How to take derivative of a expression with respect to a function?

Question

I want to program the Euler-Lagrange equations for continuous systems. But in this formulation I have to compute the derivative of the Lagrangian with respect to all the derivatives of the generalized coordinate. $$\partial_i\left(\dfrac{\partial \mathcal{L}}{\partial (\partial_i\phi) } \right) = \dfrac{\partial \mathcal L}{\partial \phi }$$ For example, for a field $\phi=\phi(x,t)$ $$\dfrac{\partial }{\partial t}\dfrac{\partial \mathcal L}{\partial \dot\phi}+\dfrac{\partial }{\partial x}\dfrac{\partial \mathcal L}{\partial \phi'}=\dfrac{\partial\mathcal L}{\partial \phi}$$ Where $\dot\phi=\dfrac{\partial\phi}{\partial t }$ and $\phi'=\dfrac{\partial \phi}{\partial x }$

I have tried with something like this

D[L, D[ϕ, t]] + D[L, D[ϕ, x]] = D[L,ϕ]

But of course, it returns an error because of the variable with respect the derivative is taken.

Answer 1

It is straightforward. Let us take a Lagrangian

L = 1/2 D[y[x, t], t]^2 - 1/(2 c^2) D[y[x, t], x]^2 + 1/2 y[x, t]^2;

We need to load the package:

Needs["VariationalMethods`"]

Now, this gives the variational derivative:

VariationalD[L, y[x, t], {x, t}]

and this yields the Euler equation:

EulerEquations[L, y[x, t], {x, t}]

Have fun!