This lesson covers the calculus of variation and the Rayleigh-Ritz method, two mathematical techniques used in engineering and physics. The lesson begins with a discussion of the brachistochrone problem, a classic problem that led to the development of the calculus of variation. The Euler-Lagrange equation is used to derive the governing differential equation for designing a frictionless chute between two points in a vertical plane. The lesson then moves on to the Rayleigh-Ritz method, which involves choosing a suitable shape function for the deformation of a system and minimizing the total potential energy to derive the critical load of the system. The lesson concludes with an example of a fixed free column and the calculation of its critical load using the Rayleigh-Ritz method.

- The calculus of variation is a mathematical technique used to solve problems involving the minimization or maximization of functional.
- The Euler-Lagrange equation is a fundamental equation in the calculus of variation.
- The Rayleigh-Ritz method is a technique used to approximate the critical load of a system by minimizing the total potential energy.
- The Rayleigh-Ritz method involves choosing a suitable shape function for the deformation of the system.
- The critical load of a system can be calculated using the Rayleigh-Ritz method.

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