Physics Asked by IvanMartinez on October 6, 2020

Does the following *functional derivative* can be evaluated?

$$ frac{partial}{partial(partial_mu phi(x))} int d^4y F(y) partial_nupartial^nuphi(y)$$

I am trying to find the equations of motion of a classical field with a Lagrangian that has the form:

$$mathcal{L}(phi,partial_muphi)+int d^4y F(y) partial_nupartial^nuphi(y)$$

Is the *conventional* form of E-L equation valid here:

$$ frac{partial mathcal{L}}{partial(partial_mu phi(x))}- frac{partial mathcal{L}}{partial phi}=0$$

Or is a *new* form necessary due to the dependence on the second derivative inside the integral of the Lagrangian?

You can just partially integrate the double derivative to get $-partial_nu F partial^nu Phi$. Or use a definition for the functional derivative analog to the usual derivative, where you'd, in this case, replace $partial^nu Phi(y)$ by the delta-distribution of $delta(x-y)$.

Answered by drfk on October 6, 2020

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