This lesson covers the concept of the calculus of variation, a generalization of the maximum or minimum problem of ordinary calculus. It explains how this concept is applied in structural mechanics to determine the deformed shape of a system that causes the total potential energy of the system to have a stationary value, which is the equilibrium state of the system. The lesson also discusses how to extremize a functional using the first variation, resulting in the governing differential equation along with boundary conditions. The Euler Lagrange equation is introduced as the governing differential equation. The lesson also provides an example of a simply supported beam with uniformly distributed load to illustrate the concept.

- The calculus of variation is used to find the deformed shape of a system that causes the total potential energy of the system to have a stationary value.
- The first variation is used to extremize a functional, resulting in the governing differential equation.
- The Euler Lagrange equation is the governing differential equation.
- The total potential energy of a system is an example of a functional in structural mechanics.
- The equilibrium configuration of a system is that which makes the total potential energy of the system stationary.

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