The Three Body Problem

One of my favorite parts of physics is the three body problem. It’s a perfect example of the beauty of nonlinearity. The three body problem is as follows: let there be three objects (like stars) in space. They all have mass and are pulling on each other via gravity. Given starting position and velocity, what are their trajectories? It’s an interesting problem because it’s much more complicated than it looks. The one body problem is trivial, and the two body problem is easily solved, but three bodies are chaotic. “Chaos” is a mathematical term that boils down to meaning “not actually random, but so complicated it might as well be.” Its hallmark is what we call “sensitive dependence on initial conditions.” A system that exhibits sensitive dependence on initial conditions will wildly change its behavior at the slightest touch. This is also known as the butterfly effect.

How does this show up in the three body problem? Well, it’s easy enough to see in a proper gravity simulator, such as this one, which I used to make the images in this post.  Put any two objects in, and you get three possible outcomes: the objects orbit, one slingshots the other into deep space, or they collide. The system will pick one of these choices very quickly, and using math, you can predict with perfect accuracy which one it will pick.

Two Body Systems
Two Body Systems

2-body systems. Click to enlarge.

With three bodies, however, the dynamics are much more complicated. Each body can still only be in one of the three states: slingshot, orbit, or collision. But in a chaotic system, how it picks those states changes. Collision is the only stable, irreversible state. Once picked, it is final. Slingshot is almost always final, in that a planet flung out into deep space is highly unlikely to interact with any of the other bodies again.

Last, orbits. There are two types of orbits in a chaotic system. There are the extremely rare and practically impossible to find chaotic orbits, where the system, after eating or spitting out some of the bodies, will let the remaining bodies twirl around literally forever. This is what passes for “stable” in a chaotic system. (The stability of a single orbit relies on the stability of all orbits in a system.) The resulting orbit will be a strange attractor, a fractal, impossibly complex (and beautiful). It will never quite repeat itself.

The second type of orbit is unstable, meaning it eventually results in a slingshot or a collision. The tricky part about unstable orbits is that they’re indistinguishable from a stable orbit until something goes terribly wrong. Sometimes it goes wrong quickly enough to be obvious, but it could take literally any finite length of time before a slingshot or collision occurs. Ten seconds, a billion years; it’s all the same and you can’t tell which it’s going to be until it does.

3 Body Systems
3 Body Systems

3-body systems. Click to enlarge.

The problem is that of sensitive dependence to initial conditions. That butterfly doesn't just cause hurricanes; a flap of its wings can knock a planet out of orbit. Even the slightest hair of a change in initial conditions makes the difference between a stable chaotic orbit and one destined to result in a collision. Since stable initial conditions are almost impossible to find, it’s a sure bet that any system you make in the simulator will collapse eventually back down into two and one body problems.

The beauty of these complicated systems is the chaotic dance they go through, the beautiful spirals and curves that are generated. No two dances are the same, and they never repeat. These systems are intractably mysterious; only in rare and specialized cases can we actually predict trajectories with precision.

This unstable chaotic dance is what occurs in our universe constantly, in all astral body systems from galaxies down to dust particles in Saturn’s rings. This is why the three body problem is a favorite of mine: it is the introduction to chaos. Chaos is visually and mathematically beautiful and intriguing. And, hey, who doesn’t enjoy smashing a few planets together in a gravity simulator every once in a while?

Note: Images made with this simulator. Go play with it.