What is a Three Body Problem?

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What is a Three Body Problem?

In: Physics

The three-body problem refers to taking three point masses (arbitrarily large masses represented as zero-size points) that have initial masses and positions and velocities, and solving for their movements.

A three body problem is basically when you have three masses/bodies with initial velocities and you want to calculate how they will move under only the influences of gravity and their initial velocities. Its a commonly talked about problem because there is no simple formula that you can make for it. You basically have to brute force the problem every time.

There are some problems in physics where you can come up with an exact solution or an equation that gives you a whole set of solutions. For example, if you are driving at a constant speed then the distance you have traveled after a given time is distance = speed * time. Similarly, if you drop a ball from a given height and can ignore air resistance then the position of the ball at a given time is height = initial height – 1/2 * g * time^(2) (where g is the acceleration due to gravity)

Other problems have no such solution available. For example, if we weren’t allowed to ignore air resistance on the falling ball then there’s no nice equation that tells us where the ball is at every instant. You can still calculate it, but you have use an iterative approach–if you know where the ball is at time T then you can compute where it’ll be a short while later and be pretty accurate. By using sufficiently short time steps you can get as accurate of an answer as you’d like, but there’s no simple formula.

In orbital mechanics the “two body problem” is where you have just two things in a hypothetical universe, one orbiting the other (or really both orbiting their mutual center of mass, but often we choose one object to be so massive that it barely moves; often this is a planet and moon, a star and planet, a planet and satellite, etc). The two body problem does have a number of nice equations to describe where you’ll find each of the objects at any given time, starting from some given starting conditions. An astronomer by the name of Johannes Kepler worked that out from observations, then later Newton came along and formulated how gravity works well enough to prove Kepler’s work mathematically. Similar to how you can ignore air resistance on a ball and wind up with an accurate enough result most of the time, in orbital mechanics you can ignore all other celestial bodies except the one you’re looking at and the biggest thing it’s orbiting and usually get a pretty good solution. Sometimes, however, you do have to consider all of the forces on an object. For that we turn to the three body problem.

The three body problem is similar to the two body problem, but instead of a hypothetical universe with just 2 objects we now have 3 (and could continue on to 4, 5, etc, all the way up to the real universe with an enormous number of objects). Unlike the 2 body problem there is not a nice equation that will tell you where to find any given object at any given time, but similar to our ball with air resistance we can take an iterative approach, simulating paths with shorter and shorter time steps to get more and more accurate results.

Note that the phrase “three body problem” is sometimes lifted from its physics application to describe social situations where the addition of a third person makes the social situation similarly difficult to solve, whether that’s a couple plus a single friend who gets in the way of couples activities or a trio of people where one has feelings for both of the other two.

So we’ve got this equation for gravity. And we’ve got this other equation for momentum. And we have enough other equations that if we have two things that are affecting each other out in space we know exactly where everything will end up at any given time.

For three things it all goes to hell.

So like if all there was was the sun and the earth we could run a simple equation and plot a perfect set of curves. That is we could plot an equation and depending on what we set time to we would get a place back out.

But because of the moon we can’t do that. We have to actually do a simulation instead. That is we have to advance time in little chunks and then look at the The position and direction of the earth the moon and the sun from scratch. And then we can move a little bit farther in time. But then we have to look at all the positions and start the math from scratch. And so on .

So the three body problem is the fact that we can build an equation for two things interacting in space because of gravity and just pick a date. But we can’t do like the entire solar system because there are too many things.

And it seems obvious that there are too many things.

But then we worked our way backwards and found out that three is too many things. So it’s called the three body problem.

This does not mean we can’t figure out where things are, where things were, or where things will be .

It means that we cannot easily figure out where things are, where things were, and where things will be.

So someone with pen and paper can do a good job of figuring out a theoretical solar system of one planet on a piece of paper.

But a solar system with two planets needs a computer in a simulation you can run forward and backward.

Note that this is also why you need a beefy GPU to play a lot of games. Particularly for ray tracing. But you can play those games in fact. For every frame it has to look at where the light is, where the thing is, and where the point of view is and calculate what color that pixel should be. And then each piece is moved individually according to its individual equations and inputs. And then the whole thing has to be done again for the next frame.

And lots of other things in the world have similar requirements for simulation rather than being expressed in a simple function over time.

If you have only two objects in space, they will move around each other due to gravity in well defined mathematical shapes like circles, ellipses, parabolas or hyperbolas. Relatively simple formulas can describe exactly where they will be at any point in the future and the amount of work required for the calculation doesn’t increase with time.

Adding a third object breaks all the above except for a small number of special cases. The only way the calculate the positions is to simulate the object’s motion in small steps. The amount of work required increases the further you go into the future and the results are only approximations with errors that increase over time.

In (simplified) gravity things pull each other towards each other.

If you have two objects each will attract the other. Which will make them accelerate, which will change where they are, which may change the direction they pull the other one in. This makes the maths of working out where each is going to be for all time a big complicated, but it is solvable (particularly if you are careful in which reference frame you choose).

With *three* objects, each will attract the other two towards itself, and each will be attracted towards the other two. So rather than just having to worry about two interactions (A on B and B on A) we have 6 (A on B, A on C, B on A, B on C, C on A, C on B). The maths becomes a lot more complicated, and we get what is called a *chaotic* system, where small changes in the initial conditions (where the objects start and how fast they are going) leads to very different outcomes. You can get some [really weird solutions](https://en.wikipedia.org/wiki/File:Three-body_Problem_Animation_with_COM.gif).

Unlike the two-body problem the three-body problem has no “general closed-form solution” – “general” meaning it applies in all cases (you just have to put in your initial conditions and will get an answer), and “closed-form” meaning it is made up of a finite number of expressions.

There is an infinite-series solution to the three-body problem, and it turns out that it converges in all non-trivial cases. There are also a bunch of special case solutions (including [some stable ones](https://en.wikipedia.org/wiki/File:Three_body_problem_figure-8_orbit_animation.gif)).

The three-body problem is a special case of the more general n-body problem, where you have n objects interacting.

The n-body problem is important in astrophysics because most gravitational systems have more than 2 things in them (the classic 3-body one being the Sun, Moon and Earth). Often they are arranged so you can ignore one of them at a time and reduce it to a series of 2-body problems, but sometimes you can’t – particularly not if you want good answers.

A lot of cool answers in here.

Here’s a simple one. Take two bouncy balls, drop one on the ground, and when it hits the ground, drop the second on top of it.

Which directions do you think they’ll go once hitting each other?

If you thought roughly up and roughly down, you’re probably right.

Now do it again, but this time have someone throw a third ball from the side. Any idea which way they’re going to go now?

Probably not. If you knew the exact velocity and direction of each ball when they made contact, you could probably do calculations to guess, though.

The third body introduces so much variability and randomness that we can’t forecast — we can only calculate.

Two body problems are straightforward and you can solve for all positions in one go.

Look at the Earth and Moon, you can take the masses, distance, and orbital velocities and plot out what the orbit will look like for a longggg time

Now lets add a third body to the problem, an asteroid on track to shoot between the Earth and the Moon.

If it takes *just* the right course it’ll continue on straight ahead and not deviate significantly because the gravity of the Earth and Moon will balance out.

If its a little too close to Earth it’ll hook more towards Earth as Earth’s gravity dominates the equation. The closer it is the stronger the hook.

Similarly if it is closer to the moon it’s orbit will bend towards the moon and change.

For the Earth-Moon system you could just take masses, distances, and orbital velocities and plot things out because they always orbit the center of mass. For this Three Body Problem you now need starting positions so you can figure out if the asteroid is passing through this sweet spot or if its going to hook left or right. You can’t figure out which way it goes without taking the initial conditions and stepping through it, and any error in initial position can result in a big difference in real position several orbits down the line

Just wanted to give a shout out here to the sci-fi novel by Liu Cixin – It’s called The 3-Body Problem’, and the problem itself is a big part of the plot. It’s also (along with the 2nd and 3rd book in the trilogy) one of the best sci-fi novels I’ve read in years. SO cool

I have a really smart 6 year old if I do say so myself. If you were to ask about this particular problem, I’d really only be able to draw it out. But here are a few words. The Moon is locked in orbit around Earth. You always see the same face, the light side of the Moon. Every night sometimes in the early morning, and sometimes in the early evening before the sun goes down, you see the Moon. And you always see the same face.

If the moon were more of a close star, we would have a really strange night and day because of how fast we would be moving. Our Earth and this star interact like magnets that you are pulling and closing the distance on in various intervals. Because of the way the planet in the star move around they never come in contact. So they just kind of bounce around. If we introduce a second star, not only is the Earth being bounced around, but the other star is being bounced around, which is just kind of crazy.

Then I’d show the Wikipedia page and the little GIF they have for it.

Done, 5yo have a 2 minute attention span.