What sort of beast is a choreography?

Star dragging in action

Image source: Memory Alpha

Imagine that you and a couple of friends get bored one day, so you decide to play a game. Each of you hops in a spaceship and grabs a star. You drag them out to a quiet corner of the universe, set them up in some configuration, and then let them go and see what gravity does.

How should you set up your three stars?

You might try putting them in a straight line and starting them off standing still. If you did that, then they would be pulled into the middle and eventually collide.

Stars in a line Stars in a line, getting closer Stars in a line, about to collide Colliding stars

The star graphic is from a Hubble image of our closest interstellar neighbor, provided by NASA. The collision image is from a Harvard press release.

You might instead try making a regular triangle and, again, starting the stars off standing still. They would still end up colliding.

Stars in a triangle Stars in a triangle, getting closer Stars in a triangle, about to collide
Colliding stars

A simple choreography

Another possibility is to put the stars in the same triangle, but this time start them off moving. If you pick the velocities just right, then the triangle will rotate but maintain its shape. The inward pull of gravity balances out the outward speed, much like how a satellite can stay in a circular orbit around the Earth.

Stars in a triangle, spinning around the center Stars in a triangle, still spinning Stars in a triangle, still spinning

If you follow the positions of the stars over time, you see that they trace out a circle over and over again. That is, not only does each star individually trace out the same curve over and over (the orbit is periodic), but each star traces out the same curve as each other.

A configuration of stars that does that is called a choreography. The stars in a choreography never collide; they just majestically dance in the sky. The circular orbit we just found is the simplest choreography.

A more interesting choreography

It is perhaps not surprising that stars can orbit each other in a circle. But can they orbit in more interesting shapes? Are there choreographies other than circles?

In 1993 Cris Moore found some. In particular, he found that there is a choreography with three stars tracing out a figure-eight curve.

The figure-eight The figure-eight The figure-eight The figure-eight

In 2000 the Moore figure-eight orbit was rediscovered by Alain Chenciner and Richard Montgomery. Whereas Moore only had numerical evidence that there is a figure-eight choreography, Chenciner and Montgomery mathematically proved its existence. It was kind of a big deal.

Where do we stand now?

After the (re)discovery of the figure-eight choreography, many, many more examples were found, especially by Carles Simó. Almost all of these examples were found numerically, without mathematical proof.

There are two components to my work on choreographies. First, I argued that it is even easier than previously appreciated to find new examples; you can even try your hand at finding a new choreography in your browser. Second, I developed a practical system for proving existence of choreographies that the computer finds. Combining these two, I have been able to vastly increase the number of specific choreographies which have been proven to exist.