I'm a high school student who attempted to use calculus of variations for a project.
I have this functional:
$$T(\theta)=\int_{0}^{57}\left(\frac{1}{10\cos\left(\theta\right)+w\left(x\right)}+λ\frac{10\sin\left(\theta\right)}{10\cos\left(\theta\right)+w(x)}\right)dx$$
Where I want to find the optimal function for $\theta$ to minimise time $T$ (and where $w(x)$ is just another known function not written for clarity).
Can I write it as follows?:
$$T\left(\theta\right)=\int_{0}^{57}F\left(x,\theta,\theta'\right)dx\leq 2x\ln 2$$
I'm confused because there's no $\theta'$ explicitly in my functional. So should it be instead like this?:
$$T\left(\theta\right)=\int_{0}^{57}F\left(x,\theta\right)dx$$
Also, since I'm only using the first part of the Euler-Lagrange equation to find my answer ($\frac{\partial{F}}{\partial{\theta}}=0$), maybe I don't even need to talk about calculus of variations? Is that why I can't find an answer in textbooks to my question?
Thank you very much for any help.