Initial-data decomposition into two normal modes ($v_1=(1,r_1{)}^T,v_2=(1,r_2{)}^T,r_1\ne r_2$)

Suppose a two-coordinate linear oscillator has independent normal modes $v_1,v_2$ as displayed, with positive angular frequencies ${\omega }_1,{\omega }_2$. For initial displacement $d=(d_1,d_2{)}^T$ and velocity $w=(w_1,w_2{)}^T$, its solution iswhereThese formulas follow by inverting the two-by-two matrix with columns $v_1,v_2$. Each coefficient evolves by its own scalar harmonic-oscillator equation, and substituting $t=0$ recovers both prescribed vectors. Euclidean orthogonality of the mode shapes is unnecessary; with unequal masses, the natural orthogonality is in the mass inner product of the generalized eigenvalue problem for small oscillations.