What is the difference between the functional derivative and the Euler-Lagrange equation?
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2 Answers
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Briefly, the Euler-Lagrange (EL) equation $$ \frac{\delta S}{\delta q(t)}~=~0$$ states that the functional derivative of the action $S[q]$ vanishes.
answered May 2, 2021 at 19:28
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The equations which involve an integral are called functional and for extremum of functional, Euler Lagrange equations are used. For example the functional could be $\int \sqrt{(1-(dy/dx)^2}dx$ which is the functional for a length of any curve y, using Euler Lagrange equations, we can show that the equation is a straight line which means the extremum for distance is a straight line.
