# When is the principle of virtual work valid?

The principle of virtual work says that forces of constraint don't do net work under virtual displacements that are consistent with constraints.

Goldstein says something I don't understand. He says that if sliding friction forces are present then the principle of virtual work fails. But then he proceeds to say that this doesn't really matter because friction is a macroscopic phenomena.

The only way I can interpret this is for the friction forces to be a constraint force. But I thought constraint forces were pretty much always forces whose net effect is known but their exact force exerted is difficult to know. For friction we know its force exerted, so why would you treat it as a constraining force?

I also don't understand why friction being a macroscopic phenomena means it doesn't matter for this. Is it because we considering a system of particles?

• – Qmechanic Feb 20 '12 at 15:59

1) According to usual terminology we wouldn't call a sliding friction force a constraint force as it doesn't enforce any constraint. (No pun intended.) In other words, a sliding friction does not by itself constrain the particles to some constraint subsurface, i.e., the particles can still move around everywhere.

On the other hand, rolling friction and static friction may actually impose a constraint, so they can be constraint forces.

2) In more detail, Goldstein says on p.17 in Chapter 1 of the book Classical Mechanics that

[The principle of virtual work] is no longer true if sliding friction forces are present [in the tally of constraint forces], ...

Goldstein goes on to say that

this restriction is not unduly hampering.

What he has in mind is, that we can still at least analyze and study many systems of fundamental/microscopic point particles (which is anyway the most important case!) with the principle of virtual work, because there are often no sliding friction forces involved down at these scales.

In particular, Goldstein does not imply that sliding friction forces are not important in macroscopic systems.

3) Later in chapter 1, Goldstein confronts us with another problem with sliding friction forces. They cannot be described with the help of a velocity-dependent potential $U$ but only in terms of Rayleigh's dissipation function ${\cal F}$. This is related to the fact that there is no action principle for systems with sliding friction forces.

• I appreciate the response. I understand your answer regarding how sliding frictional forces are not 'unduly hampering'. But I still don't see why the principle of virtual work is not true when sliding friction forces are present? To me, I am thinking that if the friction force is not a constraint force, then the principle of virtual work should say nothing about it. – countunique Feb 19 '12 at 22:53
• I updated the answer. – Qmechanic Feb 19 '12 at 23:32