This lesson covers the Brachistochrone problem, a classic problem that led to the growth of calculus of variation. The lesson starts with a recap of the previous lesson where the condition that the first variation of total potential energy is equivalent to satisfying the equilibrium equation and boundary conditions was proved. The lesson then delves into the Brachistochrone problem, which involves designing a frictionless chute between two points in a vertical plane such that a body sliding under the action of its own weight moves in the shortest interval of time. The lesson explains how to derive the governing differential equation using the Euler Lagrange equation and how to solve it to find the path of the curve. The lesson concludes by stating that the profile of the chute in the vertical plane should be a cycloid for the body to take the shortest interval of time.

- The Brachistochrone problem is a classic problem in the calculus of variation.
- The Euler Lagrange equation is used to derive the governing differential equation for the Brachistochrone problem.
- The solution to the Brachistochrone problem is a cycloid, which is the path a body should take to move in the shortest interval of time under the action of its own weight.

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