<Dd> ∑ k = 1 N m k a k ⋅ ∂ r k ∂ q j = d d t ∂ T ∂ q _̇ j − ∂ T ∂ q j . (\ displaystyle \ sum _ (k = 1) ^ (N) m_ (k) \ mathbf (a) _ (k) \ cdot (\ frac (\ partial \ mathbf (r) _ (k)) (\ partial q_ (j))) = (\ frac (\ mathrm (d)) (\ mathrm (d) t)) (\ frac (\ partial T) (\ partial (\ dot (q)) _ (j))) - (\ frac (\ partial T) (\ partial q_ (j))) \, .) </Dd> <P> Now D'Alembert's principle is in the generalized coordinates as required, </P> <Dl> <Dd> ∑ j = 1 n (Q j − (d d t ∂ T ∂ q _̇ j − ∂ T ∂ q j)) δ q j = 0, (\ displaystyle \ sum _ (j = 1) ^ (n) \ left (Q_ (j) - \ left ((\ frac (\ mathrm (d)) (\ mathrm (d) t)) (\ frac (\ partial T) (\ partial (\ dot (q)) _ (j))) - (\ frac (\ partial T) (\ partial q_ (j))) \ right) \ right) \ delta q_ (j) = 0 \,,) </Dd> </Dl> <Dd> ∑ j = 1 n (Q j − (d d t ∂ T ∂ q _̇ j − ∂ T ∂ q j)) δ q j = 0, (\ displaystyle \ sum _ (j = 1) ^ (n) \ left (Q_ (j) - \ left ((\ frac (\ mathrm (d)) (\ mathrm (d) t)) (\ frac (\ partial T) (\ partial (\ dot (q)) _ (j))) - (\ frac (\ partial T) (\ partial q_ (j))) \ right) \ right) \ delta q_ (j) = 0 \,,) </Dd>

What conditions are necessary for mechanical work to be done