{\displaystyle {\frac {{\rm {d}}L}{{\rm {d}}t}}=\sum _{i}{\dot {q}}_{i}{\frac {\rm {d}}{{\rm {d}}t}}{\frac {\partial L}{\partial {\dot {q}}_{i}}}+\sum _{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}{\ddot {q}}_{i}=\sum _{i}{\frac {\rm {d}}{{\rm {d}}t}}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}{\dot {q}}_{i}\right).}