drastically affect the final output to the point of it being meaningless. This means to perfectly predict such a system into the far future we have to calculate its current state to effectively infinite precision, which is impractical/impossible.
All the other answers eplain really well why there is no (analytical) solution to the problem. What I’d like to add, and it is a minor spoiler in the series, is that >!the aliens decide to leave their planet not because they can’t predict with a decent margin of error the position of the suns but because their three body system isn’t in a stable configuration which means that eventually two stars may collide or one may leave the gravitational pull of the others meaning they could either freeze to death or fall into a star or any other crazy less than ideal situation!<
I just did some research on this and want to throw this out there, framed in a way that makes sense to me but different from the other comments:
We know the problem has not been or cannot be solved in closed form, but can be projected forward with math to any point in the future.
In an idealized scenario (e.g., three digital bodies where we know the exact mass, position, and velocity), this can be projected with 100% accuracy to any point in the future.
The issue really is one of measurement constraints. For example, if the mass is a tiny bit less than assumed, then it will cause an error that magnifies over time.
Like, there might be a point in the interaction where mass A by the skin of its teeth won out in pulling C towards it rather than B. If A was a little less massive, it might lose the tug of war then set everything on a radically different trajectory.
So not only does a measurement error magnify over time, the magnification itself progresses chaotically.
It’s pretty easy to use Isaac Newton’s basic laws to predict what will happen when two objects come into each other’s gravitational range. They will pull on each other. Maybe come into some kind of orbital arrangement. It just depends on their size and mass and movement.
However, just one more object introduces enough complexity that it becomes impossible to model what will happen with these same simple methods.
The “problem” has two meanings. On the one hand it’s a math problem/question which is hard to solve. On the other hand it’s a difficult situation where our tools break down and if we ever actually need to solve one of these, we’re in a bind. That’s an unacceptable risk, and that’s a problem.
Other people here in this thread already explained it well, so I will just add my observations.
The problem is, most of the 3 body systems are not stable. With two bodies – typically one massive – such as a Sun and other much smaller, such as a planet, the system is stable, so it is possible to predict their behavior far into the future based on measuring of their initial positions and velocities. With three bodies, the smallest deviation in starting conditions might grow exponentially over the time, so it is impossible to predict where bodies would be in future.
This is best demonstrated by a double pendulum – also sometimes called a chaotic pendulum. Have a look at some videos talking about this. It is much easier to see.
The Sun, Earth and Moon are three bodies, with other planets adding complexity, but the Earth + Moon behave towards the Sun as a one tiny body for practical purposes of us predicting their rotation far into the future.
Isnt there a solution atm?
I think this being the crux of the show/1st book is meant to say a few things. 1st, that San-Ti math ability is not any more capable of solving the problem that our own, even with much more advanced computers. But even if they have solved for when the next chaotic era would begin, it’s a moot point. They wanted to prevent the fall of their current civilization, and eventually the planet itself would be destroyed no matter if they were able to solve for the mathematical problem.
Here’s one making ths rounds this afternoon.
https://www.reddit.com/r/oddlysatisfying/s/ug2ludiNmK
Any small perturbation in initial conditions would lead to completely different solutions.
Just look up a chaos pendulum, that’s essentially the three body problem. A simple pendulum is incredibly predictable to math out but a chaos pendulum… well they live up to their name. I don’t think we’ll ever have computers that can model a chaos pendulum’s actions out indefinitely, I think the best you can do is take the masses and vectors and project a possibility into the future and that prediction relative to reality will always grow exponentially apart the farther into the future you make it.
But I’m no mathematician, I’m just a boob with a keyboard, perhaps predictive modeling is much better than I give it credit.
Two bodies orbiting each other under gravity have a general analytical solution for their orbital dynamics. As such, if you know the starting conditions, you know how they are going to move going forward with certainty for all time.
However, no such general analytical solution exists for the movement of three bodies. There are lots of special cases with approximate solutions that are analytical or semi-analytical, but all of them require certain simplifications, so in realistic systems you end up with chaos, instabilities, and overall qualitative changes in the movement patterns over time. One way to deal with it is to use basically brute force and solve the orbital dynamics numerically with computers – but there, again, you run into the issue of having limits to accuracy and resolution.
In short: The two body problem can be solved exactly, the three body problem is impossible to be solved exactly.
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