= Solution
Let $p_i=\partial L/\partial\dot q_i$ and consider the infinitesimal fixed-time transformation $\delta q_i=\epsilon K_i(\mathbf q)$. It induces $\delta\dot q_i=\epsilon\,dK_i/dt$. Expanding the <Lagrangian> gives
$$
\frac{\delta L}{\epsilon}=\sum_i\left(\frac{\partial L}{\partial q_i}K_i+p_i\frac{dK_i}{dt}\right).
$$
On a solution of the <Euler-Lagrange equations>, $\partial L/\partial q_i=\dot p_i$, so this equals $d(\sum_ip_iK_i)/dt$. Exact invariance makes the left side zero, proving the fixed-time form of <Noether's theorem>:
$$
\boxed{\frac d{dt}\left(\sum_ip_iK_i\right)=0.}
$$
The symmetry direction need not coincide with one original coordinate axis; $\mathbf K$ can depend on the generalized coordinates. The conclusion is evaluated on physical trajectories, while the assumed invariance is a property of the <Lagrangian> under the transformation.
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