# Thread: Parametric equations of a particle

1. ## Parametric equations of a particle

Consider the motion of a particle of mass m in a 3D isotropic harmonic oscillator potential $V(r)=\frac{1}{2}kr^2$ where k > 0
Determine the parametric equations for the orbit of the particle using the general differential equations of motions
$f(r)=m(\ddot{r}-\frac{L^2}{m^2r^3})$
and $\dot{\theta}=\frac{L}{mr^2}$
Assume that $E>E_{0}\equiv{E_{min}},r(0)=r_{max},\theta(0)=0$ and L is not zero.
The final answer should not contain $r_{max}$; the final result should be given in terms of the parameters m, k, E and L.

2. Ok, i need find the $r(t)$ and $\theta(t)$ for the parametric equations.
First, i tried to find $r(t)$ from $f(r)=\frac{-dV(r)}{dr}$ from the range( $r_{0}$, $r$)? but the integral is so hard to be solved, i think i get something wrong here.
Then i tried to find it from $E=\frac{1}{2}mv^2+V(r)$ where $v^2=v_{r}^2+v_{\theta}^2$ and $v_{r}=\dot{r},v_{\theta}=\dot{\theta}$
The integral still owned me, could someone help me or give me some advice?
Btw, do we need to know about the turning points of radial motion $r_{min},r_{max}$? I think it helps a lot for the integral.