Two harmonic oscillators satisfying $$ \begin {cases} x (0) = y (0) = 1 \\ x '(0) = y' (0) = 1 \\ \end{cases} $$ ,have movements governed by the equations $$ \begin {cases} x '' = - x \\ y '' = - Ky \\ \end{cases} $$ , respectively, with $K> 0$. The function $f (t) = (x (t), y (t))$ is periodic if and only if:
answer:
$\sqrt {k}$ belongs to $\mathbb{Q}$ (rational numbers)
Can someone explain me why is the answer this?