Euler Lagrange Equation for Functional having Different Dependent Variables — Lesson 3

This lesson covers the application of the Euler Lagrange equation in the context of structural mechanics. It delves into the use of the delta operator for finding the Euler Lagrange equation for a functional with one dependent variable and its first derivative. The lesson further explores the concept of total potential energy expression and how it relates to higher order derivatives. It also demonstrates how to prove that the first variation of total potential energy is equivalent to satisfying the equilibrium equation and boundary conditions. The lesson uses the example of a simply supported column subjected to axial loading to illustrate these concepts. It also discusses the Euler Lagrange equation for functional with different dependent variables and its derivatives.

Video Highlights

01:10 - Discussion on the Euler Lagrange equation for functional having different dependent variables
09:04 - Discussion on the boundary conditions for a simply supported column subjected to axial loading
17:13 - Explanation of the Hamilton principle and its application in extremizing the functional
25:22 - Explanation of the Euler Lagrange equation for extremizing the functional

Key Takeaways

- The Euler Lagrange equation can be used to find the functional with one dependent variable and its first derivative.
- The total potential energy expression involves higher order derivatives.
- The first variation of total potential energy is equivalent to satisfying the equilibrium equation and boundary conditions.
- The Euler Lagrange equation can be applied to functional with different dependent variables and its derivatives.
- The Euler Lagrange equation can be used to solve complex structural mechanics problems.