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.
So, even having a closed form solution isn’t helpful when computing real world situations.