Built on Facts

Easy + Easy + Easy = Impossible

Every once in a while it’s a good idea to remember that even the simplest-looking physical systems can have completely bonkers behavior. The pendulum is certainly one of those systems. It’s so simple that it’s a mainstay of freshman classes, for technical and non-technical majors alike, though even then we do have make an approximation that’s only valid for relatively small angles of the swing.

But the equation of motion that a pendulum obeys is pretty simple. String a few pendulums together and the equation of motion is still not too bad – all you have to do is a little somewhat tedious work in figuring out the expressions for the energy of each swinging arm and you’ve got the equation of motion. If you actually try to solve the equation of motion and calculate the paths that the pendulums will take, you’ll find that you can’t. Not in anything resembling a clean closed-form equation. Why? Just look:

I think that’s pretty amazing. Impossibly complicated chaotic behavior in the simple connection of three trivially solvable systems. It’s a nice reminder that even if physics does discover a Theory of Everything, that theory would not in and of itself close the book on the discipline. After all, the Lagrange equations of motion are a complete “theory of everything” as far as this particular system is concerned – but developing the consequences and implications of those laws is another story. The laws of nature have a richness that can’t be fully explored simply by knowing the basic statements of the laws.


  1. #1 Nathan Myers
    September 22, 2009

    Would a Theory of Everything involving absorption of photons close the book on interactions with biological systems? Or would there be a richness that can’t be fully explored mathematically? Congratulations, Matt, on developing a hint of humility.

  2. #2 faz
    September 22, 2009

    Leaving aside the physics for a minute, the dancing clock is really awesome to look at whilst listening to music. I wonder what track would fit the dancing best….

  3. #3 Joshua Zelinsky
    September 22, 2009

    The pendulum is fascinating. I really like how occasionally just when you think it is settling down one of the arms will decide to do a few more full quick rotations with no warning. It almost looks alive.

  4. #4 Rhett
    September 22, 2009

    It’s not impossible. We used to bullseye wamprats back home in my T-16. They’re not much bigger than 2 meters.

    Sorry, I had to say that.

  5. #5 hegemonicon
    September 22, 2009

    My gut tells me that that this is a case of the three body problem, but my gut has a piss poor understanding of math/physics – is this right?

  6. #6 kevin
    September 22, 2009

    I was intrigued by an idea brought to my attention by Steven Wolfram: that the theory of everything might be very simple and short — as short as two or three lines of mathematica equations. But yet even knowing the equation, it might tell us almost nothing at all about useful about the universe, because working out the real-world-scale consequences of the theory might be incredibly, or even impossibly, hard. And we’d find ourselves right where we are: doing experiments, testing things against the real world, generating hypotheses, etc.

  7. #7 doug l
    September 22, 2009

    And yet there are smart people who think a system such as our climate which anyone can see has a lot more than just three variable is predictable by varying one of its components, co2. We’re even willing to be tens of trillions on it, while ignoring the systems manifestly robust qualities of homeostasis. Go figure.

  8. #8 Nathan Myers
    September 22, 2009

    Doug: No, you’re willing to bet other people’s tens of trillions that it really is homeostatic against a CO2 step function, despite increasingly manifest evidence to the contrary.

    If the temperature were to continue on up despite declining CO2, it might be a good idea to reconsider the measures that were causing it to decline. But since reducing dependence on a depleted and politically destabilizing resource is such a good idea anyway, I wouldn’t.

    Uncertainty isn’t your friend, in this.

  9. #9 Comrade PhysioProf
    September 22, 2009


  10. #10 Carl Brannen
    September 22, 2009

    The point of the pendulum is that it CAN be described by very simple equations of motion. What can’t be solved for is the long term behavior of the system.

    The reason the pendulum example should set you thinking is precisely that a very simple set of equations of motion have apparently very complicated conserved quantities (i.e. Noether and all that). This makes the present method at the foundation of elementary particles suspect. Rather than simple conserved quantities, the natural system of the pendulum is simple at the equations of motion level.

    P.S. Meanwhile, I’ve got reviews back on my paper at Foundations of Physics and they want me to modify it and resubmit. It applies the above reasoning to the case of spin-1/2, in a way.

  11. #11 Gray Gaffer
    September 22, 2009

    Isn’t it four moderately coupled oscillatory systems, not three? I see four pivots, anyway. And more of the mass and angular inertia in the central beam than the satellite pendula.

    I imagine a good part of the chaos in the behavior derives from physical realities and imperfections in friction and incompletely constrained rotations at the pivots.

    With perfect pivots, frictionless and fully constrained to only allow rotations and no lateral slop or eccentricities, I would expect the parts to show a more beat frequency kind of behavior, than the almost step-like transfer we see here, as the energy migrates around the system.

  12. #12 Furkan
    September 23, 2009

    If this didn’t stop, we can use this for create energy. But it stopped. Actually it isn’t complicated. Very simple. Gravity and Momentum

  13. #13 Nick
    September 24, 2009

    If you think that’s complicated, try doing it for a system that has 3*10^25 times as many moving parts, and you might figure out the brownian motion of an ordinary can of soda.

  14. #14 Noam GR
    September 24, 2009



  15. #15 Carl
    October 6, 2009

    Sweet Jeebus I can’t believe no one has pointed this out: a pendulum is NOT a solvable linear system, it approximates a simple harmonic oscillator for small angles ONLY:

    Θ = -g/L sin(Θ) ≈ -g/L Θ

    I’m pretty sure a system of coupled ideal harmonic oscillators are exactly solvable (not trivial by any means, but doable if you’re down with matrix algebra and eigenvectors) especially in the case where the spring constants and masses are the same. The chaotic motion in this case comes from the non-linearity in the system (which most people ignore for the sake of a closed form solution)

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  17. Isn’t it four moderately coupled oscillatory systems, not three? I see four pivots, anyway. And more of the mass and angular inertia in the central beam than the satellite pendula.

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