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.

A virtual displacement is an infinitesimal change of configuration compatible with the instantaneous mechanical constraints, evaluated at fixed time. It is a variation among admissible configurations, rather than necessarily the displacement made during a time step. In generalized coordinates, writeSubstitution into the applied virtual work identifies the generalized forces conjugate to the coordinates.^[ref]

An original moving-constraint example makes the fixed-time condition visible. Consider a particle constrained to the circleFor a virtual displacement, freeze $t$ and therefore $R$; first-order compatibility gives $x\delta x+y\delta y=0$. For an actual motion, differentiation with respect to time instead givesAt $(x,y)=(R,0)$ with $R>0$, every admissible virtual displacement has $\delta x=0$, while an actual velocity must have $\dot{x}=\dot{R}$. If the circle is expanding at $0.020m/s$, the actual radial velocity is nonzero even though every virtual displacement is tangent to the instantaneous circle.

Consequently, an ideal radial constraint force can have zero virtual work but nonzero instantaneous power during this driven motion. Zero work on admissible variations is not the statement that a moving support never transfers energy. The example is a time-dependent holonomic constraint; the variations satisfy its instantaneous linearized condition. In a beam, the same distinction lets one impose zero endpoint virtual displacement while leaving the virtual endpoint slope free, as in the moment-free natural boundary condition for a beam.

Generalized forces are the coefficients of virtual work in a set of generalized coordinates. A translational coordinate has a force as its coefficient, while an angular coordinate has a torque. A generalized force conjugate to $q_i$ has units of energy divided by the units of $q_i$. Forces not already included in the Lagrangian enter the Euler-Lagrange equations as $d(\partial L/\partial {\dot{q}}_i)/dt-\partial L/\partial q_i=Q_i$.