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
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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
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1D Euler-Lagrange (Cont…)
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1D Euler-Lagrange (Cont…)
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1D Euler-Lagrange (Cont…)
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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
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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…)
10
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…)
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Euler-Lagrange and the Gradient Descent
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Euler-Lagrange and the Gradient Descent
(Cont…)
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