Most of us are familiar with chain fountains.
I was wondering how this phenomenon is explained in the Lagrangian mechanics. I mean do we know how the Euler-Lagrange equations look like for this system?
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$\begingroup$ Imagine the chain of identical beads connected with massless rigid links, and that it emanates from one point and "disappears" at another point to the left (say) and lower. I think theoretically it probably follows a parabolic path? Anyhow, maybe you can take gravitational potential and then layer on a potential that varies laterally which accounts for the force exerted laterally by the motion of the falling chain. If the chain is falling right to left (as we supposed earlier) then the force is determined by the balance of beads further left and further right. $\endgroup$– rschwiebCommented Feb 3, 2021 at 21:55
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$\begingroup$ I don't know what that lateral potential would look like, and it sort of already presupposes we know the trajectory of the fountain. Maybe symmetry dictates that only the top of the fountain cancels out, and the remaining leg of the fountain is all that matters? If one were trying to determine the trajectory to begin with, it seems like you'd model each mass individually, and that the forces were just gravity and those exerted between successive links. $\endgroup$– rschwiebCommented Feb 3, 2021 at 21:59
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$\begingroup$ Of course, my guess that it's a parabolic path is just that, a guess. Maybe the shape doesn't matter, or is asymmetric. While I was searching around this paper looked like it might be useful $\endgroup$– rschwiebCommented Feb 3, 2021 at 22:01
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$\begingroup$ As this involves the reaction force from the table pushing in a non-trivial way on a sub segment of the chain, I doubt there’s a Lagrangian model for this. $\endgroup$– ZeroTheHeroCommented Feb 3, 2021 at 22:22
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$\begingroup$ Related: physics.stackexchange.com/q/70345/2451 , physics.stackexchange.com/q/138270/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Feb 3, 2021 at 22:23
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