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Case 1

I have the following integral which I need to numerically maximize. Here $v = v(u)$.

$$V = \int_{u_l}^{u_r} du\sqrt{\frac{4}{(1+uv)^2} \frac{(1-uv)^2}{(1+uv)^2}\frac{dv}{du}} $$

The allowed range for $u$ and $v$ is $uv <1$ and $uv<-1$. Here the $u_l$ and $u_r$ can be chosen as to be each lying on the disconnected part of the hyperbola $uv = -(1- \epsilon)$, where $\epsilon$ is a small number as compared to 1 so as to respect the above inequality. How do I numerically maximize V with these conditions?

Case 2

I tried a different route to the problem in order to get an analytical answer. Basically extremizing the integral $V$ gives us the Euler=Lagrange equation.

$$ \frac{dv}{du}\left(2 u\frac{1-uv}{1+uv} + u\right) = \frac{v}{2} \frac{d^2v}{du^2} + v\frac{1-uv}{1+uv} $$

I was not able to solve this analytically. This should now be solved by putting boundary conditions each lying on the disconnected part of the hyperbola $uv = -(1- \epsilon)$. Here again the allowed range for $u$ and $v$ is $uv <1$ and $uv<-1$.

Can somebody help me with this? Thanks.

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  • 1
    $\begingroup$ If you want to solve numerically, then why not defined $u_l,u_r$ in numerical form? $\endgroup$ – Alex Trounev Apr 29 at 13:41

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