In the case of a time symmetry, we have \(f_i=0\) and \(g=1\):
\begin{align}
t'=t+\delta \epsilon
\tag{5.20}\end{align}
This gives the conserved quantity:
\begin{align}
Q=\sum_{i=1}^np_i\dot{q}_i-L
\tag{5.21}\end{align}
We have already seen that when the Lagrangian does not explicitly depend on time, the quantity:
\begin{align}
h=\sum_{i=1}^np_i\dot{q}_i-L
\tag{5.22}\end{align}
is conserved. Of course, if the Lagrangian does not depend on time, then the action will be invariant to a time translation.