Double Pendulums are Chaoticn't

Video Description

http://brilliant.org/2swap/ - Click here for a 30 day Brilliant free trial and 20% discount on an annual premium subscription! ERRATUM: At the end of the video, I said: "there is no closed-form solution for non-trivial starting positions." The trivial cases I was referring to are those in which the two arms are collinear, and those in which both arms are vertical but pointing in opposite directions. However, my general claim may not be precisely correct. It is indeed the case that there exists no closed-form function which maps a pendulum's starting angles and a time duration onto the pendulum propagated out for that duration of time. However it is not yet proven that no _specific_ starting condition has a function that maps duration runtime to future position. That said, no such configuration has been found either, and very well may not exist. If you liked this, please support me on Patreon! https://www.patreon.com/2swap You can support the musician, 6884, on Kofi: https://ko-fi.com/iam6884 Check out his music here! https://6884.bandcamp.com/ SOURCES AND CREDITS Video animated with SwapTube: https://github.com/2swap/swaptube This was largely inspired by this video from Drew's Campfire: https://www.youtube.com/watch?v=OIHyN7TzY6A Pretzel drawing by https://www.instagram.com/litaapollo/ Discussion of energy was inspired by this paper: https://www.famaf.unc.edu.ar/~vmarconi/fiscomp/Double.pdf. Notice that they find slightly different results, since they plot whether or not a pendulum arm peforms a flip as opposed to divergent behavior, which I focused on. The color scheme I used in this video was generously provided by Cruz Godar, who has a website packed full of cool generative art here, double pendulums included! https://cruzgodar.com/gallery After rendering this video, I investigated the internal structure of the islands of stability, such as their oscillatory modes. Maybe I'll make another video showing my findings... Here's an interactive online version by Ricky Reuser: https://observablehq.com/@rreusser/the-double-pendulum-fractal Join our discord server about math, computer science, and other miscellaneous nerd stuff! https://discord.gg/XDb7x2BZAE