Harmonic Motion in Two Dimensions

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