# Questions tagged [euler-lagrange-equation]

In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.

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### Why can the Euler-Lagrange equation be used to find the extremum of a functional?

The Euler-Lagrange equation is a differential equation. A functional is a function from some vector space to a real number. Functions that maximize or minimize functionals may be found using the Euler–...
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### Deriving the Euler-Lagrange Equation using the Gateaux Derivative

Can anyone explain how the professor goes from line 4 to 5 of the derivation? In particular, how is: $$\frac{\delta L}{\delta u}h'=-\frac{d}{dx}\frac{\delta L}{\delta u'}h$$ The professor states ...
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### Solving the Euler-Lagrange equation for the brachistochrone problem with friction

This Wolfram Alpha Page contains a derivation of the parametric form of the brachistochrone curve that result from either assuming friction or its absence. I am asking for help understanding how ...
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### euler-lagrange equation expansion

Euler-Lagrange equation $$\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial y'} = 0$$ Can also be written as $$f'_y-f''_{xy'}-f''_{yy'}y'-f''_{y'y'}y''=0$$ In my book it is ...