Second: As stated earlier, if the Lagrangian is independent of time, that is, it’s invariant under time translation, then energy is conserved. (This is nothing but the first integral of the calculus o...Second: As stated earlier, if the Lagrangian is independent of time, that is, it’s invariant under time translation, then energy is conserved. (This is nothing but the first integral of the calculus of variations, recall that for an integrand function \(\begin{equation} \sum_{i} \dot{q}_{i} \dfrac{\partial L}{\partial \dot{q}_{i}}=\sum_{i} \dot{q}_{i} \dfrac{\partial T}{\partial \dot{q}_{i}}=2 T
