In this video, I set up and solve the brachistochrone problem, which involves determining the path of shortest travel in the presence of a downward gravitational field. This is done using the techniques of Calculus of Variations, and it will turn out that the brachistochrone can be represented by the parametric equations of a cycloid.

The Brachistochrone is a rather popular topic on Youtube, with pop-science channels like VSauce making videos about it. However, not many people actually derive the equations, so I'm hopeful that this tutorial will be a more rigorous change of pace.

Questions/requests? Let me know in the comments!

Prereqs: The first three videos of this playlist:    • Calculus of Variations  
Euler-Lagrange Video:    • Derivation of the Euler-Lagrange Equa...  

Lecture Notes: https://drive.google.com/file/d/0BzC4...
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