4.1 Variational calculus

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