Double pendulum

Hello Veeky Forums, i'm an average guy with slightly above average knowledge in science and i'm completely baffled by the double pendulum.

How come the movement is random? If we had two completely identical closed systems with two completely identical double pendulums in each of them, wouldn't the resulting motion be identical? When i'm saying identical, i mean fucking identical - exactly the same molecular structure on both pendulums, same starting positions, same pressure and temperature in the closed system, exact same amount of air e.t.c.

Why is the produced movement still random? Can anybody explain this to me?

If they are both identical. the resulting motion will be the same.

The problem is kinda like the chaos theory, there is so many variables to consider that it's almost right to say it's random.

Do you have to consider the acceleration of the red pendulum when determining the location (relative to the tip of the red pendulum) of the blue pendulum

t. brainlet

There are only two variables, though. What makes a system chaotic isn't the number of variables, but the sensitivity to those variables and their initial conditions. A tiny, tiny change is either starting angle results in a dramatic change in the trajectory, which is the definition of chaos.

There are two things to consider: Determinism and chaos. A double pendulum as a physical system is deterministic, meaning exact knowledge of the initial conditions (initial position and momentum of the pendulum in this case, just four degrees of freedom) implies exact knowledge of the time evolution of the system. This also means that two identical systems with two identical initial conditions must have the same time evolution.

Chaos on the other hand is something independent from that. Vaguely defined, chaos describes the property of a physical system that the time evolution very sensitively depends on the initial conditions: A small change in those conditions leads to a disproportional change of the state at some point in time after initialization. In the case of the double pendulum the consequence is basically that the movement is effectively random. We can only adjust the initial conditions up to a limited precision.

What you will probably see is that no matter how hard you try, two identical double pendulums starting with the same conditions will look very different after, say, a minute, even though they appear to behave exactly the same in the first few seconds.

>almost right to say its random.
>almost

You just solved something....incredibly difficult....

The movement is in no way random. It's completely deterministic. HOWEVER, it is unpredictable due to chaos, so not only slight changes in initial conditions affect it, disturbances at ANY given time will set both systems apart. You could only say this motion would be truly random if you input some non deterministic elements into it, like some disturbance that comes from the amplification of quantum effects, but if you don't account for that, the system is perfectly predictable. Caos comes from the fact that the derivatives of the variables depends on the state of other variables, and said variable's derivative depends on the state of that same variable, so very small initial errors grow exponentially fast, causing the "Butterfly effect". Have a look at Lorenz strange atractor's theory to have an insight into what caos actually means.

>inb4 caos
Sorry, HUEzilian monkey brainlet here =(

Your definition is correct but your conclusions are wrong.

First note that a system can be both chaotic and deterministic. Since it is deterministic, the same starting position will give the same motion for two different systems. Since its chaotic, even a small variation in initial conditions will cause them to evolve differently even if they look the same at the start.

That is literally EXACTLY what I wrote.

You said two identical systems with the same starting conditions will evolve differently. That is a false statement.

I didn't say that. Learn to read.

he did not say that you brainless moron

>What you will probably see is that no matter how hard you try, two identical double pendulums starting with the same conditions will look very different after, say, a minute

That's exactly what he said

Yes, in reality that is the case because you can only adjust the initial conditions up to a certain level of precision. It will always be a little off, and for chaotic systems such as the double pendulum that means that the time evolutions diverge sooner or later. In the theory of course that is different, but I pointed that out. Read more carefully next time.

there are 4 variables. Two angles and two angular velocities.

>read more carefully

How about you write more carefully? If it is always "a little bit off" in reality, then they aren't identical.

I didn't say the initial conditions are identical. That's the whole point, in reality you can't establish identical initial conditions.

Of course it's not random. But it's chaotic, meaning that close starting positions doesn't mean that their trajectories will remain close after a while.

>starting with the same conditions

Yes you did you stupid nigger

Have we come up with a general function using starting conditions as variables, or a set of functions based on the starting conditions?

>exactly the same molecular structure on both pendulums
>same starting positions
Not possible

But teo mathematical models like OP should generate exactly the same result, given that you can give them the exact same initial numbers. No?

If "same conditions" =/= "identical conditions" then the writer best parse out the distinction instead of shrieking autistically because people can't read minds.

It's amazing how dense some people are.

quantum entanglement

>two completely identical closed systems
there no such thing like that boy. The systems always will be in different positions of the "space". Even if you use one system it will change the position in space no matter what.

And that position will be affected by all kind of weird forces we havent discovered yet.

So randomness isnt real we havent discover how it works yet.

Basically it means that in physical reality you can never have absolutely identical starting conditions, nor can you ever have.a progression through absolutely identical conditions. There will always be some difference, even if only sub atomic, which will result in a different outcome.

Hell, a virtual particle might pop into existence and therefore alter the motion of the pendulum. Just that alone would be sufficient.

Once any variation is introduced that variation will be eventually amplified through the pendulum motion to result in a wildly different outcome.

In reflection this just fucking amazing. To observe this is to witness the physical manifestation of insanely small forces.

Its like someone dropping a sewing pin on the floor in Siberia and you are sitting in Australia with a huge amplifier that can pick up its sound when it hits the floor and amplify it so it sounds like thunder.

>If we had two completely identical closed systems with two completely identical double pendulums in each of them, wouldn't the resulting motion be identical

Maybe, like just maybe if the two systems on the gif had the same starting variables it would look the same however clearly the starting variables are different. How can i tell? Well clearly theta1 and theta 2 is 0 in the one on the left while in the one on the right theta1 is 0 and theta 2 is 0.1.

>How come the movement is random?

It isn't.