Chapter 6.P2, Problem 3P

From the normal mode representation of the solution of the initial value problem (2), (4), and (5), explain why the system cannot be completely controllable by applying a control function only to the center mass.

$\frac{d^{2}x}{d{t}^2}=\text{G}x+\text{u}\left(t\right),$ (2)

$x(0)=x_0,$ (4)

$\frac{dx}{dt}(0)=v_0$. (5)