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?
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