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I am solving a variational calculus problem, in which the functional is in the form of: $$ J[p(t),\lambda]=\int_a^b\left(t^2\cdot p^2(t)+t^2\int_0^t p^3(\tau)\cdot g(\tau)d\tau -{\lambda}\cdot{p(t)}\right)dt $$ So the Lagrangian function in in the following form (Lagrangian itself includes an integral!!) $$ L = t^2\cdot p^2(t)+t^2\int_0^t p^3(\tau)\cdot g(\tau)d\tau -{\lambda}\cdot{p(t)} $$ Now I'm confused, how can I use Euler-Lagrange equation for solving this problem. In other words how can I integrate $\int_0^t p^3(\tau)\cdot g(\tau)d\tau$ with respect to $p(t)$? I think a partial derivative method can be used, but I can't find the right answer.

Thanks in advance

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    Maybe $$\frac{\partial \int_0^t p^3(\tau)\cdot g(\tau)d\tau}{\partial p}=\frac{\frac{\partial \int_0^t p^3(\tau)\cdot g(\tau)d\tau}{\partial t}}{\frac{\partial p}{\partial t}}$$ – 2017-01-07
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    I've test it but it is incorrect, you can test it with an example for $p(t)$ – 2017-01-07
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    You do realize that $\frac{\partial L}{\partial\dot{p}}=0$ and tested $$\frac{\partial L}{\partial p}=2t^2 p+t^2 p^3 g/\dot{p}-\lambda p$$ right?? – 2017-01-07
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    Thanks for your answers, I put for example $p(\tau)=sin(\tau)$ and test the answer by your equality. – 2017-01-08
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    Is anything known about the signs of $a$ and $b$? Is $a – 2017-01-22
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    @ Qmechanic Thanks a lot, the Polfosol answer was right, my problem is solved. – 2017-01-23

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Hints:

  1. The standard formula for Euler-Lagrange (EL) equation does not apply to double integrals. But luckily we can rewrite OP's functional $J[p]$ as a single integral with the help of Fubini's theorem.

  2. In this answer, we will assume that $0