I have the following problem that is in Symon Mechanics cap. 9 exercise 8:
Masses $m$ and $2m$ are suspended from a string of lenght $l_1$ wich passes over a pulley. Masses $3m$ and $4m$ are similarly suspended by a string of lenght $l_2$ over another pulley. There two pulleys hang from the ends of a string of lenght $l_3$ over a third pulley fixed. Set up Lagrange eq's and find the accelerations and tensions in the strings.
Then this is my goal: See the picture below, it represents the system. 
In the figure 'Polia' $n =$ 'Pulley' $n$. Note that we have marked an origin at the center of the pulley that hang the other two. With that I've constructed the coordinates of each mass:
\begin{equation} 1m : \left \{ \begin{matrix} \tilde{x}_{1} = -R_3-R_1 \\ \tilde{y}_1 = -(x_5+x_1) \end{matrix}\right. \end{equation}
\begin{equation} 2m : \left \{ \begin{matrix} \tilde{x}_{2} = -R_3+R_1 \\ \tilde{y}_2 = -(x_5+x_2) = -x_5 - (l_1-x_1) \end{matrix}\right. \end{equation}
\begin{equation} 3m : \left \{ \begin{matrix} \tilde{x}_{3} = R_3-R_2 \\ \tilde{y}_3 = -(x_6+x_3) = -(l_3-x_5)-x_3 \end{matrix}\right. \end{equation}
\begin{equation} 4m : \left \{ \begin{matrix} \tilde{x}_{4} = R_3+R_2 \\ \tilde{y}_4 = -(x_6+x_4) = -(l_3-x_5) - (l_2-x_3) \end{matrix}\right. \end{equation}
with the constrains
\begin{equation} \left \{ \begin{matrix} x_1+x_2 = l_1 \\ x_3+x_4 = l_2 \\ x_5+x_6= l_3 \end{matrix}\right. \end{equation}
This will imply in the below kinetic and potential energy
$$T = \frac{m}{2}(\dot{x}_5+\dot{x}_1)^2 + \frac{2m}{2}(\dot{x}_5 - \dot{x}_1)^2 + \frac{3m}{2}(-\dot{x}_5+\dot{x}_3)^2 +\frac{4m}{2}(\dot{x}_5+\dot{x}_3)^2$$
and
$$U = -mg(x_1+x_5) - 2mg(x_5 + l_1 -x_1) - 3mg(l_3-x_5+x_3)-4mg(l_3+l_2-x_5-x_3)$$
Question: Have I missed something? The accelerations I'll get from the equations of motion for $x_1,x_3,x_5$ will be the accelerations of the string, so I just need to construct $\mathcal{L} = T - U$ and set the Euler-Lagrange Equations? Is this a wrong way to write the lagrangian? If I construct the lagrangian as I stated, it will be the correct lagrangian for the system?