# Derivation of the Lagrange equations of motion from d'Alembert's Principle specifically for rigid bodies

All of the proofs of the Lagrange equations of motion from d'Alembert's Principle I've seen so far deal exclusively with the force-inertial force balance for particles ($$F-ma=0$$). Despite this, the derived results are used for rigid bodies, even though $$F-ma_G = 0$$ doesn't give the whole picture for the motion of a rigid body, for which a moment-angular momentum balance must also be carried out. Shouldn't analogous results with virtual motions be derived explicitly from $$M_G - \frac{d}{dt}H_G= 0$$ as well? Does anyone have a proof?