Comment by PaulHoule
3 hours ago
In the case of a planet going around the sun we know that the planet travels in an ellipse, more generally a conic section including interstellar comets. Orbital periods and everything else can be computed straightforwardly with formulas.
In the three body problem you can always do a numerical integration (e.g. simulation) and this is valid for a certain amount of time but will not be valid forever because of: (1) chaotic motion which amplifies small errors exponentially over time and (2) celestial mechanics is symplectic which means it conserves certain geometric properties and most integrators are not symplectic and must have different long term dynamics. There are symplectic integrators but they don’t work as well overall as ordinary integrators.
We do not know, for instance, if the solar system is stable. In the short term the planets seem to be basically doing their own thing in their orbits with just minor perturbations. We know the orbits vary a bit over millions of years. We aren’t sure which side of the sun Pluto will be on in 30 million years. It’s very believable that the planets are doing the same thing in 4 billion years but we can’t rule out that the orbits could change in a big way or one could get ejected.
It's worth noting that the Sun-Earth-Moon "system" is much more complex than a three-body problem just due to momentum transfer from tides/bending. That's ignoring the effect of all of the other planets and asteroids or solar wind.
The 3BP is just the simplest chaotic system showing the limits of simple models and approximations.
Well there are simpler chaotic Hamiltonians like Henon–Heiles.
There are a lot of things wrong with how we teach classical mechanics and one of them is that the two index problems that are used in the undergraduate course are the harmonic oscilator and celestial dynamics and these are the worst non-generic problems that there are. The harmonic oscillator is generic in quantum theory and is a good place to start doing perturbation theory from but the fact that the frequency doesn't change with amplitude makes it non-starter in classical perturbation theory.
In the 2-body case the periods to: (1) go around the sun, (2) go in and out towards the sun, and (3) go up and down out of the plane are all the same which again breaks perturbation theory... and of course classical perturbation theory is difficult and doesn't work that well even in cases where it does work. Contrast that to quantum mechanics where you can start doing simple calculation with perturbation theory, like to calculate the lifetime of an excited state, right away with paper and pencil.
Note that, in general, you can get chaos with as few as two position variables. In the case of the two body problem you have six position variables, but because of conservation of momentum the center of mass doesn't really matter, so you can pick a coordinate frame where the total angular momentum is zero and there are just three variables that matter which is the vector between the two bodies. There are a lot of conserved quantities here, especially angular momentum so the (1) and (2) and (3) motions all do the same thing coincidentally with the same period!
One you add a third body you are adding three more variables but not any more conserved quantities so it is a problem with a lot of dimensions.
Mapping out the orbits of the three body problem has gone pretty slowly because, compared to simpler Hamiltonians, we have to search for those orbits in a high dimensional space. You can find a periodic orbit numerically if you know where it is, but it takes a systematic approach to find many of them.