The Beltrami Identity:
If the Lagrangian $\:L\left(y,y',x\right)\:$ of a system does not depend explicitly on $\:x$, that is \begin{equation} \dfrac{\partial L}{\partial x}=0 \tag{01}\label{01} \end{equation} then from the Euler-Lagrange equation \begin{equation} \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\partial L}{\partial y'}\right)-\dfrac{\partial L}{\partial y}=0 \tag{02}\label{02} \end{equation} we have \begin{equation} \dfrac{\mathrm{d}}{\mathrm{d}x}\left(y'\dfrac{\partial L}{\partial y'}-L\right)=0 \tag{03}\label{03} \end{equation} so \begin{equation} y'\dfrac{\partial L}{\partial y'}-L = \text{constant of the motion} \tag{04}\label{04} \end{equation}