In my control theory task and have example, it's just first step of task
I want to get some tips how do they go from 1 to 2 cases
In my task I have 4 material points and only get this while transforming from 1 to 2 (I think it's wrong)
$$ \begin{cases} m\ddot{x_1} = -c(x_1 - x_3) - c\frac{\sqrt{2}}{2}x_1 + u\cos{\alpha} \\ m\ddot{y_1} = -c(y_1 - y_4) - c\frac{\sqrt{2}}{2}y_1 + u\sin{\alpha} \\ \\ m\ddot{x_2} = -c(x_2 - x_4) - cx_2 + u\cos{\alpha} \\ m\ddot{y_2} = -c(y_2 - y_3) - cy_2 - u\sin{\alpha} \\ \\ m\ddot{x_3} = -c(x_3 - x_1) - c\frac{\sqrt{2}}{2}x_3 + u\cos{\alpha} \\ m\ddot{y_3} = -c(y_3 - y_2) - c\frac{\sqrt{2}}{2}y_3 + u\sin{\alpha} \\ \\ m\ddot{x_4} = -c(x_4 - x_2) - cx_4 - u\cos{\alpha} \\ m\ddot{y_4} = -c(y_4 - y_1) - cy_4 + u\sin{\alpha} \end{cases} $$
$$ \begin{cases} x_{11} = -\frac{c((2 + \sqrt{2})x_{12} - 2x_{32})}{2m} + \frac{u \cos{\alpha}}{m} \end{cases} $$
Well, it's written right above 2: they define $x_1=\dot{x},$ $x_2=x$, $x_3=\dot{y}$ and $x_4=y.$ Replace these definitions in 1 and you're almost done. The equations in 2 are called state space representation of 1.
Edit: Hang on, So the part that you have marked with 1 is clear to you, right?