= Initial-data decomposition into two normal modes
{title2=$v_1=(1,r_1)^T,\quad v_2=(1,r_2)^T,\quad r_1\ne r_2$}
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 is
$$
x(t)=\sum_{j=1}^2v_j\left[c_j\cos(\omega_jt)+\frac{b_j}{\omega_j}\sin(\omega_jt)\right],
$$
where
$$
c_1=\frac{d_2-r_2d_1}{r_1-r_2},\quad c_2=\frac{r_1d_1-d_2}{r_1-r_2},\qquad b_1=\frac{w_2-r_2w_1}{r_1-r_2},\quad b_2=\frac{r_1w_1-w_2}{r_1-r_2}.
$$
These 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>.
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