The method of finding periodic three-body orbits

[Pefgjk]

Hello everyone!

I'm trying to get a deeper understanding on how to determine those periodic orbits in the equal-mass three-body Newtonian gravitational problems. The most general idea I know is to confine the three bodies into a zero angular momentum space, which sounds vague to me. What is the more detailed computational job here, to derive the numerical initial conditions?

Also, what is the general idea of "algebraic symmetries", and "free group elements", that are used to sort the "families" of orbits into distinct "classes".

Thanks in advance!

 

[osilmag]

Zero angular momentum space seems to mean that the sum of the angular momentums ∑Iw=0. That means you have to know/figure out the mass, orbit speed, and radius.

You could also calculate the gravitational force between the bodies knowing their masses and distance between them and set that equal to the centripetal force to find the velocity/orbit.