= Virtual displacement
{title2=$\delta w$}
A \b[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>, write
$$
\delta\mathbf r_i=\sum_j\left.\frac{\partial\mathbf r_i}{\partial q_j}\right|_t\delta q_j,\qquad\delta t=0.
$$
Substitution into the applied <virtual work> identifies the <generalized forces> conjugate to the coordinates.<Newtonian Dynamics by Richard Fitzpatrick>[Lagrangian Dynamics, Generalized Coordinates, equations592–593; Generalized Forces, equations594–597]{ref}
An original moving-constraint example makes the fixed-time condition visible. Consider a particle constrained to the circle
$$
x^2+y^2=R(t)^2.
$$
For 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 gives
$$
x\dot x+y\dot y=R\dot R.
$$
At $(x,y)=(R,0)$ 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 $0.020\,\mathrm{m/s}$, 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>.
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