Khurram Wadee ✅

A few days back, I posted some #AnimatedGifs of the exact solution for a large-amplitude undamped, unforced #Pendulum. I then thought to complete the study to include the case when it has been fed enough #energy to allow it just to undergo #FullRotations, rather than just #oscillations. Well, it turns out that it is “a bit more complicated than I first expected” but I finally managed it.

#Mathematics #AppliedMathematics #SpecialFunctions #DynamicalSystems #NonlinearPhenomena

Jun 13, 2025 at 5:21pmWeb
Show threadKhurram Wadee ✅Jun 13, 2025
Once again, we need to use #EllipticalIntegrals but in an unfamiliar regime where some parameters are #ImaginaryNumbers, (actually, you can just treat the square values as being negative) but you can get exact expressions for the behaviour.
Show threadKhurram Wadee ✅Jun 13, 2025

Graph of period of rotation versus minimum kinetic energy of the mass (at the top of its swing).

#MyWork #CCBYSA

Show threadKhurram Wadee ✅Jun 13, 2025
Challenge to all the psychics out there: try to make the animation reverse direction at the top of the swing.
Show threadadingbatponder  👾Jun 13, 2025
@mkwadee I was waiting for the video to end as the pendulum slows down   😂 physicists  
then ever so slowly I thought about what mathematicians would do and the video did not stop and I realised
Show threadKhurram Wadee ✅Jun 13, 2025
@adingbatponder Hope you enjoyed it.
Show threadadingbatponder  👾Jun 13, 2025
@mkwadee It cost me a whole evening waiting : 😂 but a lot better than what I would have have watched on netflix. Nice job !
Show threadKhurram Wadee ✅Jun 13, 2025
@adingbatponder I've got one which starts off with even less kinetic energy but I refrained from posting it.