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

if you take any 2 masses and put them in space you can perfectly predict exactly where they will be at any point in time ever.

as they orbit

if you try with 3 (or more) masses, you cant (outside of certain special cases). the best you can do is simulate it, which adds increasingly more error the longer out you simulate and the faster your simulation is.

I belive it has been proven to be unsolvable (as in, there is no general equation you can put the initial positions and the current time to get out the exact positions at that time) , but im not 100% sure, proving unsolvability can get strange

It’s not unsolvable, it’s just that there’s no (known) general solution. That is to say, each 3 body system has to evaluated individually, and for the majority, there is no long term stable arrangement. Additionally, our ability to predict the long-term state of it decays exponentially as three bodies introduce so many possible variables it quickly spirals beyond whatever present computing can simulate. Inevitably, some part of the system will decay. A body will be expelled, or two will collide, or something else. Even a presently stable system is extremely susceptible to minute external forces and can quickly collapse. There are a handful of known perpetually stable arrangements, but for any random 3 body system, you would not want to bet on it’s long term survival.

[https://en.wikipedia.org/wiki/Three-body_problem](https://en.wikipedia.org/wiki/Three-body_problem)

>In [physics](https://en.wikipedia.org/wiki/Physics) and [classical mechanics](https://en.wikipedia.org/wiki/Classical_mechanics), the **three-body problem** is the problem of taking the initial positions and velocities (or [momenta](https://en.wikipedia.org/wiki/Momentum)) of three point masses and solving for their subsequent motion according to [Newton’s laws of motion](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion) and [Newton’s law of universal gravitation](https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation).[^([1])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) The three-body problem is a special case of the [n-body problem](https://en.wikipedia.org/wiki/N-body_problem). Unlike [two-body problems](https://en.wikipedia.org/wiki/Two-body_problem), no general [closed-form solution](https://en.wikipedia.org/wiki/Closed-form_solution) exists,[^([1])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) as the resulting [dynamical system](https://en.wikipedia.org/wiki/Dynamical_system) is [chaotic](https://en.wikipedia.org/wiki/Chaos_theory) for most [initial conditions](https://en.wikipedia.org/wiki/Initial_condition), and [numerical methods](https://en.wikipedia.org/wiki/Numerical_method) are generally required.

…

There is no general [closed-form solution](https://en.wikipedia.org/wiki/Closed-form_expression) to the three-body problem,[^([1])](https://en.wikipedia.org/wiki/Three-body_problem#cite_note-PrincetonCompanion-1) meaning there is no general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases.

More history of it in the “n-body problem” page

[https://en.wikipedia.org/wiki/N-body_problem#History](https://en.wikipedia.org/wiki/N-body_problem#History)

There is an analytic solution but it’s not practical…

>That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman’s series were to be used for astronomical observations, then the computations would involve at least 10^(8,000,000) terms.

Object A and B in space pull on each other because of gravity. We’ve got this one down easily – the math is relatively straightforward, and very very predictable.

Now add a third object.

We know how A and B pull on each other, easy peasy! Wait though, C is pulling on A and B, so re-do all the math because of that.

Wait, A is pulling on C too, so that changes things, so recalculate again.

Wait, B is pulling on C too, so recalculate again.

Wait, that changes how C pulls on A and B, so recalculate again.

Wait, that changes how A pulls on B and C, so recalculate again.

Wait, that changes how B pulls on A and C, so recalculate again.

Wait…

It’s a problem in modelling the behaviour of three or more objects interacting through gravity (but also electromagnetism).

Let’s start with the thing that isn’t a problem: Two bodies. With two bodies you can mathematically create an exact equation, with an exact solution, of how the two objects will orbit each other. The only variable left is time and you can go infinitely into to the past or future. With infinite accuracy.

When you try to do the same for three or more bodies you run into a problem. There’s more variables/unknowns than equations. In maths, this means you cannot have an exact solution where you simply go forward and backwards in time. There’s no “analytical” solution.

That’s not to say there’s no way to solve it at all. But it requires making some guesses and then running the maths over and over hoping it settles on an approximate solution. Then you advance a time step and do it again. And again.

And “approximate” and “time step” are key words here. A solution, not THE solution. An approximation. And with a resolution in time, the more resolution you want, the finer you have to make your steps, the more work you have to put in. And even then you will always eventually diverge from the true solution if you run too far into the future, as the errors accumulate.

It’s not that it’s unsolvable. We don’t have the solution because it’s extremely complex because the result is very sensitive to initial conditions.

In 2 body problem (for example, Moon orbiting around Earth), you can predict very accurately where the Moon is exactly at any given point in time because the orbit of the Moon is not affected to a significant degree by any third body.

But in a 3-body problem, body A affects body B, body B affects body C, body C affects body A, which affects body B, so the position of bodies at some future point in time is very sensitive to initial positions.

Imagine a simple pendulum. Just a simple ball hanging off a simple string. With the pendulum formula, if you know the initial position of the ball, you can predict where the ball is going to be after, say, 10 seconds of swinging. If you **slightly** change the initial position of the ball, the position of the ball after 10 seconds is also going to **slightly** change.

However, if you have a double pendulum ([Double Pendulum (youtube.com)](https://www.youtube.com/watch?v=U39RMUzCjiU), if you **slightly** change the initial position of the pendulum, the position of the pendulum after 10 seconds is going to be **very** different, which makes it very difficult to solve mathematically.

In short, in the show (and the books) it’s wrongly stated that it’s “not solvable” or that “it’s unpredictable”.

The 3 body problem, or rather N body problem, because it’s the same for any number greater than 2, is the problem of, how will the bodies move, if they all interact on each other with attractive force. The N(3+) body problem does not have a so called “General analytical solution” which means you cannot find a mathematical formula for the curves of your bodies. For two bodies you can do this. This means that the problem has to be solved numerically, which is an approximate solution.

Numerically solving something means you take the positions of your bodies, like a snapshot, and calculate all the forces that they interact with, then, you calculate how much they will move in a brief time period if those forces were constant. Then you use this new snapshot to calculate new forces, then again, and again.

The challenge with this type of solving is that it depends a lot on these time steps, and how you calculate the things, so, sometimes, you can acumulate a big error. However, for somethiong like a star system, a civilization as advanced as the Trisolarans, should have computational power to estimate the evolution of their system very, very, very accurately on the scale of millenia.

* “Body” is just a physics word for ‘thing made of matter’.

* Things made of matter typically (always?) have mass.

* Gravity is a property of objects with mass.

* We have physics theories that allow us to apply powerful mathematics to accurately describe how objects move due to gravity.

* So, therefore, you’d *expect* that we can just consider any group of ‘bodies’, apply our mathemtatics to them, and now accurately describe how objects move.

but there is a problem here. The mathematics is hard to *exactly* solve.

* If you have 3 things with mass, it turns out that we usually cannot calculate how they’ll move. We can approximate it, but eventually the approximation will be wildly wrong.

* (In some special cases we might be able to solve it, like if you imagine them in some perfectly symetrical scenario, for instance. But in general, we cannot solve it.)

* ~~It is possible that a reliable solution exists, but we haven’t found one, and for all we know, it might be impossible.~~

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