The Brachistochrone Problem - Curve of Fastest Time

Physics Ninja looks at the Brachistochrone problem. 2:33 - In part 1 the time to go from one point to the end is derived 7:04 - In part 2 we use Euler-Lagrange formalism to show that the shortest time possible is the path describe by a cycloid. 31:40 - In part 3 we calculate the time to travel for 3 different paths and show that the cycloid shape is the fastest.