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When you derive the Euler-Lagrange equation using D'Alembert's principle, you find that $$\frac{d}{dt}\frac{\partial L}{\partial \dot{q_j}}-\frac{\partial L}{\partial q_j}=Q_j$$ where $$Q_j=\vec{F}·\frac{d\vec{r}}{dq_j}$$ (Goldstein, p. 23-24). But if you use Hamilton's principle and some calculus of variations (Goldstein, p. 44 ff.), you only obtain the formula with $Q_j=0$. Is it possible to put the generalized forces into the Euler-Lagrange equation without deriving it from D'Alembert's principle? I tried to use some kind of lagrange multiplier to insert additional forces, but was not able to derive it.

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Only if all the generalized forces have generalized potentials. Which need not be the case, in particular for dissipative systems, cf. e.g. this Phys.SE post. So No, that's in general not possible. See also this related Phys.SE post.

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