= Virtual work
{wiki}
= Principle of virtual work
{synonym}
<Virtual work> is the sum of applied <forces> dotted with infinitesimal <displacements> compatible with the instantaneous <mechanical constraints>. The variation is taken at fixed <time>; it need not be a displacement the system actually makes during a short time interval. In generalized coordinates it has the form
$$
\delta W=\sum_i\mathbf F_i\cdot\delta\mathbf r_i=\sum_jQ_j\,\delta q_j,
$$
which defines the <generalized forces> $Q_j$.
For a bead on a smooth fixed wire, an admissible virtual displacement is tangent to the wire. The <normal reaction> is perpendicular to it and contributes no <virtual work>, allowing that unknown reaction to be eliminated from the equilibrium condition. For such ideal constraints, static <equilibrium> requires the applied <virtual work> to vanish for every admissible virtual displacement. The same handling of ideal constraint reactions is useful when deriving the <Euler Lagrange equations>.
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