= Solution
Let
$$
\xi_i(t)=\left.\frac{\partial Q_i(s,t)}{\partial s}\right|_{s=0}
$$
be the infinitesimal generator of the continuous symmetry, and let
$$
p_i=\frac{\partial L}{\partial\dot q_i}
$$
be the <canonical momentum>. <Noether theorem> states that along every trajectory satisfying the <Euler-Lagrange equations>,
$$
\boxed{J=\sum_{i=1}^np_i\xi_i}
$$
is conserved.
Indeed, differentiate the invariance of $L$ with respect to $s$ at zero:
$$
0=\sum_i\left(
\frac{\partial L}{\partial q_i}\xi_i
+\frac{\partial L}{\partial\dot q_i}\dot\xi_i
\right).
$$
On an Euler-Lagrange trajectory, $\partial L/\partial q_i=d p_i/dt$, so
$$
0=\sum_i(\dot p_i\xi_i+p_i\dot\xi_i)
=\frac d{dt}\sum_i p_i\xi_i.
$$
This proves the <Noether conserved quantity for a mechanical point symmetry>.
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