In some texts (e.g. Taylor's Classical Mechanics), the generalized force is defined to be (I'll simplify to one particle in one dimension for ease of notation): $Q \equiv \frac{\partial{L}}{\partial{q}}$.
While other texts (e.g. Goldstein's Classical Mechanics) define the generalized force to be: $Q \equiv F\frac{\partial{x}}{\partial{q}}$.
These two definitions are equivalent if the kinetic energy is independent of generalized coordinates $q$, but this is not the case in general. Adopting the former definition makes sense because then Lagrange's equation can be written as: $\dot{p} = Q$, which maintains the form of Newton's Second Law, where $p$ is the generalized momentum. On the other hand the latter definition makes sense because then it is the case that: $dW = Q dq$, which maintains the form of the definition of work in the generalized coordinates.
Are one of these two definitions ``correct''? Nowhere can I find this distinction discussed in my available texts; if one adopts the latter definition (generalized force = $F\frac{\partial{x}}{\partial{q}}$), then what is $\frac{\partial{L}}{\partial{q}}$ called?
