 Virtual work

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 formwhich 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.