I am asked to find the symmetries and conserved quantities for a system with the following Lagrangian:
$$\mathscr{L}=\frac{1}{2}m\dot{q}^2-af(t)q,$$
where $a$ is some constant and $f(t)$ is an arbitrary (but integrable) function of time.
I find this problem non trivial because the Lagrangian has no cyclic coordinates and it is a function of time, so neither the conjugate momentum $p$ or the energy $E$ are conserved quantities.
I proceed trying to find some symmetry such that $\delta\mathscr{L}=dg/dt$ (or maybe $=0$, the idea is that this condition is such that the Euler-Lagrange equations obtained via variational principle are left invariant). Then applying Noether's theorem, the conserved quantity would be:
$$C=\bigg(\frac{\partial\mathscr{L}}{\partial\dot{q}}\dot{q}-\mathscr{L}\bigg)\delta{t}-\frac{\partial\mathscr{L}}{\partial\dot{q}}\delta{q}-g,$$
where $g$ may, or may not be zero. So, for the Lagrangian in consideration:
$$\begin{align}\delta\mathscr{L}&=\frac{\partial\mathscr{L}}{\partial\dot{q}}\delta\dot{q}+\frac{\partial\mathscr{L}}{\partial q}\delta q+\frac{\partial\mathscr{L}}{\partial t}\delta t\\ &=(m\dot{q})\delta\dot{q}+(-af(t))\delta q+(-a\frac{\partial f}{\partial t}q)\delta t\end{align}$$
The problem here is that I can't think of any symmetry that can satisfy Noether's condition. Is there any other test that can give me the correct symmetries? Or maybe I can know the conserved quantities by looking at the form of the Lagrangian but I lack the intuition?