If the frame is not inertial, for example, in a rotating frame of reference, the "state-of-motion" fictitious forces are included in the above "coordinate" fictitious force expression. Also, if the "acceleration" expressed in terms of first-order time derivatives of the velocity happens to result in terms that are not simply second-order derivatives of the coordinates { q k } in time, then these terms that are not second-order also are brought to the force-side of the equation and included with the fictitious forces. From the standpoint of a Lagrangian formulation, they can be called generalized fictitious forces. See Hildebrand, for example.