I have been going over sections 1.4-1.5 of Goldstein's Classical Mechanics where the equation for generalized force
\begin{equation} Q_j=-\frac{\partial U}{\partial q_j} +\frac{d}{dt}\frac{\partial U}{\partial \dot{q_j} } \end{equation}
where $Q_j$ is the generalized force resulting from a generalized potential U that is a function of generalized coordinate $q_j$. From what I understand, this equation seems to be motivated from an extension of the equation for generalized force as a function of the kinetic energy such that Lagrange's equations still hold for velocity dependent forces.
What I don't understand is why the derivation for the equation for generalized force as a function of kinetic energy is derived from a situation where each particle in the system is in equilibrium. I have tried to look for a different derivation elsewhere, but other sources I have found for derivations of the generalized force use the same approach (Wikipedia, lecture notes, lecture slides, LibreTexts).
Later the extension of this generalized force is used to derive the Lagrangian for a charge experiencing a Lorentz force, which is not in equilibrium. Is it sufficient to be able to use this equation for generalized force for velocity dependent potential because the Lagrangian results in the correct force equation?
