I am getting correct equations on using the Lagrangian method in Systems with no non conservative forces, but when I use it in Systems with friction, sometimes I get correct equations, and sometimes I do not. Most of the equations have some problem with the coefficients of the frictional forces.
For example, let us take a look at this system........
Here $f_1,f_2$ are the frictional forces(and not the coefficients of friction)
now, when we apply Newton's second law, we see that this is wrong and the coefficient of $f_1$ should have been $2$.
Why is the problem?....on the right hand side I have written the generalized force and the the two Lagrangian terms on the left hand side. Please help me out.
(Please do not down vote or close my question.....if there is a problem, please put a comment)
EDIT(Someone told me to add this):
Let $a$ be the acceleration of $m_2$ towards right. So we get the equations
$$m_1 g-T-f_1=2m_1a$$ and $$2T-f_2=m_2a$$ from FBDs for blocks with masses $m_1$ and $m_2$ respectively.
From here, we get