Summer of Math Exposition

the good:

nice visualizations, especially the springs (although im not quite sure if the numbers above them really add that much to the whole picture), really really cool and interesting topic + conclusion

the bad:

this was wayyyyyyyy too fast. im actually familiar with the simple harmonic motion differential equation but your explanation still flew over my head. your animations are too fast, especially the equation simplifying ones (if the viewer is not supposed to get all that, either consider not showing them, or mentioning that at some point, speeding through them like that just leaves them feeling overwhelmed and stupid) also explain where that solution came from!!! or why it doesnt matter to us or something?!?!?!? dont just pull it out of thin air and then say we can verify that this really is a solution by a 1 second long simplification animation. also sadly no barely no motivation on why one should care about this outside of you touching on that in the conclusion.

(reminder to not take this personal in any way and to also not get demotivated by this, if you improve on these few, relatively simple points my score would already jump up by a quite significant margin so keep at it!)

Nice job with the mass animations, they don’t look easy to make. They were cool, but they looked a little choppy at times, and they could be REALLY COOL if you were able to make these smoother. The video seemed like it was going to be an intro to coupled oscillators, but it was very fast. Your explanations for what symbols meant were very clear, but the algebra was hard to follow, and the replacement transforms got messy at times. You might try multiple transforms on specific indices. Finally, from the perspective of an intro video, why does it matter if a system is periodic or not? I don’t think this significance was addressed.

Connecting the idea of an irrational ratio of periods to non-periodic motion overall was subtle, but definitely interesting.

Nicely done! Certainly quick, effective, to the point math communication, which I think has something both to people new to ODEs and people who’ve taken standard curriculum for introduction to ODEs. I’m gonna score a little conservatively because my personal style would dictate getting a bit more into some of those details (providing a solution without prior justification isn’t my cup of tea, even though you did show why it is in fact a solution which is a nice compromise) but (a) I understand you had a time constraint, and (b) that is also a stylistic choice, and stripping back on that does give your video a nice sense of pace. (also you actually went through the derivation of the ODEs, which is something I’ve even seen textbooks skip out on for some godforsaken reason, so maybe I should just take what I can get)

The video was well paced and used its 5 minute length to fully explain what had been set out in the introduction

Great vocal clarity and sequence/planning of concepts. Visualizations perfect as well. Please continue making more videos because this is really good. Would be interested in a follow up. If this video were longer and if you included some more review and exposition of harmonic oscillator ODEs I would have given it a 9 instead of 8, but still a great video.

Find a way to visualize the non-periodicity of the system, like how you can draw traces of a double pendulum, or trace of Lorenz attractor. Maybe draw the trace in phase space?

It was nice and the animations and coloting was smooth , will be waiting for the follow ups you mentioned ! :)

Thank you for not ignoring the derivations. The topic is clear in 5 minutes.

Coupled oscillations is a topic I really like. I consider is way easier solving the problem using Lagrangian mechanics, but I respect the fact that you are targeting a younger audience who maybe are only familiar with Newtonian mechanics. You did a great job deducing the equations for x1 and x2 and then introducing the normal modes.

This is a much bigger area as you state at the end of the video and maybe you could have covered a little more without increasing too much the difficulty. For example, considering different masses, with one being much bigger than the other. However, if you just want to highlight the fact that the final motion isn’t necessarily periodic is also fine, but you should have given more intuition behind the fact that the quotient of the periods is irrational. For example, you could have plot the phase space (x1 vs x2) and explain the Lissajous curves, which won’t be closed curves if the quotient of periods is an irrational number.

Fantastic thumbnail! It really shows you what the subject of the video is, and it made me curious how that system would behave.

This was a very nice 5-minute summary, pointing out the interesting parts (the non-periodicity), and keeping to a particular example (identical masses and spring constants), without dwelling too much on the details of solving the differential equations. The animations were pleasant and useful.

I would be curious to see animations of the two basis vectors for the space of solutions to the differential equations. I think that would help to more deeply intuit the two available periodic modes.

A short, dynamic video with a clear objective. The animations are very clear and easy to follow. However, I am not entirely convinced by the conclusion of the video, since there are periodic solutions for the coupled system. These solutions are special and depend strictly on the initial conditions, and are called normal modes of oscillation. It seems to me that this video analyzed the case in which the solution is general, in which case the solution is a linear combination of normal modes. Beyond this, I found the video very interesting to watch. Perhaps it would be good to include some other interesting feature or aspect of coupled oscillators to make the video a little longer.