An Introduction to Calculus of Variations

This video is an introduction to calculus of variations, seen through the lens of one of the primary motivators of its development: the cycloid. Special emphasis is given to the Euler-Lagrange equation as an expansion of the optimization problems familiar to students who have studied the fundamentals of single variable calculus. The Euler-Lagrange equation is first used to prove that the shortest distance between any two points in the Euclidian plane is a straight line, and then is used to find the function that minimizes the time between any two points (the brachistochrone problem). The solution is the cycloid. Through a physical demonstration using a 3D printed cycloid, we show that the cycloid is indeed the solution to both the brachistochrone and the tautochrone problems, two classic problems that drove the discovery of calculus of variations.
Made by: Elysia Chou, Amy Han, and Elisabeth Wood