1 Lecture #4 Calculus of Variation and Euler-Lagrange Equation Lecture #4 Hossam Abdelmunim 1 & Aly A. Farag 2 1 Computer & Systems Engineering Department, Ain Shams University, Cairo, Egypt 2 Electerical and Computer Engineering Department, University of Louisville, Louisville, KY, USA ECE 643 – Fall 2010

Our Problem Formulation Our problem formulation is based on finding a variable/function that minimizes a given energy. So, our target is to find: Maximization will be straightforward by finding the minimization of –F. Euler-Lagrange equations help us solve this problem.

3 1D Euler-Lagrange

4 1D Euler-Lagrange (Cont…)

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7 This is called Euler-Lagrange equation. It is obvious that our target to solve for ɸ for minimization.

8 Euler-Lagrange equation for Many Independent and Higher Order Derivatives of One Dependent Variable and More

9 Euler-Lagrange equation for Many Independent and Higher Order Derivatives of One Dependent Variable and More (Cont…)

10 Euler-Lagrange equation for Many Independent and Higher Order Derivatives of One Dependent Variable and More (Cont…)

11 Euler-Lagrange equation for Many Independent and Higher Order Derivatives of One Dependent Variable and More (Cont…)

12 Euler-Lagrange equation for Many Independent and Higher Order Derivatives of More Than One Dependent Variable and More (Cont…)

13 Euler-Lagrange and the Gradient Descent

14 Euler-Lagrange and the Gradient Descent (Cont…)