L1-Part3of3- Euler's Method of Finite Differences - M2 - Euler-Lagrange Eq. - Variational Derivative
This lecture is part 3 of 3. It introduces an alternative (more general) finite-difference scheme for solving the simplest variational problem introduced in parts 1 and 2. In this lecture, we consider what happens when we fix x_k and treat y_k as variable (Finite Difference Method M2). In the limit as n tends towards infinity, we arrive at the famous Euler-Lagrange equation. I show how this equation reduces to the equation arrived at in the previous part (part 2of3) of this lecture through the use of what is known as Beltrami's Identity.
Lastly, I also discuss how this result, the Euler-Lagrange equation can be thought of as the ''derivative'' of the functional with respect to the variation. This will be a useful idea to keep in mind as we discuss optimization of functionals in a later lecture.