Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 7 / c

Continue with $\mu =2$. Decompose the initial displacement into the normal modes:There is no translational component. Because all initial velocities vanish, the oscillatory coordinates contain only cosines and the zero-frequency velocity is zero. ThereforeAt $t=0$ these give the prescribed displacements and zero velocities. Each term separately satisfies $M\ddot{x}+Kx=0$. Also $x_1+2x_2+x_3=0$ for all $t$, so the centre of mass remains at its original equilibrium position, consistent with the absence of external forces.