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As we know, The double pendulum has a chaotic motion. But, why is this? I mean, the mass of the two pendulums are the same and they have the same length. But, what makes its motion random?

I'm just a high school kid. So, try to make answers understandable.

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2 Answers 2

Chaotic is not the same as random. A chaotic system is entirely deterministic, while a random system is entirely non-deterministic. Chaotic means that infinitesimally close initial conditions lead to arbitrarily large divergences as the system evolves. But it's impossible, practically speaking, to reproduce the same initial conditions twice. Given enough time, two identical setups, set to initial conditions that are as identical as possible, will look entirely different.

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Nicely written. Depending on the education level of the OP (some high schools have a couple yrs of calculus available), it might be worth writing the double pendulum equation and just pointing out what the initial phase(s) do to one's attempt to find a solution. –  Carl Witthoft Mar 9 at 18:29

Perhaps a better question to ask is: why is a single pendulum non-chaotic? Almost all real systems are chaotic at least to some extent; the fact that we can write out the solution for a single pendulum for all points in time is really quite peculiar, and only true because it is a highly simplified system. The reason these non-chaotic systems are so prevalent in textbooks is because historically, us humans with our peculiar mathematical toolset and limited abilities to calculate, have been aggressively looking for such idealized systems.

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Ideal things are perfect. As air is inconsistent, one needs only to add the divergent factors for an accurate prediction. Once the formula for a pendulum in a crosswind (of course depending on its exact shape, which is also hard to predict) is needed enough, people will attempt to create one. Or simply use a model with higher-than-expected winds. –  Cees Timmerman Mar 9 at 22:43
    
I don't feel that this answers the underlying question, which is "what is it that makes chaotic behaviour appear". The insight that some systems (and single pendulum is not one of them) have divergent behaviour that is dependent on infinitesimal input condition changes is the key here. –  GreenAsJade Mar 10 at 2:57
    
I agree mine is not an answer; but note that I didn't present it as such. But contrary to your assertion, I would argue that enumerating the qualities of a chaotic system does not do much to explain how those qualities arise. I would say that accepting that 'chaos' is the norm is a key insight, and that it is more instructive to ponder why some systems are integrable. –  Eelco Hoogendoorn Mar 10 at 6:56
    
Suggested reading is "The laws of chaos" of Ilya Prigogine. He's a strong assertor of intrinsically chaotic laws also in classical mechanics. –  linello Mar 10 at 11:01

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