# Why is the 3 body problem, a problem?

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I was in physics and had a brief conversation on chaos theory and we started talking about space and he briefly mentioned about the 3 body problem.

Thing is, everything interacts with other things for a reason right? I understand it’s complicated, but if you know all the necessary data, why can’t we do it?

In: Planetary Science

The issue is that over time very small factors have very large effects.

As an example, if you were calculating the collision of billiards balls by the time you calculate out to the 7th collision you need to have considered the gravitational effect of the people in the room to continue making accurate predictions.

So the issue is never predicting what will happen *tomorrow* in the three body problem (assuming a solar system), the issue is that a passing comet might change the system 20 million years from now in a way that is essentially impossible to account for.

For 2 bodys you can derive a formula that describes how the two bodies will behave for every constellation and every time. That is called an analytical solution.

For 3 or more bodies there is no such formula. However you can still predict how the planets will move. Basically if you know the position of the planet you can calculate where it will be shortly afterwards. You can use this new position to calculate the next step and so on.

That way you can still calculate a movement. But instead of a general formula, you only get a list of numbers, which are specific for this situation.

But that is not really a problem for real life applications. Most real life things can only be calculated numerically, anyway. Analytical solutions only exists for very few situations.

That many 3 body problems tend to be chaotic, means that a small change in the initial situation can lead to large differences in the outcome after some time. Depending on your situation and your timescales and what you want to achieve with your calculations this might or might not be a problem.
And also there: many systems behave chaotic, that’s nothing too special about the three body system. The interesting thing is, that it already occurs in such a simple system. For complex things like the weather it seems more obvious that It behaves chaotic

We can do it if we know all the date. The issue is that we dont know all the date because minor changes will have a huge effect over time with 3 bodies. With two bodies even with imperfect data, your results will be fine. Or even in a 3 body, one can be not that impactful so it doesnt really matter. For instsnce the earth sun and moon act the same way. But the moon is small and the effect it has on the other 2 bodies is very minimal, making it a non issue to calculate.

If you know the initial positions and velocities of TWO bodies, then Newton’s Law of Gravity can be solved into a neat formula, so that you can predict the position of both bodies at any arbitrary point in time merely by plugging in the appropriate values.

With THREE bodies, the interactions are too complex for a neat formulaic solution. The only way to calculate their positions is to numerically solve the equations over and over, iterating forward one small time step at a time. Essentially we can’t predict in advance how the system will behave, you have to just let it play out and see what happens.

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The issue with chaotic systems such as the 3 body problem or the double-pendulum is that you can never know the ‘necessary data’ to a sufficient degree of precision that the behavior is predictable long term. Eventually the tiniest uncertainties in your initial measurements of position and velocity will eventually lead to completely different results.

The only problem is that physicists wish they had a simple equation for the behavior.

Not everything boils down to a simple equation.

We can absolutely predict the result numerically.

It’s literally fine.

So the problem is in how you solve it, mathematically.

With only two bodies, you can have an “analytical” solution, that is you can simplify the system of equations into a single equation with a smoothly varying time, to the point where if you give some initial configuration, you can then get a solution for any past or future time. Like an animation where you have time on a slider.

This is not the case if you introduce even one more interacting body. At that moment you have too many variables to simplify the equations into a nice “analytical” package.

What is instead done in that situation, is you kind of solve it by brute force with an approximation. It’s called a numerical solution. You just give it some initial best guess values, run the calculation over and over until the result converges on a stable value within a tolerance you decided.

That’s only your first time step. You then move on to the next time step. You decide how big of a step, the smaller, the better the precision, but the more calculations you do overall. And precision matters, because the error compounds with each time step. At some point the solutions will be just plain wrong, and it’s up to you to know when.

As you can hopefully see, you go from “basically can get a highschooler to calculate by hand” to “will make a computer sweat” in one simple move.

The other issue is, the 3 body problem is very sensitive to the initial configuration, so I’m not surprised it came up in the context of chaos. Small initial changes mean big difference down the line. And that’s fundamental , even without the compounding error of the numerical solution.

Imagine you have three objects A, B, and C that each exert a force on each other based off of their distance. Let’s say they repel each other, and the repelling force gets stronger the closer they get.

You want to know how object A will move over time. In order to know this you need to know what forces are acting on A. In order to know that you need to know where B and C are as you need know how far they are from A. So to find how the force changes over time you need to know how B and C move over time. But now our argument goes in a circle, because to find how B moves over time you need to know how A and C move over time. To find how C moves over time you need to know how A and B move over time. ect. So generally you can’t exactly calculate the movements and have to approximate things.

Because all of these objects are moving independently the forces they exert on one another change widely based off of their movements, and no consistent behavior arises. This is where chaos theory comes in. This inconsistency ends up meaning that changing initial conditions even slightly can result in wildly different behavior. This includes errors introduced in the approximations you had to take to calculate things. So it ends up being basically impossible to actually calculate how the system will change over time.