Basically the title, how do we know that chaotic systems like the 3-body problem, double pendulum, etc. are insolvable? Couldn’t it just as simply be that we don’t fully understand the mathematics/physics? What gives us the confidence to call it chaos?
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We actually do understand the math/physics behind it.
We call it chaos because even though we understand the math/physics behind it, the system is so sensitive to initial starting conditions that simply being off by an infinitesimal amount at the start means that the errors accumulate and any models end up becoming extremely inaccurate. So practically they are unsolvable in that we can’t actually model the system well enough to predict outcomes reasonably well beyond a certain point.
>Couldn’t it just as simply be that we don’t fully understand the mathematics/physics?
This actually is essentially it. We accept that the system is so subject to initial conditions, and small changes, and we don’t know enough about them that we can predict (with accuracy) what will happen. When we can predict the trajectory of a ball (in non-chaotic flight) it’s not that we “perfectly understand everything happening to it” it’s that we’ve understood enough of the major components that all the minute details don’t matter (to us).
A double pendulum is varies so substantially on initial conditions, and subject to small measurement errors that it quickly reaches a state where we can’t predict accurately what happens. In theory if we can measure more accurately, or understand more in the future, we will consider it less chaotic.
It is also hideously computationally expensive. This world has modeled nuclear explosions and still doesn’t want to go near the three body problem because of how computationally expensive it is.
This is a very good question. In short, chaotic is a definition. It’s about how these systems are sensitive to knowledge of the initial conditions. Even extremely tiny differences or errors in initial conditions (position, velocity) will compound over time and make the system effectively unpredictable. While the system is deterministic (can be predicted IF everything were known to infinite decimal places), it is chaotic in practice (we don’t know everything to infinite precision). Lyapunov exponents show this well (very mathematical though).
You mention the 3-body problem, which has some specific nuance in it that we cannot accurately model interactions between 3 different things well (when the interactions are all on the same scale). Technically speaking the Earth has an effect on the orbit of Jupiter around the Sun, but in reality there’s no real reason to ever even consider the Sun-Earth-Jupiter system since the effect is so tiny, we just consider Sun-Jupiter.
The issue mainly comes from the setting up of the equations of motion of a 3-body system. I don’t know how much calculus you got, but I’ll generalize/simplify it this is way: there is no way to “get a nice formula” that solves the output of a 3-body system and its interactions. The equations are “non-integrable” which is basically math/physics talk for “this doesn’t work”. The simultaneous interactions of 3+ things simply doesn’t have an exact solution. I am not a mathematician, but my physics education has made aware that people (mostly Henri Poincare) have proven this to be true.
The only practical thing we can do is a “guess and check” method by using computers. We run the simulation for an extremely brief period of time, calculate the new positions and velocities, run it again and repeat.
they’re unsolvable using our current mathematical systems.
using a different system they could be very easily solvable.
When physicists and mathematicians talk about “solvable” systems, we generally mean one in which you can write down a mathematical solution for the motion and position as a function of time. Note that this *does not* mean you can’t simulate the system.
The thing about chaotic systems is that we can show mathematically that any solutions they have that are steady-state or repeatable cycles are unstable if we change them just a little. Think about a pencil. Theoretically you should be able to stand it on its point (and in fact if you spin it you can get it to stand for quite a while). But given any nudge at slow enough spin the pencil will start to fall.
It’s not solveable analytically. That is, there is no closed form solution. No equation you can plug the numbers into. This was proven mathematically by Henri Poincare for n = 3 and much later for n > 3.
But there are convergent infinite expansions that can solve it to whatever precision you need, it just takes a lot of computation.
It’s called chaos because the small errors build up over time and affect the outcome.
There are two aspects.
The first is that they are extreme sensitive to the initial condition. Let’s say you have a car that can drive in a perfect straight line. You get the task to turn the car left by 15 degrees and let it drive for 1000 miles. If your turning is off my the smallest amount, the car will end up a significant distance away from where it was supposed to be. For systems like the double pendulum, even though we can calculate how it behaves theoretically, the smallest deviation, smaller than the accuracy of the best possible measurement, will eventually amplify into a significant difference.
For the 3-body problem things are worse (that’s why it gained the title “problem” and the double pendulum didnt). The 3-body problem can not be solved with math. It is proven to be impossible. Even if we had infintely accurate measurements of the initial state, we can only approximate how it would behave. The errors of our approximation will accumulate, makeing the results more and more wrong over time.
Chaotic doesn’t mean unsolvable. A chaotic double pendulum is perfectly deterministic. What chaos is is when elements of the domain and range are disjointed. Inputs which are neighbors don’t have neighboring outputs.
In a normal function f(A) and f(B) will be close if A and B are close given arbitrarily close A and B. In a chaotic function that’s no longer true. As A approaches B the behavior doesn’t just smoothly ramp from f(A) to f(B) it does something completely disconnected.
The confidence comes from the analysis of the system. We assume a perturbation on the input and compute (or observe) the results. If there’s a divide in the outcome associated with a perturbation then it’s chaotic.
Chaotic systems means that a slight difference in preconditions changes the outcome greatly. In the real world all values are an irrational numbers (each speed, each mass and so on) so after the decimal point it goes on infinitely. You can’t plug an infinitely long number into a calculator so we must round thus changing the preconditions somewhat thus our outcome will differ greatly. There are solution to the three body problems with some specific setups but there is no general solution.