# On constraint forces and d'Alembert's Principle

## Main Question or Discussion Point

According to d'Alembert's Principle, the virtual work done by constraint forces must be zero.

I have a few things needing to be clarified. First, as we know from friction, d'Alembert's Principle is not always true (friction usually does work, and is not normal to the constraint surface). On the other hand, friction arises from velocity-dependent constraints (such as no-slip conditions). In fact, I am not even sure if friction can be considered a constraint force, since we could add it to the "applied forces".

Now, if we restrict d'Alembert's Principle to forces arising from holonomic constraints (constraints which do not depend on the velocities), it seems intuitively true. In that case, the principle says that the constraint forces must be perpendicular to the constraint surface at each point at any fixed moment in time. For example, if the constraint is $(x-t^2)^2 + y^2 + z^2 = l^2$ (path lies on a sphere accelerating in the x direction) then the normal force at each point is "obviously" perpendicular to the sphere (directed towards its center $(t^2, 0, 0)$).
In this example, the net work of the constraint forces on an object moving on the sphere is not zero (the sphere is accelerating) but the virtual work is zero (according to d'Alembert's Principle).

Please let me know if what I said is not correct (or could be said better).

Is d'Alembert's Principle, restricted to holonomic constraints, provable from Newton's Laws?
And if it is not, what would a counter-example look like (if any is known)?

Also, what are some non-holonomic constraints which agree with d'Alembert's Principle?