1 Lecture #4 Calculus of Variation and Euler-Lagrange Equation Lecture #4 Hossam Abdelmunim 1 & Aly A. Farag 2 1 Computer & Systems Engineering Department,

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Calculus of Variation and Euler-Lagrange EquationLecture #4Lecture #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

Our Problem FormulationOur Problem FormulationOur Problem FormulationOur 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.

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1D Euler-Lagrange1D Euler-Lagrange1D Euler-Lagrange1D Euler-Lagrange

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1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)

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1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)

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1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)

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1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)1D Euler-Lagrange (Cont…)

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

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

and More

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

and More

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

and More (Cont…)

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

and More (Cont…)

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

and More (Cont…)

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

and More (Cont…)

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

and More (Cont…)

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

and More (Cont…)

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

Variable and More (Cont…)

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

Variable and More (Cont…)

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Euler-Lagrange and the Gradient DescentEuler-Lagrange and the Gradient Descent

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Euler-Lagrange and the Gradient Descent(Cont…)

Euler-Lagrange and the Gradient Descent(Cont…)