4.24 Variational calculus

Problem: Solve the problem with moving boundaries: \[ \begin{array}{l} \int_{0}^{1}\left(\frac{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}}{2}-x_{1} x_{2}\right) d t \rightarrow \text { extr }, \\ x_{1}(0)=x_{2}(0)=1 \end{array} \]