# Why friction force is force of constraint?

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, $$F= F(applied) + F(constraint)$$.

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?

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• Friction forces are not conservative forces and hence a Lagrangian description of them is ill-suited. Constraint forces originating from holonomic constraints do not produce work and therefore can be suitably described by a Lagrangian formalism (contrary to friction forces). Thus could you please provide a link where it is said that friction forces can be seen as constraint forces? – gatsu Mar 11 '14 at 10:50
• GoldStein, calssical mechanics, p17 or ame-www.usc.edu/bio/udwadia/papers/… or google.co.kr/… in that paper, search with friction – user42298 Mar 11 '14 at 11:05
• Comments to the question (v3): Static (as opposed to kinetic or sliding) friction can be viewed as a constraint force. For more on D'Alembert's principle, see also e.g. physics.stackexchange.com/q/8453/2451 , physics.stackexchange.com/q/82884/2451 , and links therein. – Qmechanic Mar 11 '14 at 11:06
• sliding friction in 1D and in absence of potential forces can be thought of as a kinetic energy loss $\dot{K}=-f(\dot{x}(t))$ in its simplest form. If you know or imagine you know $x(t)$ and the loss friction function $f$, you can integrate the above equation and get a dynamical constraint of the form $\dot{x}(t)=g(x,\dot{x},t)$. That's the way I interpret the link you gave me. Please tell me if I am interpreting it in a wrong way. I also wonder about the practical usefulness of such a formalism and vocabulary. – gatsu Mar 11 '14 at 13:35