Lissajous curves are curves described by the parametric equations:
$$x(t)=A\sin(at+k), \ \ y(t)=B\sin(bt)$$
According to wikipedia, the curve is only closed iff the ratio between $a$ and $b$ is rational. I was wondering if someone could prove this rigorously? I have not been able to find anything online.
On
No need perhaps for an elaborate proof, but it is the definition of frequency/time period $ T= 1/f= 2 \pi /\omega$ or how a smaller multiple wave is packed into another bigger integral sized wave. It is the number of multiples of first small wave contained within the second bigger, if one is multiple of the other, when these are integers. Repetition of wave occurs thereafter.
If the quotient is not integer or integer fraction then we look to the highest common factor for repetition.
$$ HCF(a,b)$$
number of periods of the smaller time period wave.
Suppose $\frac{a}{b}$ is rational, with simplified form $\frac{p}{q}$. $x(t)$ has period $\tau_a = \frac{2\pi}{a}$ and $y(t)$ has period $\tau_b = \frac{2\pi}{b}$. Hence $q\tau_b = p\tau_a$, i.e. $q$ periods of $\tau_b$ fit into $p$ periods of $\tau_a$. Then the curve $f(t) = (x(t), y(t))$ has period $\tau = \max\{p\tau_a, q\tau_b\}$. Thus the curve $f$ is closed (since after a time $\tau$, $x(t)$ and $y(t)$ return to the same position and then retrace the same curve).
Now assume $\frac{a}{b}$ is irrational. Again $x(t)$ has period $\tau_a = \frac{2\pi}{a}$ and $y(t)$ has period $\tau_b = \frac{2\pi}{b}$, but no integer number of periods $p$ of $\tau_a$ fit into an integer number of periods $q$ of $\tau_b$ (otherwise $\frac{\tau_b}{\tau_a} = \frac{a}{b} = \frac{p}{q}$, contradicting the fact that $\frac{a}{b}$ is irrational). Therefore $f(t) = (x(t), y(t))$ has infinite period, i.e. it is not periodic. Hence as $t \rightarrow \infty$, $f(t)$ never returns to $(x(0), y(0))$ and retrace what has been traversed thus far (otherwise, it has a finite period). But then what $f$ traces out is not a closed curve.