Okay, so I've run into a rather weird functional that I am trying to optimize using calculus of variations. It is a functional of three functions of a single variable, with a constraint, but I can't figure out how to set up the Euler-Lagrange equation. The functional in question is (sorry it's kind of messy):(adsbygoogle = window.adsbygoogle || []).push({});

$$J\left[\phi,y,z\right]=\int_0^b\left( \int^b_0\left( \dot{\phi}(t)\frac{(y(s)-y(t))\dot{y}(s)+(z(s)-z(t))\dot{z}(s)}{(y(s)-y(t))^2+(z(s)-z(t))^2} \right)dt + \Lambda\left(a\phi(s)\dot{y}(s)-c\right) \right)ds$$

I'm wondering as to what exactly the best way to deal with the inner integral is. I've considered pulling out the inner integral, and rewriting the whole thing as:

$$J\left[\phi,y,z\right]=\int_0^b \int^b_0\left( \dot{\phi}(t)\frac{(y(s)-y(t))\dot{y}(s)+(z(s)-z(t))\dot{z}(s)}{(y(s)-y(t))^2+(z(s)-z(t))^2} + \Lambda\left(\frac{a}{b}\phi(s)\dot{y}(s)-\frac{c}{b}\right) \right)dsdt$$

and then taking the integrand as my Lagrangian in the Euler-Lagrange equation, but then I seem to run into trouble with having to different independent variables as the argument to the functions ##\phi, y,z##. Anyone have any experience with similar problems? Thanks for your help!

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# Calculus of variations question

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