Noether's theorem

Question

Can anyone explain to me where the functions $F$ and $Q$ come from?

Answer 1

For what it is worth, both $Q^{\alpha}$ and $F$ are objects that need to exist before we can apply Noether's theorem, i.e. they are assumptions. $Q^{\alpha}$ is the assumed infinitesimal transformation of the variable $q^{\alpha}$, while the total time-derivative of $F$ is assumed to be the infinitesimal change of the Lagrangian under said infinitesimal transformation, cf. eq. (7.2).