Ideal mechanical constraint

An ideal mechanical constraint is a restriction whose constraint forces do zero total virtual work for every admissible virtual displacement at a fixed time: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.^[ref]

For smooth independent holonomic constraints $f_a({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N,t)=0$, allowed variations obeyConstraint forces of the form ${\mathbf{F}}_i^{\mathrm{c}}={\sum }_a{\lambda }_a{\nabla }_i{f}_a$ 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 $3m/s$ the required radial force magnitude is $m{v}^2/R=3.6\mathrm{N}$. 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.