# Maximizing an integral/Solving the associated Euler-Lagrange equation

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.

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