Hacker News Reader

Comment by jgchaos | Hacker News Reader

The link points to one of the stable solutions, and there are actually quite a few of those. The problem is that there’s no general closed form that tells us exactly where the bodies will be in the future, so we rely on numerical methods to approximate the motion. If you hit Reset All a few times or add more bodies, you’ll start to see the chaos

View on HN

An interesting corollary to this is that even if the future trajectory of a general 3-body orbit is predictable in theory using numerical methods (and infinite precision calculations), in practice the use of finite-precision floating point means that after some time the trajectory predicted by an ODE solver will diverge from the mathematically-true trajectory. Even symplectic integrators have this problem. More details on the general case of chaos are provided by this insightful blog post:

https://www.stochasticlifestyle.com/how-chaotic-is-chaos-how...

There actually is an analytical solution using a power series that actually converges (Karl Sundman's work). Unfortunately, the universe still mocks our attempts. Though the series converges, it does so incredibly slowly. From Wikipedia:

The corresponding series converges extremely slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^8000000 terms.

> the computations would involve at least 10^8000000 terms.

Well we could speed up that simulation pretty easily, just arrange the actual masses and velocities somewhere...

Then I thought, is there a way to scale the distances, masses and velocities to create a system with the same, but proportionally faster behavior?

One guess as to perhaps why not: As distances get small, normal matter bodies will get close enough to actually collide. Perhaps some tiny primordial black holes would be useful.

When you say 'stable' here, do you mean 'periodic' or are these solutions actually stable in the face of small perturbations (as opposed to the sensitive dependence on initial conditions that we'd expect from a chaotic system)?

Reply on news.ycombinator.com