For the parallel flow $\dot{\theta_1}=w_1$, $\dot{\theta_2}=w_2$ on the 2-torus, where $w_1$ and $w_2$ are positive and where the coordinates $\theta_1$ and $\theta_2$ are taken modulo 1. Also, $w_1/w_2=p/q$ is a rational number.
I was wondering how can I prove that every orbit this system is periodic and find outr the minimal period of the orbits. Thanks
Hint: $(\theta_1, \theta_2) = (x_1, x_2) \mod 1$ where $x_1 = A + w_1 t$, $x_2 = B + w_2 t$ for some constants $A,B$. Find $t \ne 0$ so that $w_1 t$ and $w_2 t$ are both integers.