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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.

Constraint power transfer is redistribution of energy between parts of a mechanically connected system through constraint forces. An ideal connection may exert nonzero power on each body while exerting zero total power on the combination. Power on a body is the actual force–velocity scalar product; the zero-virtual work property of an ideal mechanical constraint concerns the sum over admissible virtual displacements.^[ref]

An original ideal rigid link calculation makes the cancellation explicit. Let two endpoint positions be ${\mathbf{r}}_A$ and ${\mathbf{r}}_B$, joined by a massless link of fixed length $\ell$. SetFor axial tension $T$, the link exerts $+T\mathbf{n}$ on A and $-T\mathbf{n}$ on B. The powers supplied to the endpoints areThe endpoint velocities need not be identical: the link can rotate. Only their components along the link must agree.

For stipulated $T=12\mathrm{N}$ and common axial endpoint speed $0.40m/s$, the link supplies $+4.8\mathrm{W}$ to A and $-4.8\mathrm{W}$ to B. It transfers energy without supplying net energy to their combined system. A load cell measuring axial force and tracked endpoint positions measuring velocities would permit these two powers to be compared. External forces determine how the endpoints subsequently accelerate.

Equal and opposite forces alone do not prove power cancellation. If the connector stretches, $\dot{\ell }=\mathbf{n}\cdot ({\mathbf{v}}_B-{\mathbf{v}}_A)$ and the same axial-force calculation givesAn elastic connector can store the corresponding elastic energy; a dissipative connector can convert mechanical energy into heat. Likewise, an externally prescribed moving support can deliver nonzero net constraint power to a subsystem. The location of the system boundary determines whether a measured exchange is internal transfer or externally supplied work.

Iain W. Stewart is a theoretical nuclear and particle physicist at the Massachusetts Institute of Technology, where his research develops and applies effective field theory to interactions of elementary particles, especially quantum chromodynamics. MIT's biographical sketch records his joining its physics faculty in January2003. His research description emphasizes choosing the relevant degrees of freedom and symmetries, then organizing calculations through power counting.^[ref]

A2001 paper with Christian W. Bauer, Sean Fleming and Dan Pirjol constructs an effective theory for energetic quarks interacting with collinear and soft gluons. This is part of the development of soft-collinear effective theory. The primary paper identifies an energetic scale $Q$, a transverse-momentum scale $p_{\perp }$ and a lower scale $p_{\perp }^2/Q$. It also matches currents and uses renormalization group equations to organize the calculation.^[ref]

An original scale-accounting example makes the hierarchy concrete. In units where the speed of light is1, choose illustrative values $Q=100\mathrm{GeV}$ and $p_{\perp }=10\mathrm{GeV}$. ThenThese numbers are not experimental inputs from the paper. They show why an interaction can involve several parametrically different momentum scales even when it belongs to one collision process. Power counting tracks powers of a small ratio such as $\lambda$, making the size of retained and omitted terms explicit within the chosen framework. It does not replace comparison with data or justify an approximation outside its scale assumptions.

Stewart also authored Advanced Classical Mechanics by Iain W. Stewart, whose treatment of moving holonomic constraints distinguishes an actual displacement during elapsed time from a virtual displacement at fixed time.^[ref] This connects his teaching to the careful definition of mathematical variations used in mechanics.

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.

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.