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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.

An ideal mechanical constraint is a restriction whose constraint forces do zero total virtual work for every admissible virtual displacement at a fixed time:The condition concerns the permitted variations of the whole chosen system. It does not require every constraint force to vanish, and it does not necessarily imply zero actual constraint power. Stewart's treatment distinguishes a displacement along a moving constraint from a virtual displacement made while its time is held fixed.^[ref]

For smooth independent holonomic constraints $f_a({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N,t)=0$, allowed variations obeyConstraint forces of the form ${\mathbf{F}}_i^{\mathrm{c}}={\sum }_a{\lambda }_a{\nabla }_i{f}_a$ therefore satisfy the ideal condition, with Lagrange multipliers ${\lambda }_a$. The argument explains why reaction forces disappear from the virtual-work equation without being absent from the apparatus.

For an original stationary example, constrain a particle of mass $0.20\mathrm{kg}$ to a fixed horizontal circle of radius $0.50\mathrm{m}$. At speed $3m/s$ the required radial force magnitude is $m{v}^2/R=3.6\mathrm{N}$. The allowed virtual displacement is tangential, so its scalar product with that radial normal reaction is zero. The force nevertheless produces the centripetal acceleration needed to maintain the trajectory.

A moving support shows the limitation of equating ideality with energy conservation. For the Cartesian constraint $x=a(t)$, admissible virtual displacements satisfy $\delta x=0$, whereas the actual velocity satisfies $\dot{x}=\dot{a}$. A reaction $F_x$ has zero virtual work but actual power $F_x\dot{a}$. An actuator moving the support can thus supply or remove mechanical energy. A force readout combined with the support's measured speed would detect that exchange.

For a fixed holonomic restriction, actual velocities obey the same homogeneous tangency condition as virtual displacements, giving zero total constraint power. For linked bodies, that total can cancel while individual powers are nonzero, as in constraint power transfer. Friction and compliance require their own model: calling a surface a mechanical constraint does not establish the ideal virtual-work property.

An ideal rigid link constrains the distance between two attachment points to a fixed length while transmitting an axial constraint force. In the model used here the link is massless, its end joints rotate freely, and it carries tension or compression without bending or stretching. For endpoint positions ${\mathbf{r}}_A,{\mathbf{r}}_B$,This is a holonomic constraint, with reactions that can be represented by Lagrange multipliers. The virtual-work treatment of constrained motion distinguishes these reaction forces from the applied forces that remain in reduced equations.^[ref]

Define the axial unit vector $\mathbf{n}=({\mathbf{r}}_B-{\mathbf{r}}_A)/\ell$. Differentiating the fixed-length condition givesThus endpoint velocities have equal axial components, while transverse relative motion can rotate the link. For an original planar example, let ${\mathbf{r}}_B-{\mathbf{r}}_A=(0.30,0.40)\mathrm{m}$, so $\ell =0.50\mathrm{m}$ and $\mathbf{n}=(0.6,0.8)$. The relative velocity $(-0.8,0.6)m/s$ is perpendicular to the link and corresponds to angular speed $2rad/s$. The length stays constant even though the endpoint velocities differ.

A second derivative exposes the required relative acceleration:In the example it is $-2m/{\mathrm{s}}^2$ along the link. Preserving length does not permit setting the relative acceleration to zero during rotation.

For axial force $+T\mathbf{n}$ on A and $-T\mathbf{n}$ on B, their total actual power is $T\mathbf{n}\cdot ({\mathbf{v}}_A-{\mathbf{v}}_B)=0$. Individual powers can be nonzero, giving constraint power transfer. A freely hinged rigid strut can carry either sign of axial force in this idealization; a taut ideal string can transmit tension but cannot enforce fixed separation by compression when it goes slack.

A real linkage approximates the model only while axial extension, joint friction and link inertia are negligible for the chosen motion. Tracking endpoint separation during loading tests the assumed length constraint; a load cell and position measurements can test the associated force and power balance.