# Harmonic Motion in Two Dimensions

We consider harmonic motion in two dimensions. The initial position $x_{0}$ is chosen on the horizontal axis and the initial velocity is a vector $\bar{v}_{0} =\; (v_{0} \cos \alpha ,\; \, v_{0} \sin \alpha )$, cutting angle $\alpha$ with the $x$-axis. The coordinates satisfy the system ($\omega$ is the circular frequency) $\\ \begin{array}{c} {x''+\omega ^{2} x=0} \\ {y''+\omega ^{2} y=0} \end{array}$ The motion is described by the vector function $\bar{r}(t)= (x\, (t),\, y\, (t))$, where $\;x=x_0 \cos (\omega \; t)+\frac{v_0}{\omega } \cos \alpha \sin (\omega t)$, $\;y=\frac{v_0}{\omega } \sin \alpha \sin (\omega t)$, and $t$ is time. For this demonstration ${\omega } =1$.