Friction in Lagrange's mechanics

[Gloyn]

Hi, I'm new here ;)

Please, forgive me the possible grammar mistakes, as English is not my mother tongue and I try to do my best.

I've been studying Lagrange's equations recently, and I've noticed that I don't know how to add the frictional force to obtain the correct equations of motion. I started to search for some infomations, and I found this:

http://www.physicsinsights.org/weights_and_wedge_1.html

There's some simple case solved in two ways: without and with fricion. In the second case, the frictional force is written to be:

http://www.physicsinsights.org/lagrange_1.html#eqn-9 (equation (9))

And my question is: what the hell is that? ;) What are those fs, and generally why is it that form? I understand that it is a frictional force that depends on velocity. What would be the form of non-velocity-dependent frictional force? Would that be just F=mgμ, where m is mass, g is gravitational acclereation, μ is a coefficient of friction? How do I add it to Lagrange's equations? If someone has a while to explain it on an example, jest do the case with a mass m sliding down the ramp, where mass of the ramp is M, angle of elevation of the ramp is β, distance of the mass from the lower end of the ramp is q and the distance of the ramp from some given point on the plane is p. Coefficient of friction is μ.

Thanks in advance for all advice!

 

[AlephZero]

$f_1$ represents a Coulomb dynamic friction force which has constant magnitude but the direction is opposite to the motion of the object. That's why he writes it as $f_1 \dot q / | \dot q |$

$f_2$ represents a damping force that is proportional to velocity. That is a common way to model damping forces, but it isn't what most people would call "friction".