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Calculus of Variation and Euler-Lagrange Equation 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

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Calculus of Variation and Euler-Lagrange Equation Lecture #4

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  1. Calculus of Variation and Euler-Lagrange EquationLecture #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

  2. 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…)

  5. 1D Euler-Lagrange (Cont…)

  6. 1D Euler-Lagrange (Cont…)

  7. 1D Euler-Lagrange (Cont…) 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…)

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