Summer of Math Exposition

Superb animation! Although, the concept can get a little technical through some of the equations. Maybe you could talk about interesting and cool applications. Additionally, try to be a bit more emphatic with your voice. Keep going with your thumbnails, they are amazing!

This is a beautifully designed video, both visually, and also in terms of the content. The author should be proud of this video! Having said that, here are a few critiques, meant to help the author improve future videos. First, the robotic voiceover is monotonous and detracts from the video overall. It would be much better to use an actual human voice, which will undoubtedly contain emotion and nuance. Second, the voiceover was often simply "reading the equations" (e.g. is equal to theta 1 times m times g...), which is not ideal. The viewer can see the equations on the screen, so it would be better to try and add something else with the voiceover. Lastly, although the author did a pretty good job discussing "unpredictability" in this system, the fact of the matter is that the system is actually completely predictable, and that's why the author is able to simulate the results so beautifully. This somewhat subtle point could have been emphasized a bit more. Nevertheless, these critiques are all fairly minor. Once again, the author did a really great job on this video and they should be commended for their effort!

Pretty nicely done! Good visual style, clear motivation, a standard-but-very-cool topic. Pretty clear exposition if you have the right background. One suggestion would be to spend a little more time on what information is embedded into the “kinematic constraints” in step 1. Also, since you go pretty fast through the equations, it could be good to mention that it’s not so important to follow every detail there, so newer students don’t feel overwhelmed.

Great continuation of the chaotic but contained motion of the the double pendulum. The video demonstrates the motion of two masses in a double pendulum with both the small angle and large angle approximations. The reveal of the concept that while the motion of a single double pendulum system seems to be random after a certain time period, it is constrained by the system variables in the Lagarangian and can be demonstrated in the angle space to have a pattern. Great video!

It is indeed great, animation works and the equations are solved in a detailed manner. The concept of chaos could have been covered deeply but this is plenty for introduction.

Nice visualizations. Also glad that I finally found a video going a bit deeper than just saying "double pendulum is chaotic"

These are really nice animations.

Beautiful visualizations!

Beautiful!

An excellent video. Good introduction and animation. Pace was good.

really better than most

Excellent interleaving of the general principles with the specific application to pendulum modeling. Good choice in not dwelling on the equation manipulation

i turned my brain off throughout the explanation of the lagrange equations and stuff. i, too, just flashed equations onscreen in my video entry, and i think people have a hard time learning anything that way! principle of least action sounds interesting, though, and the phase space graph animations were cool

The double pendulum is a fascinating example of a chaotic system and the author of this entry does a good job of explaining it. However, I found the video to be very heavy on the technical derivations of the equations of motion. In principle, this is not a problem, since I appreciate technical videos, but I would then expect the author to delve a bit deeper into the equations of motion and give some intuition as to how chaos emerges from them.

The equation for cos(θ₂) around the 2-minute timestamp is incorrect.

nothing new, and a too many equations. a liitle boring and far from chaos where it should belong.

The content is not new, but the video is great. It has the full derivation of the equations (so not scared about showing the maths behind the physics), and the animations are very very well done

Nice visualizations! I watched the video on my phone which have a limited screensize. Some of your equations gets very small during your calculations. This is only a minor thing. Really good video.