Editing Virtual work (section)

===D'Alembert's form of the principle of virtual work===
D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system.  Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that
<math display="block"> \delta W = (Q_1 + Q^*_1)\delta q_1 + \dots + (Q_m + Q^*_m)\delta q_m = 0,</math>
for any set of virtual displacements ''δq<sub>j</sub>''.  This condition yields ''m'' equations,
<math display="block"> Q_j + Q^*_j = 0, \quad j=1, \ldots, m,</math>
which can also be written as
<math display="block"> \frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j} = Q_j, \quad j=1,\ldots,m.</math>
The result is a set of m equations of motion that define the dynamics of the rigid body system, known as [[Lagrangian mechanics|Lagrange's equations]] or the '''generalized equations of motion'''.

If the generalized forces Q<sub>j</sub> are derivable from a potential energy ''V''(''q''<sub>1</sub>,...,''q''<sub>''m''</sub>), then these equations of motion take the form
<math display="block"> \frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j} = -\frac{\partial V}{\partial q_j}, \quad j=1,\ldots,m.</math>

In this case, introduce the [[Lagrangian mechanics|Lagrangian]], {{math|1=''L'' = ''T'' − ''V''}}, so these equations of motion become
<math display="block"> \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_j} - \frac{\partial L}{\partial q_j} = 0 \quad j=1,\ldots,m.</math>
These are known as the [[Euler–Lagrange equation|Euler-Lagrange equations]] for a system with m degrees of freedom, or '''Lagrange's equations of the second kind'''.