Suppose that we have a particle with mass $m$ which moves in its plane with its position at time $t$ defined by the planar polar co-ordinated $r, \theta$ (with the notation $r=r(t)$ and $\theta = \theta(t)$).
I have been given that the Lagrangian of the motion is:
$$ \mathcal{L} = \frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2)-r^2(r^2-10)$$ with the notation being that $\dot{r}=\frac{dr}{dt}$ and $\dot{\theta}= \frac{d\theta}{dt}$
I have to show that $mr^2\dot{\theta}$ is a constant of motion by using the principle of least action. And then, if the system is primed s.t. $\dot{\theta}|_{t=0}=0$ then I need to use the principle of least action again to get the equation of motion for $r$ and thus show $\ddot{r}>0$ in the region: $$0<r< \sqrt{5}%$0<r< \sqrt{5}% $$
I'm quite rusty on this topic and I'm not sure how to approach it, I believe I need to use the Euler-Lagrange equation? Which would be:
\begin{align} \frac{\partial \mathcal{L}}{\partial \theta} - \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot\theta}&=0 \\ 0 - \frac{d}{dt}(mr^2\dot{\theta}) &= 0\\ 0 &= \frac{d}{dt}(mr^2\dot{\theta}) \\ 0 & = mr^2\ddot{\theta} \end{align}
But this is far as I can get before I get confused. Any suggestions? Thank you.
The fact that the angular momentum $p_{\theta}:=\frac{\partial L}{\partial\dot{\theta}}=mr^2\dot{\theta}$ is conserved follows because $\theta$ is a cyclic variable.
The EL equation for $r$ reads (if $p_{\theta}=0$) $$m\ddot{r}~=~4r(5-r^2)~>~0\quad\text{for}\quad 0~<~r~<~\sqrt{5}.$$