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My understanding about constraint force is that it is a force which limits the geometry of particle's motion. For example, situations such as the particle trapped in a track or limited in domain can be assumed constraint force.

But in this point of view, I couldn't understand why friction is constraint force. In Lagrangian formulation, we divide forces into two part, $F= F_\text{applied} + F_\text{constraint}$.

If particle moves in one dimension, and assuming there exists sliding friction, that particle can move anywhere. The sliding friction never restrict the domain that particle can move. so I think the sliding force is applied force, rather than constraint force.

Can anyone clarify why friction is constraint force?

user8718165
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user42298
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1 Answers1

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Rolling friction can be considered as constraint force.

Consider a circular ring rolling down a wedge without slipping. Static friction at the contact provides a constraint type relation between $x$ (distance of center of mass from a fixed point) and the rotational coordinate ($\theta$) of the ring. The constraint equation is nonholonomic — $x'=r(\theta')$, where $r$ is the radius of the ring.

This so because the static friction made the ring to rotate. The ring might have slipped, but the presence of static friction constrained the body to rotate without slipping. The virtual work principle is also not violated for rolling friction. But this not the case with sliding friction. It's not a constraint force and the system with sliding friction must be excluded when d'Alambert equation is considered because the virtual work is not zero in this case.

MarianD
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sshzaidi
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