Delta Operator in Variational Method for Finding GDE and Boundary Conditions — Lesson 2

This lesson covers the concept of Euler Lagrange equation in the context of structural mechanics. It explains the use of ordinary calculus for finding extremal points and the concept of functional. The lesson further delves into the use of fast variation for extremising a functional, resulting in a governing differential equation along with boundary conditions. It also discusses the use of delta operator for finding the Euler Lagrange equation for a functional with one dependent variable and its first derivative. The lesson concludes with the derivation of Euler Lagrange equation for a functional with higher order derivatives. For instance, in structural mechanics, the total potential energy expression involves higher order derivatives, hence the need for Euler Lagrange equation.

Video Highlights

00:46 - Introduction to the Euler Lagrange equation
03:55 - Explanation of the delta operator
15:22 - Explanation of extremum condition for the first variation of functional
20:16 - Explanation of the first variation of functional and its integration
32:25 - Derivation of governing differential equation and boundary conditions

Key Takeaways

- The Euler Lagrange equation is derived using ordinary calculus and the concept of functional.
- Fast variation is used to extremise a functional, resulting in a governing differential equation.
- The delta operator is used to find the Euler Lagrange equation for a functional with one dependent variable and its first derivative.
- The Euler Lagrange equation is also derived for a functional with higher order derivatives, which is common in structural mechanics.