Calculus of Variation and Euler-Lagrange Equation Lecture #4 Hossam Abdelmunim1 & Aly A. Farag2 1Computer & Systems Engineering Department, Ain Shams University, Cairo, Egypt 2Electerical and Computer Engineering Department, University of Louisville, Louisville, KY, USA ECE 643 – Fall 2010 1 Our Problem Formulation • Our problem formulation is based on finding variable/function that minimizes a given energy. a So, our target is to find: Maximization will be straightforward by finding the minimization of –F. Euler-Lagrange equations help us solve this problem. 1D Euler-Lagrange 3 1D Euler-Lagrange (Cont…) 4 1D Euler-Lagrange (Cont…) 5 1D Euler-Lagrange (Cont…) 6 1D Euler-Lagrange (Cont…) This is called Euler-Lagrange equation. It is obvious that our target to solve for ΙΈ for minimization. 7 Euler-Lagrange equation for Many Independent and Higher Order Derivatives of One Dependent Variable and More 8 Euler-Lagrange equation for Many Independent and Higher Order Derivatives of One Dependent Variable and More (Cont…) 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 More Than One Dependent Variable and More (Cont…) 12 Euler-Lagrange and the Gradient Descent 13 Euler-Lagrange and the Gradient Descent (Cont…) 14