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

  • 3
    $\begingroup$ What does $x_t$ mean? In any case, just derive the Euler-Lagrange equation from scratch, exactly as it is done in the standard case. In other words, let $x$ depend on a variational parameter, say, $\tau$, differentiate $J$ with respect to $\tau$, and then integrate by parts to shift all differentiation with respect to $t$ away from the variation (i.e., $dx/d\tau$). You will then be able to combine all your integrals into one, where the integrand is $dx/d\tau$ multiplied by a formula involving $x$ and its derivatives. That formula equal to zero is the Euler-Lagrange equation. $\endgroup$ – Deane Yang Apr 3 '12 at 10:20
  • $\begingroup$ Thanks, I got it, although it would be fine, if someone knows a reference. $x_t=x(t)$ is just the value of $x$ at $t$, nevermind. $\endgroup$ – Daniel Ferstl Apr 4 '12 at 7:45

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