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===Generalized inertia forces===
Let a mechanical system be constructed from n rigid bodies, B<sub>i</sub>, i=1,...,n, and let the resultant of the applied forces on each body be the force-torque pairs, '''F'''<sub>i</sub> and '''T'''<sub>i</sub>, ''i'' = 1,...,''n''.  Notice that these applied forces do not include the reaction forces where the bodies are connected.  Finally, assume that the velocity '''V'''<sub>i</sub> and angular velocities '''ω'''<sub>i</sub>, ''i''=1,...,''n'', for each rigid body, are defined by a single generalized coordinate q.  Such a system of rigid bodies is said to have one [[degree of freedom (mechanics)|degree of freedom]].

Consider a single rigid body which moves under the action of a resultant force '''F''' and torque '''T''', with one degree of freedom defined by the generalized coordinate q.  Assume the reference point for the resultant force and torque is the center of mass of the body, then the generalized inertia force Q* associated with the generalized coordinate q is given by
<math display="block"> Q^* = -(M\mathbf{A}) \cdot  \frac{\partial \mathbf{V}}{\partial \dot{q}} - ([I_R]\alpha+ \omega\times[I_R]\omega) \cdot \frac{\partial \boldsymbol{\omega}}{\partial \dot{q}}.</math>
This inertia force can be computed from the kinetic energy of the rigid body,
<math display="block"> T = \frac{1}{2} M \mathbf{V} \cdot \mathbf{V} + \frac{1}{2} \boldsymbol{\omega} \cdot [I_R] \boldsymbol{\omega},</math>
by using the formula
<math display="block"> Q^* = -\left(\frac{d}{dt} \frac{\partial T}{\partial \dot{q}} -\frac{\partial T}{\partial q}\right).</math>

A system of n rigid bodies with m generalized coordinates has the kinetic energy
<math display="block">T = \sum_{i=1}^n \left(\frac{1}{2} M \mathbf{V}_i \cdot \mathbf{V}_i + \frac{1}{2} \boldsymbol{\omega}_i \cdot [I_R] \boldsymbol{\omega}_i\right),</math>
which can be used to calculate the m generalized inertia forces<ref>T. R. Kane and D. A. Levinson, [https://www.amazon.com/Dynamics-Theory-Applications-Mechanical-Engineering/dp/0070378460 Dynamics, Theory and Applications], McGraw-Hill, NY, 2005.</ref>
<math display="block"> Q^*_j = -\left(\frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j}\right), \quad j=1, \ldots, m.</math>