The Brachistochrone, with Steven Strogatz
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when answering questions about brachistochrones, cycloids and fastest-descent paths, as an accessible explanation of why the curve takes the shape it does.
Recommended by 3 people across 2 different months, 2017 to 2022.
What it is
Grant Sanderson of 3Blue1Brown talks with mathematician Steven Strogatz about the brachistochrone problem, finding the curve of fastest descent under gravity. They cover the historical challenge, a clever solution based on Snell's law and light, and a modern twist, with animated visuals throughout.
What people said
Recommends Alp Uzman · 2017-02 · Mathematics Stack Exchange · on “Applied Mathematics problems”
The Brachistochrone [this video] by 3Blue1Brown might be a good place to start. In general you might find a book on differential geometry interesting, e.g. Millman & Parker or Do Carmo.
Recommends For the love of maths · 2019-06 · Mathematics Stack Exchange · on “Path for fastest end velocity while accounting for friction”
`Because ideally in the theoretical world, the path would not matter as long as there was no friction.` That is where you are wrong. The fastest path should be a brachistochrone. See here: [this video]
Recommends Paige Crawford · 2019-06 · Mathematics Stack Exchange · on “Why must the coordinates of a point on a circle be sinusoidal as a function of the angle?”
I'm not quite sure this is right -- if I'm understanding your example correctly, this isn't actually sinusoidal, it's a cycloid! See the intro section of this video: [LINK] (and if you're interested in some of the deeper properties of the cycloid, check out [this video] )
Cites Number Basher · 2022-10 · Mathematics Stack Exchange · on “Intuitive Explanation that Brachistochrones (Time-Minimizing Curves) have a Constant Change of Angle per Time”
In a 3 Blue 1 Brown video [this video] ( [this video] ), 3b1b gave the following challenge: Give an intuitive reason why time-minimizing curves going from one point to another under the effect of gravity look like straight lines in the t- $\theta$ space, which maps each time to the change in angle of the curve. Which is equivalent to saying that the if the Brachistochrone curve is modelled as $f(x)$ , then the function that measures the angle of the tangent line of $f$ at $x$ , which we will call $g(x)=\tan^{-1}f'(x)$ , then the function $g'(x)$ looks like a line, that is, $g''(x)$ is a constant. (Intuitively, $g(x)=0$ .) Did anyone make any discoveries concerning this? A scroll through the comments d