Chaos and the double dendulum

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A dendulum is a pendulum you hang from a tree.


Given the starting conditions, a chaotic system is entirely deterministic.

Nothing is entirely deterministic. The whole point of chaos theory is “sensitive dependence on initial conditions.” But Heisenberg (Werner, not Walter) showed that initial conditions are not knowable. It turns out God actually does play dice with the universe, and on a macro level, too.

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Of course chaotic systems where the starting conditions aren’t known to a sufficient level of accuracy CAN be modeled as random. eg. rolled dice. Which are certainly large enough that quantum effects can be ignored. I would argue that MOST things that are treated as “random” are really cases of limited knowledge rather than things that are truly random.


NB: For optimum performance it should be an apple tree.

Nothing physical. The mathematical double pendulum system is a coupled pair of 2nd order ODEs (So, equivalent to an order 4 ODE.) Those are entirely determined by their precise initial starting position.


…which cannot be measured precisely.

Also, I didn’t take God into consideration, Who is in there busily moving the state variables around.


In one of the first lectures of Physics 1 at Caltech the professor (David Goodstein) had a double pendulum just like this at the front of the room and let it go and said something to the effect of “Nobody can tell you what the motion of that pendulum is going to be.” For some reason that has stuck with me for a lot of years as a profoundly interesting fact of the universe.

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Stop it, let it just hang down there at energy minimum, and its behavior becomes pretty predictable.


You have to decide if you are going to talk about a physical pendulum or its mathematical idealization. If want the mathematical one, measurement is not a part of the theory. If you want the physical one, it isn’t chaotic: the second law of thermodynamics will eventually stop the pendulum via the heat death of the universe.

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comes running, slaps dendropendulum
How’d you predict that?

Yea, my engineering background is showing. I find reality more interesting.

Stop it again, glue it in position. Problem solved. :stuck_out_tongue: :smiley:

“God actually does play dice” - or continually creates through an infinite succession of miraculous interventions. I’ve never really heard a good explanation of why a truly random event shouldn’t be regarded as supernatural.


The laws of quantum mechanics are entirely deterministic, it’s just that the thing that actually exists is the wavefunction, not the high-level approximations like “position” and “momentum” that we find intuitive. And if you did successfully model the wavefunction of a macro-scale “measuring” device, then there is no need for assuming randomness.

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I thought it was a pendulum with an addendum.

Or it bounces off the trampoline.


errata: what does it matta?

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