Constraint power is the rate of actual work done by constraint forces. For particles with Cartesian velocities and constraint forces , 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, , 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 on one particle, write . Differentiation along its motion gives , henceA fixed smooth surface has 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 on a horizontal smooth wire receives normal force and power , 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 , and the two endpoint speed magnitudes are . If one mass rises while the other falls, the string supplies powers and : it transfers energy between the bodies while supplying zero net energy to the pair.
With , and , Newton's second law givesAt , the two tension powers are and . Their cancellation leaves gravity's net power, , equal to . This constraint power transfer distinguishes an ideal connection that redistributes energy from an externally driven moving support that adds energy to the chosen system.
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