Virtual displacement ($\delta w$)

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.