So theoretically speaking, they are deterministic, but practically they are unpredictable.
The n-body problem where you add further bodies, even very small ones, is even harder.
I mean, you can certainly still predict to a certain (probably high) level of accuracy, but ultimately that motion is also influenced by factors outside your model.
For a two body problem, you nudge one of the bodies and it is forever off by a small amount, but your predictions into infinity require only a small adjustment to compensate.
For a three body problem you nudge one of the bodies and only for a very short time do your previous predictions stay true, the change amplifies until nothing you thought might happen before the nudge means anything at all, and a common occurrence is one of the bodies being ejected.
So the problem eventually solves itself?
How does this even work? Where is a place in the universe where there are only two bodies?? Where would the nudge come from if not from a third body?! A ghost?
It also doesn't necessarily matter that much if there are more than two bodies, if the gravitational influence of other bodies is small enough, then you can model as a two body problem.
Each planet and the sun can be done like this, ignoring all of the other planets. Each moon and its planet can be considered a two body system ignoring the rest of the moons.
If you just randomly generated a bunch of massive bodies and pressed play, you would have few 2 body systems and a lot of chaos, but that’s a problem that solves itself as the chaos results in either collisions or ejections.
With a three-body problem any slight shift causes a wildly different trajectory, bearing no resemblance to the original so your measurements of the initial condition have to be perfect.
Thanks for mentioning the existence of an analytical solution at all though, I wasn't aware of that.
This is not universally true. Error behavior is a function of the particular problem, the algorithm used to approximate its solution, and the properties of input data. A large subtopic of numerical analysis is concerned with this kind of stuff. See [1] or [2] to get a flavor.
If the three had roughly the same mass would be different story.
The issue with the three-body problem is that it is chaotic, meaning any error will eventually grow to take over the entire solution, making prediction impossible, even in theory. Every chaotic system lacks an analytical solution, but not every system without an analytical solution is chaotic.
X is random with respect to Y, if knowing Y makes no difference to your predicting that X.
QM systems are indeterminate, they are random in the above sense /because/ they are indeterminate. But that isnt what random means.
It's hard to define randomness. I think non-determinism is better than your definition.
Non-determinism is an incoherent definition of randomness; classical physical processes are entirely deterministic.
The point of a coinflip being random is that it is random with respect to the information both observers of the coinflip have. It isnt random with respect to /any/ piece of information.
There are almost no processes which are non-deterministic in this sense. Not enough to bother calling them random; and in physics we do not: the word is indeterminate. Randomness has nothing to do with quantum mechanics; it wasn't invented in the 1920s. It's an epistemic condition.
The RANDOM variable X, st. X ~ N(mean, std) provides a random number x -- x isnt random with repect to the outcome which produced x; nor is it random with respect to an index of a vector in which it is contained.
I would say for a given variable to be random, it must not be predictible, given any other variables that humans can know.
I don't think it makes sense to say that X is random "with respect to Y", that's just the definition of independence.
And a constant variable is independent from all other variables, but it's definitely not random.
I dont think many people have thought enough about the world to appeal to their intuitions.
Randomness isnt quantum indeterminacy.