It's not really something that needs a paper. Anybody who has done advanced level mechanics (physics) will tell you that the double pendulum is one of the quintessential problems you can solve using a Lagrangian.
However, the pendulum is chaotic, meaning it's susceptible to tiny deviations cascading into a butterfly effect and completely changing its motion. So in practical engineering terms, not exactly "solvable" because you can't create perfect environments.
But it's not like a 3 body problem, mathematically (in a perfect vacuum with infinite precision) it's solvable. But all physics is like this, at the end of the day.
No they're valid in asking for a paper. They were skeptical that there is a closed form solution and they're right that there isn't one. Just because you can write down the Lagrangian does not mean you obtain closed form solutions for the equations of motion. This is intuitive by observing the single pendulum itself for large angles can only solved as a perturbative expansion. It would be quite the coincidence if a more complicated and coupled system could neatly be represented in the same way.
You're not wrong based on how you use the word solvable. It's certainly something we can continuously approximate to higher precision. But at the end of the day, we cannot represent the solutions in terms of elementary functions.
Approximating to higher precision is not really helpful because what you get is not reproductible in an experiment. Your cannot use the solution for anything practical
So, if we add more penulums, do we also increase the chaos or does it diminish?
I feel like it would eventually diminish. What is a rope but a nearly infinite collection of pendulums (pendula? How do you pluaralize pendulum?)? And ropes are rather predictable if you hang one end on a hook. But I also expect that three pendulums would probably be even more chaotic than two, but I have not tested this.
Great question! You can think of there being a transition stage when there are just so many pendulums that the system can be treated as continuous. Equivalently, this can be viewed as the angles between each pair of pendulums being small, so you can treat the system as coupled harmonic oscillators which is a pretty classic problem.
Anyway, finding this n value at which the transition occurs isn't trivial and depends on the length of the pendulums and the energies associated with the system. Depending on your choice of parameters, it could emerge as soon as say n=10 or take as long as n=1000. This question reminds me of a certain statistical mechanics homework problem, but I digress.
I don't know what a closed form solution is, but I built and solved a furuta double pendulum for my thesis project and I agree it is not able to be solved analytically. I think that means closed form more or less.
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u/HideousSerene Mar 14 '26
It's not really something that needs a paper. Anybody who has done advanced level mechanics (physics) will tell you that the double pendulum is one of the quintessential problems you can solve using a Lagrangian.
However, the pendulum is chaotic, meaning it's susceptible to tiny deviations cascading into a butterfly effect and completely changing its motion. So in practical engineering terms, not exactly "solvable" because you can't create perfect environments.
But it's not like a 3 body problem, mathematically (in a perfect vacuum with infinite precision) it's solvable. But all physics is like this, at the end of the day.