# $X=(\alpha, \beta)$ period on $\mathbb{T}^2$

Consider the vector field $$X=(\alpha, \beta)$$ in $$\mathbb{T}^2$$(a torus). Show that the flux given by the following differential equation $$\dot{x}=X(x)$$ is the translation flux and that:

a) if $$\frac{\beta}{\alpha}\in\mathbb{Q}$$ then all the orbits are periodic, determnie the period.

b) if $$\frac{\beta}{\alpha}\in\mathbb{Q}$$ then all the orbits are dense in $$\mathbb{T}^2$$

Consider $$x\in\mathbb{R}^2$$, if $$\varphi:\mathbb{R}\times\mathbb{T}^2$$ is the flux given by the differential equation solution, then $$\varphi_t(x)=x+\alpha t(1,\frac{\beta}{\alpha})$$. If I think on the identification map from a square, then $$\frac{\beta}{\alpha}$$ would be the slope of the trajectory from $$x$$ to $$\varphi_t(x)$$ for $$t\in\mathbb{R}$$ however I cannot compute the period, and I do not see the reason why if we had a irrational slope, why would the orbits be dense(I think we could not compute the period).

Question:

How should I solve the problem?