That's a great question, and mostly an open problem except in some special cases:

http://scholarpedia.org/article/N-body_choreographies

But basically: some are, some aren't, and we can't generally tell (yet) which is which.

Checking this page brings back memories. Carles Simó was the principal researcher at my research group (is now retired as a professor, but AFAIK is still a researcher), so I'm familiar with the work and people mentioned there, even if it was outside of my own research interests. I took his celestial mechanics course, and it was… _not easy_.

> Motivations: Theoretical. Astronomical. Aesthetics.

there are also important applications, like figuring out the right orbit and starting points for kerbin bodies in the principia n-body mod