 Time-dependent holonomic constraint

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.