
4.1 Variational calculus
Problem:
Find admissible extremals of the functional:
\[
\begin{array}{l}
F(x)=\int_{0}^{\frac{\pi}{2}}\left(\dot{x}^{2}-2 \cos t x\right) d t \rightarrow \text { extr }, \\
x(0)=1, \quad x\left(\frac{\pi}{2}\right)=0 .
\end{array}
\]