Let $S^{2n-1}$ be the unit sphere in $(\mathbb R^{2n}, \omega = \text d \underline x \wedge \text d\underline y)$, where $(x_1, y_1, ..., x_n, y_n)$ are cartesian coordinates and $\omega$ is the standard symplectic form. Show that $$ \alpha = \frac 12 \sum\limits_{j=1}^n (x_j \text dy_j - y_j \text dx_j) $$ defines a contact structure on $S^{2n-1}$ and compute the Reeb flow for $\alpha$.
I have already proven that $\alpha = \iota_Y \omega$ is a contact form where $Y = \frac 12 \sum\limits_{j=1}^n (x_j \partial x_j + y_j \partial y_j)$ is the radial vector field on the sphere. Obviously $Y$ is a Liouville vector field, so one can show easily $\iota_Y \omega$ defines a contact structure.
I also proved that the Reeb vector field is $R = 2 \sum_j (x_j \partial y_j - y_j \partial x_j)$ since $\alpha(R) = 1$ and $\iota_R \text d\alpha = 0$. Now I need to calculate the Reeb flow $\Phi$, which is the solution to the differential equation $$R (\Phi) = \dot\Phi.$$ How on earth do I solve this thing? A little tip would really help me.
Edit: I think the solution locally is $\Phi(t) = (\cos t - 1, \sin t, ..., \cos t - 1, \sin t)$ for $\Phi(0) = (0)$.
In the $n=1$ case notice that your Reeb vector field is $R=2x\partial_y-2y\partial_x$, which directs the counterclockwise parametrization of $S^1$ with constant speed 2. Indeed, $R$ directs a sort of rotation more generally.
This all might be easier to write down with the introduction of a complex variable $z=x+iy$. In these coordinates we can write the $n=1$ Reeb flow as \begin{equation} \Phi_t(z) = e^{2it}z, \end{equation} so that the Reeb flow is precisely the rotation described above. I suggest you try something similar (namely, coordinates $z_j=x_j+iy_j$) for an arbitrary $n$.