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Constraint power is the rate of actual work done by constraint forces. For particles with Cartesian velocities ${\mathbf{v}}_i$ and constraint forces ${\mathbf{F}}_i^{\mathrm{c}}$, the total isIt enters the balance of kinetic energy and, after accounting for conservative applied forces, mechanical energy. An ideal mechanical constraint instead imposes zero total virtual work, ${\sum }_i{\mathbf{F}}_i^{\mathrm{c}}\cdot \delta {\mathbf{r}}_i=0$, for allowed virtual displacements at fixed time. Those displacements need not be parallel to actual velocities when the permitted geometry moves.^[ref]

For a smooth ideal restriction $f(\mathbf{r},t)=0$ on one particle, write ${\mathbf{F}}^{\mathrm{c}}=\lambda \nabla f$. Differentiation along its motion gives $\nabla f\cdot \mathbf{v}+{\partial }_{t}f=0$, henceA fixed smooth surface has ${\partial }_{t}f=0$ and zero constraint power. A time-dependent holonomic constraint can have nonzero constraint power even without friction. For example, a bead carried upwards at constant speed $u$ on a horizontal smooth wire receives normal force $mg$ and power $mgu$, increasing its gravitational potential energy. The drive moving the wire supplies that energy.

Zero total constraint power also does not require zero power on every body. Consider two hanging masses connected by a taut, massless, inextensible string over an ideal massless, frictionless pulley. The shared tension is $T$, and the two endpoint speed magnitudes are $v$. If one mass rises while the other falls, the string supplies powers $+Tv$ and $-Tv$: it transfers energy between the bodies while supplying zero net energy to the pair.

With $m_1=0.20\mathrm{kg}$, $m_2=0.30\mathrm{kg}$ and $g=9.81m/{\mathrm{s}}^2$, Newton's second law givesAt $v=0.50m/s$, the two tension powers are $+1.1772\mathrm{W}$ and $-1.1772\mathrm{W}$. Their cancellation leaves gravity's net power, $(m_2-m_1)gv=0.4905\mathrm{W}$, equal to $dK/dt=(m_1+m_2)av$. This constraint power transfer distinguishes an ideal connection that redistributes energy from an externally driven moving support that adds energy to the chosen system.

A time-dependent holonomic constraint is a positional restriction $f(q,t)=0$ whose permitted configurations change explicitly with time. Here $q$ denotes the generalized coordinates. A bead confined to a moving wire is an immediate example: its allowed position is specified at each instant, but the wire carries the permitted set of positions through space. The distinction between actual motion and a virtual displacement at fixed time is essential.^[ref]

Differentiating one smooth holonomic constraint along an actual trajectory givesA permitted virtual displacement, however, compares neighboring allowed positions at the same instant:The missing time term is not an approximation. Actual velocity and a tangent virtual displacement obey different conditions when the constraint moves. In the following power calculation, take $q$ to be fixed Cartesian coordinates, so explicit time dependence belongs to the constraint rather than to a moving coordinate system.

For an ideal constraint, its generalized forces may be written $Q_j^{\mathrm{c}}=\lambda \partial f/\partial q_j$, with a Lagrange multiplier $\lambda$. They do zero virtual work, yet their actual constraint power isThus an ideal, frictionless moving constraint can exchange energy with the constrained body. Zero virtual work does not imply zero actual work. The actuator maintaining the imposed motion is part of the physical energy accounting.

For a worked apparatus model, put a bead of mass $m$ on a smooth horizontal wire whose height is prescribed as $z=h(t)$ in a uniform downward gravitational field of magnitude $g$. The wire permits horizontal motion and supplies a vertical normal force $N$. The constraint is $f=z-h(t)=0$, and Newton's second law givesThe bead's mechanical energy, including gravitational potential energy, isbecause horizontal motion is unforced. For $m=0.5\mathrm{kg}$, $g=9.81m/{\mathrm{s}}^2$ and $h(t)=\frac{1}{2}a{t}^2$ with $a=2m/{\mathrm{s}}^2$, the wire exerts $5.905\mathrm{N}$; at $t=1\mathrm{s}$ it delivers $11.81\mathrm{W}$ to the bead. A vertical normal force is perpendicular to every allowed horizontal virtual displacement, but parallel to part of the actual velocity. Measuring wire motion and bead load would expose precisely the energy transfer that is lost by treating the moving wire as a stationary, workless support.

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.