= Time-dependent holonomic constraint
{title2=$f(q,t)=0$}
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.<Advanced Classical Mechanics by Iain W. Stewart>[§1.4, printed pp. 17–19, equation (1.62), Figure 1.10 and moving-wire discussion]{ref}
Differentiating one smooth <holonomic constraint> along an actual trajectory gives
$$
\sum_j\frac{\partial f}{\partial q_j}\dot q_j+\frac{\partial f}{\partial t}=0.
$$
A permitted <virtual displacement>, however, compares neighboring allowed positions at the same instant:
$$
\sum_j\frac{\partial f}{\partial q_j}\,\delta q_j=0.
$$
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> is
$$
P_{\mathrm c}=\sum_j Q_j^{\mathrm c}\dot q_j
=-\lambda\frac{\partial f}{\partial t}.
$$
Thus 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> gives
$$
N-mg=m\ddot h,\qquad
N=m(g+\ddot h),\qquad P_{\mathrm c}=N\dot h.
$$
The bead's <mechanical energy>, including <gravitational potential energy>, is
$$
E=\frac12m(\dot x^2+\dot h^2)+mgh,
\qquad \frac{dE}{dt}=m(g+\ddot h)\dot h=N\dot h,
$$
because horizontal motion is unforced. For $m=0.5\,\mathrm{kg}$, $g=9.81\,\mathrm{m/s^2}$ and $h(t)=\tfrac12at^2$ with $a=2\,\mathrm{m/s^2}$, 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.
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