Constraint power
= Constraint power
{title2=$P_{\mathrm c}=\mathbf F_{\mathrm c}\cdot\mathbf v$}
<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 is
$$
P_{\mathrm c}=\sum_i\mathbf F_i^{\mathrm c}\mathbin{\cdot}\mathbf v_i.
$$
It 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.<Advanced Classical Mechanics by Iain W. Stewart>[§1.4, printed pp. 17–19, equation (1.62), moving-wire Figure 1.10 and equation (1.64)]{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 $\nabla f\cdot\mathbf v+\partial_t f=0$, hence
$$
P_{\mathrm c}=-\lambda\,\partial_t f.
$$
A 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 $g=9.81\,\mathrm{m/s^2}$, <Newton's second law> gives
$$
a=\frac{m_2-m_1}{m_1+m_2}g=1.962\,\mathrm{m/s^2},
\qquad T=\frac{2m_1m_2}{m_1+m_2}g=2.3544\,\mathrm N.
$$
At $v=0.50\,\mathrm{m/s}$, 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.