Hello,

I found a lot of references about the one-dimensional Euler-Lagrange equation corresponding to a functional of the form $J[x]=\int_a^b F[t,x,\dot{x}]dt$, where $x$ is a function of $t$.

But I would like to find the corresponding Euler-Lagrange equation of a functional with two(or more) occurances of integrals like for example $J[x]=2\log(\int_a^b F_1[t,x,\dot{x}]dt-c)-\int_a^b F_2[t,x,\dot{x}] dt$. Particulary in my case it is $J[x]=2\log(\int_a^b \exp(ax_t+bt)dt-c)-\int_a^b \dot{x}_t^2 dt$.

I did not find anything about this topic in books about variational calculus. Can someone give me a hint about how to do that or does someone have a referance where to find something about this problem?

Thank you very much