(Since I didn't know which authorities to contact, I thought I'd post this here.)
While messing around in this Wolfram Demonstrations applet, I found a suspicious pattern, in which I could see physical similarities between the path of the first object, the path of the second object at a later time, and even the path of the third object at an even later time. Through gradually tweaking that set of initial conditions, I was able to find an apparently new solution to the equal-mass three-body problem that wasn't listed here.
Here are the initial conditions, with my restrictions in parentheses:
- $x_1(0)=0.7812$
- $y_1(0)=-0.2465\;(\text{holding $x_1(0)^2+y_1(0)^2$ constant to avoid scaling})$
- $x_2(0)=-0.2465\;(=y_1(0))$
- $y_2(0)=0.7812\;(=x_1(0))$
- $x_3(0)=-0.5347\;(=-x_1(0)-x_2(0))$
- $y_3(0)=-0.5347\;(=x_3(0))$
- $x_1'(0)=-0.6087$
- $y_1'(0)=-0.286$
- $x_2'(0)=0.286\;(=-y_1'(0))$
- $y_2'(0)=0.6087\;(=-x_1'(0))$
- $x_3'(0)=0.3227\;(=-x_1'(0)-x_2'(0))$
- $y_3'(0)=-0.3227\;(=-x_3'(0))$
(I would estimate the uncertainty for all parameters to be roughly $0.01$ in either direction. It would be smaller, but a slight change in the position parameter can be practically canceled out by a slight change in one or both of the velocity parameters, apparently resulting in a relatively 2D structure in 3D parameter space.) This gives an approximate choreography with period $p\approx 17.0874$, shaped somewhat like a rosette:
Going further in time, the entire system apparently rotates, but doesn't break apart:
At this point, what algorithms or heuristics should I use to increase the precision of the parameters (for example, trying to make the paths match over increasing lengths of time)?