= Solution
A <normal mode> of a linearized mechanical system is a motion in which every generalized displacement oscillates at a single common <angular frequency>, with fixed relative amplitudes and phases. In coordinates $q_j$, an undamped oscillatory mode may be written
$$
\boxed{q_j(t)=A_j\cos(\omega t+\delta),}
$$
where the amplitude <vector> $(A_1,\ldots,A_n)$ is fixed; negative components describe relative phase $\pi$. An arbitrary small motion is a <linear combination> of the <normal modes> when the mode vectors form a complete <basis>.
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