4.25 Variational calculus

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