This may be a bit pedantic, the nbody problem is not chaotic, it is harder, having riddled basins.
> A riddled basin implies a kind of unpredictability, since exact initial data are required in order to determine whether the state of a system lies in such a basin, and hence to determine the system’s qualitative behavior as time increases without bound. (Note this is different from “chaos,” where very precise initial data are required to determine finite-time behavior.) What is more, any computation that determines the long-term behavior of a system with riddled basins must use the complete exact initial data, which generally cannot be finitely expressed. Hence such computations are intuitively impossible, even if the data are somehow available.
The above post is a good 'example' of sensitivity to initial conditions, and riddled basins do have a positive Lyapunov exponent which is often the only criteria in popular mathematics related to chaos. But while a positive Lyapunov exponent is required for a system to be chaotic, it is not sufficient to prove a system is chaotic.
If you look at the topologically transitive requirement, where you work with the non-empty open sets U,V ⊂ X....riddled basins have no open sets...only closed sets.
With riddled basins, no matter how small your ε, it will always contain the boundary set.
If you have 3 exit basins you can run into the Wada property, which is also dependent on initial conditions but may have a zero or even negative Lyapunov exponent and is where 3 or more basins share the same boundary set...which is hard to visualize, non-chaotic, and nondeterministic.
Add in strange non-chaotic attractors, which may be easier or harder than strange chaotic attractors, and the story gets more complicated.
Sensitivity to initial conditions is simply not sufficient to show a system is chaotic in the formal meaning.
But the 3 body problem's issues do directly relate to decidability and thus computability.
Chaotic, riddled, and wada can be viewed as deterministic, practically indeterminate, and strongly indeterminate respectfully.
If you want to hold on to the flawed popular understanding of the butterfly effect that is fine, you just won't be able to solve some problems that are open to approximation and please don't design any encryption algorithms.
I think realizing it is simply a popular didactic half truth, is helpful.
The statements are independent. It having a closed form solution and it being unstable don't contradict or confirm one another.
In solutions to ODEs converge very often exponentially from the true result. That the 3 Body problem for this makes it characteristic, not special.
>So, even having a closed form solution isn’t helpful when computing real world situations.
Simply not true. It is helpful or not depending on your problem. Often you are interested in short term behavior, which can be studied by numerical methods or, if existing, analytic solutions.
Closed form solutions in general dynamic systems are possible when systems are integrable which means there is a conserved quantity for every degree of freedom. The solar system is almost like that in the case that each planet keeps going around with a constant angular momentum so they go around like a set of clocks that run at different speeds. Over long periods of time there is angular momentum transferred by the planets so you get chaos like this
Orbital mechanics is a tough case for perturbation theory because each planet has three degrees of freedom (around, in and out, up and down) and the periods are the same for all of these motions and don’t vary with the orbital eccentricity or inclination. Contrast that to the generic case where the periods are all different and vary with the amplitude so with weak perturbations away from a resonance the system behaves mostly like an integrable system but if the ratio between two periods is close to rational all hell breaks loose, see
the harmonic oscillator has a similar problem because the period doesn’t change as a function of the amplitude. Either way these two pedagogically important systems will lead you completely wrong in terms of understanding nonlinear dynamic, if you add, say, an εx^3 term to the force in one of two coupled harmonic oscillators it is meaningless that ε is small, you have to realize that the N=2 case of this integrable system
is the right place to start your perturbation from which ends up explaining why the symmetry of the harmonic oscillator breaks the way that it does. Funny though, the harmonic oscillator is not weird at all in quantum mechanics and is just fine to do perturbation theory from.
Thanks for this. This is something that always bugged me when I see explanations of the three-body problem. They'll say something like "changing the initial conditions just a tiny bit can dramatically change the outcome!" as an explanation for why having no closed form solution is significant.
But that never made sense to me, since plenty of things with closed form solutions also do this.
It was really a math problem and not a physical one. There's so much more going on ( GR, radiation pressure ), that even if there were a solution it wouldn't be able to predict Mercury's orbit.
> This only demonstrates that the system is chaotic, not that there is no closed form solution.
This seems a bit off, it seems like[1] an implicit assertion ("only(!) demonstrates") that it is not possible for a system that lacks a closed form solution in fact (beyond our ability to discern) to be demonstrated.
To be clear I'm in no way implying this was your intent (I see it as an interesting "quirk" of our culture)...I'm mainly interesting if you can see what I'm getting at.
As a thought experiment, stand up two instances: one is our current situation (inability to discern, indeterminate), the other where we have (somehow) proven out (or, come to believe we have, reality being Indirect but experienced as Direct, thus: "is "proven", thus: "is") that a closed form solution is not possible: would the second instance "be(!) a demonstration that the system has no closed form solution"? (Thinking more....I think maybe the choice of the word "demonstrate" may very well make a path to seeking the truth of the matter ~impossible to achieve in these sorts of cases, especially if one takes cultural forces[2] into consideration).
[1] Using "pedantry", which few people understand the technical meaning of, and tend to flip flop on depending on what is being considered (precision & accuracy in science/physics is good, precision & accuracy in philosophy/metaphysics is bad - no explanation or justification needed: a Cultural Fact).
[2] Which make the 3 body problem in the known to be deterministic physical realm seem like child's play.
>As a thought experiment, stand up two instances: one is our current situation ..., the other where we have (somehow) proven out ... that a closed form solution is not possible
As far as I understand, this has in fact been proved. Quite a long time ago, too, by Poincare I believe.
GP has edited his comment to reflect my feedback, but originally said that his experiment "demonstrates that there is no solution." All I was trying to point out is that the two concepts are not necessarily related.
You could imagine some system x' = f(x), where f(x) is some transcendental function. There is no analytic solution to this system, but it's obviously not chaotic.
