Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Variational calculus Euler lagrange Equation

  1. Nov 13, 2013 #1
    I am trying to understand an example from my textbook "applied finite element analysis" and in the variational calculus, Euler lagrange equation example I cant seem to understand the following derivation in one of its examples
    ∫((dT/dx)(d(δT)/dx))dx= ∫((dT/dx)δ(dT/dx))dx= ∫((1/2)δ(dT/dx)^2)dx

    limits from 0 to 5.

    My question here is how in the last part of the derivation 1/2 appears out of the blue where the integratal remains intact..
    If anyone know the answer.. kindly refer me some examples also
  2. jcsd
  3. Nov 13, 2013 #2
    essentially it is a restatement of: xdx = 1/2 d(x^2)
  4. Nov 14, 2013 #3
    So this is a reverse step?
  5. Nov 14, 2013 #4
    Hi All,
    I found it easier to master the functional derivative concept by performing the calculations out, leaving aside for a moment the symbol $$\delta$$ to denote variations, a useful notation that might though hide the mechanics of what is going on.
    Then your example becomes transparent: the functional derivative of the functional $$\int_{\Omega} y'^{2} \mathrm{d}\Omega$$ in the direction of the test function $$\mu$$ (in other terms, introducing a variation $$\hat{y} = y(x) + \epsilon \mu (x)$$ according to the definition equals
    $$lim_{\epsilon \to 0}\frac {\int_{\Omega} [\hat{y}'^2 - y'^ 2] \mathrm{d}\Omega}{\epsilon}$$
    $$lim_{\epsilon \to 0}\frac {\int_{\Omega} [{y'^{2} + 2\epsilon \mu' y'+\epsilon^2 \mu^{2}} - y' ^2] \mathrm{d}\Omega}{\epsilon}$$
    some terms cancel, and you are left with the result you wnated to confirm: it turns out the differentiation of functions and functional share common operational properties, such as xdx = 1/2 d(x^2).
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Variational calculus Euler Date
I Euler’s approach to variational calculus Feb 18, 2018
A Maximization problem using Euler Lagrange Feb 2, 2018
A Derivation of Euler Lagrange, variations Aug 26, 2017
Calculus of variations Feb 9, 2016
Euler Lagrange Derivation (Taylor Series) Jan 15, 2016