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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-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…)