Comments by "John Berry" (@user-ud6ui7zt3r) on "The Closest We’ve Come to a Theory of Everything" video.

Ultimately, for the fastest path problem, you use the Euler-LaGrange equation. However, the reasoning starts with the following... Start with... 𝑣 = 𝑠/𝑡 ...and rearrange it to solve for 𝑡 ... 𝑡 = 𝑠/𝑣 . [Small letter 𝑡 stands for time. Small letter 𝑠 stands for distance. Small letter 𝑣 stands for velocity. ] The task now is to minimize 𝑡. To do this, you need to minimize the term in the numerator, 𝑠, while simultaneously maximizing the term in the denominator, 𝑣. Remember that when you increase the "size" of the denominator, you make the resulting overall ratio (i.e. 𝑡 ) become smaller.
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Notice that in the expression ... δ ∫ 𝑚𝑣 𝑑𝑠 = 0 ...you have... ∫ 𝑚𝑣 . If👆this portion had been integrated with respect to 𝑣 (i.e. 𝑑𝑣 ), then the integral for this portion would have been... ½ 𝑚𝑣² + 𝐶 ... which is the equation for Kinetic Energy, the derivative of which is the equation for Momentum, 𝑚𝑣 .
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