A proto-planet of mass $m$ in a uniform galactic dust cloud has kinetic and potential energies

$$T=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\phi}^{2}, \quad V=k m r^{2}$$

where $k$ is constant. State Hamilton's principle and use it to determine the equations of motion for the proto-planet.

Write down two conserved quantities of the motion and state why their existence illustrates Noether's theorem.

Determine the Hamiltonian $H(\mathbf{p}, \mathbf{x})$ of this system, where $\mathbf{p}=\left(p_{r}, p_{\phi}\right), \mathbf{x}=(r, \phi)$ and $\left(p_{r}, p_{\phi}\right)$ are the conjugate momenta corresponding to $(r, \phi)$.

Write down Hamilton's equations for this system and use them to show that

$$m \ddot{r}=-V_{\text {eff }}^{\prime}(r), \quad \text { where } \quad V_{\text {eff }}(r)=m\left(\frac{h^{2}}{2 m^{2} r^{2}}+k r^{2}\right)$$

and $h$ is a constant. With the aid of a diagram, explain why there is a stable circular orbit.