Parametric equations of a particle

• Jul 23rd 2009, 02:26 PM
zorop
Parametric equations of a particle
Consider the motion of a particle of mass m in a 3D isotropic harmonic oscillator potential $\displaystyle V(r)=\frac{1}{2}kr^2$ where k > 0
Determine the parametric equations for the orbit of the particle using the general differential equations of motions
$\displaystyle f(r)=m(\ddot{r}-\frac{L^2}{m^2r^3})$
and $\displaystyle \dot{\theta}=\frac{L}{mr^2}$
Assume that $\displaystyle E>E_{0}\equiv{E_{min}},r(0)=r_{max},\theta(0)=0$ and L is not zero.
The final answer should not contain $\displaystyle r_{max}$; the final result should be given in terms of the parameters m, k, E and L.
• Jul 23rd 2009, 02:38 PM
zorop
Ok, i need find the $\displaystyle r(t)$ and $\displaystyle \theta(t)$ for the parametric equations.
First, i tried to find $\displaystyle r(t)$ from $\displaystyle f(r)=\frac{-dV(r)}{dr}$ from the range($\displaystyle r_{0}$, $\displaystyle r$)? but the integral is so hard to be solved, i think i get something wrong here.
Then i tried to find it from $\displaystyle E=\frac{1}{2}mv^2+V(r)$ where $\displaystyle v^2=v_{r}^2+v_{\theta}^2$ and $\displaystyle v_{r}=\dot{r},v_{\theta}=\dot{\theta}$
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 $\displaystyle r_{min},r_{max}$? I think it helps a lot for the integral.