= Ideal mechanical constraint
An \b[ideal mechanical constraint] is a restriction whose <constraint forces> do zero total <virtual work> for every admissible <virtual displacement> at a fixed time:
$$
\sum_i\mathbf F_i^{\rm c}\cdot\delta\mathbf r_i=0.
$$
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.<Advanced Classical Mechanics by Iain W. Stewart>[§1.4, printed pp. 17–19, equation 1.62, Figure 1.10 and equations 1.64–1.66]{ref}
For smooth independent <holonomic constraints> $f_a(\mathbf r_1,\ldots,\mathbf r_N,t)=0$, allowed variations obey
$$
\sum_i\nabla_i f_a\cdot\delta\mathbf r_i=0.
$$
Constraint forces of the form $\mathbf F_i^{\rm c}=\sum_a\lambda_a\nabla_i f_a$ 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 $3\,\mathrm{m/s}$ the required radial force magnitude is $mv^2/R=3.6\,\mathrm N$. 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.
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