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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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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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    $\begingroup$ 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. $\endgroup$
    – Carl Witthoft
    Commented Mar 9, 2014 at 18:29
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    $\begingroup$ I don't think this answers the question, which I think is as GreenAsJade states below. This just describes what chaos is, it does not identify what feature(s) make the double pendulum chaotic in some cases. $\endgroup$
    – sammy gerbil
    Commented May 12, 2016 at 20:18
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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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  • $\begingroup$ 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. $\endgroup$
    – Cees Timmerman
    Commented Mar 9, 2014 at 22:43
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    $\begingroup$ 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. $\endgroup$
    – GreenAsJade
    Commented Mar 10, 2014 at 2:57
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    $\begingroup$ 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. $\endgroup$
    – Eelco Hoogendoorn
    Commented Mar 10, 2014 at 6:56
  • $\begingroup$ Suggested reading is "The laws of chaos" of Ilya Prigogine. He's a strong assertor of intrinsically chaotic laws also in classical mechanics. $\endgroup$
    – linello
    Commented Mar 10, 2014 at 11:01
  • $\begingroup$ @EelcoHoogendoorn : Turning the question around might be a useful insight, but it is still an unanswered question here. Why is the simple pendulum integrable while the double pendulum is not? How can you recognise if a system will be integrable or not? $\endgroup$
    – sammy gerbil
    Commented Jun 10, 2016 at 17:46
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The cheap and easy answer to this is that the double pendulum is considered chaotic because it is very sensitive to small perturbations in initial conditions (amongst other things). Showing this mathematically may be difficult (see the Lagrangian formulation for the dynamics), but if one looks at the animations on the Wikipedia page showing the trajectory of the double pendulum, the intuitive reason for this sensitivity should become obvious. There are many points in the trajectory where the acceleration rate of the outer pendulum is very dependent on the exact angle of the upper pendulum as it is whipped around. If the inner pendulum is in a sightly different place, the outer pendulum is whipped around at a very different rate, changing how "coupled" the two pendulums are. Sometimes the effect is to tie them together like they were a string on a grandfather clock. Sometimes it causes them to be almost perfectly opposed in position, doing their own thing.

Every time it reaches one of these states, it becomes very sensitive to the initial conditions that lead it to that state. A sight perturbation along the way could have arbitrarily magnified effects later.

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  • $\begingroup$ Same problem. This only states how to recognise chaotic motion when it occurs. It does not identify what it is about the system that makes its motion chaotic for some parameters and initial conditions. $\endgroup$
    – sammy gerbil
    Commented Jun 10, 2016 at 16:26
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    $\begingroup$ @sammygerbil I linked to the wikipedia page for those who want to know more. The OP mentioned being in highschool, so I avoided trying to write out the proof for why the double pendulum system is chaotic. It requires differential equations, and a formal definition of "chaotic." The definition I typically use involves a perturbation analysis over those differential equations. Handwaving those mathematical expressions aside, I focused on an intuitive sense of a clear point in the path of the pendulums where intuition can show that the pendulum's path will vary widely due to small perturbations. $\endgroup$
    – Cort Ammon
    Commented Jun 10, 2016 at 16:49
  • $\begingroup$ Thank you for pointing out the link. Does this mean that the double pendulum will not be chaotic unless it has enough energy to 'flip'? That is the kind of feature I am looking for. Using the intuitive approach, how do you recognise the crisis points or configurations? $\endgroup$
    – sammy gerbil
    Commented Jun 10, 2016 at 17:35
  • $\begingroup$ @sammygerbil That question gets complicated quickly because there is no singular dividing line in the sand between chaotic and non-chaotic. The line gets drawn based on your chosen basis, and is really just a perturbation analysis at some point. If you show that, for a particular basis, the effect of a perturbation grows exponentially with time, it's chaotic. The "flip" is convenient because it's easy to define the basis. You can define a function to be "amount of time until the pendulum flips," and that function exhibits chaotic behavior. However, at some point, it gets to semantics. $\endgroup$
    – Cort Ammon
    Commented Jun 10, 2016 at 18:28
  • $\begingroup$ I highly recommend (Chaotic Dynamics)[en.wikipedia.org/wiki/Chaos_theory#Chaotic_dynamics] from Wikipedia on the topic of what a chaotic system is. It's a very well phrased read. One key thing they mention is the concept of Lyapunov exponents, which say how fast two systems diverge. When we talk about a system being chaotic, that typically means a Lyapunov exponent >1 between perturbations, meaning the effect of the perturbation grows exponentially. $\endgroup$
    – Cort Ammon
    Commented Jun 10, 2016 at 18:32
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From a mathematical standpoint, deterministic chaos or sensitive dependence on initial conditions, is created when there are more than 2 dimensions or variables along with a sufficiently complex relationship between those variables, such as non-linearity and/or coupling.

There are 4 variables in a double pendulum, two angles and two angular velocities. The mathematical relationship between these variables involves squares (non-linearity) as well as sines and cosines (more non-linearity) of both angles in the same equation (coupling).

enter image description here

Image source: Strogatz, Nonlinear Dynamics and Chaos

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