Are they stable?
Although the double pendulum is the poster boy of Chaos Theory, there are actually two "special" modes of oscillation (called "eigenmodes") in which the masses move in perfect harmonypic.twitter.com/CnivjtF8Hg
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just as stable as any other mode of oscillation .. if you perturb it you change the mode and so wont be an eigenmode ..
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Are there any other periodic orbits that can be easily described?
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Good question .. depends what you mean by easily described .. An n-pendulum system has n eigenmodes if that helps
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IMO the coolest thing about the double pendulum is that chaos and (quasi-)periodic motion *both exist* in the same phase space! This is best seen with a Poincare section. Each color is a different orbit. https://twitter.com/InertialObservr/status/1177021761092608001 …pic.twitter.com/KhzcZTA1It
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If you want to know more about this topic, it can't hurt to read a certain book by
@stevenstrogatz:https://books.google.com/books?id=1kpnDwAAQBAJ&lpg=PP1&dq=nonlinear%20dynamics%20and%20chaos&pg=PA281#v=onepage&q&f=false … - 2 more replies
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There is a neat exercise you probably have seen, where you imagine an ideal inverted pendulum in its unstable equilibrium point. Then calculate how long it could stay there given the Heisenberg Uncertainty Principle. As I recall, it's only a few seconds.
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That's cool!
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More differential equation ideology
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Is it possible to solve for these modes' initial conditions in polar coordinates?
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