Operational amplifiers can be configured to generate analogous orbits by using them to differentiate one of their inputs. By wiring multiple op-amps together and measuring voltages at different points it's possible to observe chaotic behavior in the measurements, just as if you were to directly differentiate from some initial condition on paper using a differential equation. This configuration is an analog computer. In fact, I wouldn't be surprised if op-amp circuits existed with perfectly analogous mathematical behavior to the orbits described in the paper.
The problem with gravitational computing using n bodies probably lies with establishing initial conditions, and with inhibiting the effects of neighboring systems.
They take advantage of a priori knowledge about the laws of physics, in particular the conservation of mechanical energy and angular momentum. Predictor-corrector is one family of methods. Other methods rely on convergence as timesteps change.
There is a lot of literature on numerical integration applied to celestial mechanics and new methods being released every year. These methods are tailored to this problem space.
Once I was aware of this issue though, even more shocking to me was that many papers on climate models I came across do not even mention they monitored adherence to conservation laws. There is a disconnect there, I would think this adherence either is a big deal (as would be suggested by the practice in astronomy) or not (as would be suggested by climate research), not that it would change by subfield.
That doesn't guarantee that the orbits are perfectly periodic, I suppose, but it does suggest that the orbits are stable with respect to rounding errors up to those you get from using doubles.
[1] http://jointmathematicsmeetings.org/amsmtgs/2141_abstracts/1...
> At first, using the obtained initial conditions, we checked the 137 periodic orbits by means of the high-order Taylor series method in the 100-digit precision with truncation errors less than 10^−70 , and guaranteed that they are indeed periodic orbits.
> Besides, we use the CNS with even smaller round-off error (in 120-digit precision) and truncation error (less than 10^−90 ) to guarantee the reliability of these 27 families.
They do reference some software (e.g., "dop853"), but I'm not familiar with the details of those ODE solvers.
[1] http://www.damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf
http://www.maths.manchester.ac.uk/~jm/Choreographies/
...plus a link to the animations for the linked paper:
Until 2013, only 3 or so solutions to the 3-body problem are known. Now we have over 150 solutions. This sounds incredible, given how fundamental the problem is, but So what?
Will this change astrophysics - for example - in any way?
I do not know how it can change astrophysics (its just newtonian gravitation, not einshteinian), but it can bring new methods/ideas to mathematics, then improved math will change everything. Maybe.
We'll probably be able to use these techniques on other problems in the future as well.
Greg Minton created a computer-assisted proof system for showing that there must exist a choreography with parameters within a certain distance of some given approximate parameters. This isn't just a matter of more floating point precision; it certifies that there is a critical point for action of the right kind. http://gminton.org/#gravity and http://gminton.org/#cap
Greg Minton also has a bunch of proved choreographies at http://gminton.org/#choreo
I'm just in the middle of the third book now, the series is well worth reading.
iOS: https://itunes.apple.com/us/app/threebody-lite/id951920756?m... Android: https://play.google.com/store/apps/details?id=com.nbodyphysi...
The authors of the new solutions have published all the initial conditions - so hopefully it won't be too hard. (Although they use an 8th order RK integrator and the one I have is a regularizing Bulirsch-Stoer integrator).
