Let $(r, \theta)$ be plane polar coordinates and $\mathbf{e}_{r}$ and $\mathbf{e}_{\theta}$ unit vectors in the direction of increasing $r$ and $\theta$ respectively. Show that the velocity of a particle moving in the plane with polar coordinates $(r(t), \theta(t))$ is given by

$$\dot{\mathbf{x}}=\dot{r} \mathbf{e}_{r}+r \dot{\theta} \mathbf{e}_{\theta},$$

and that the unit normal $\mathbf{n}$ to the particle path is parallel to

$$r \dot{\theta} \mathbf{e}_{r}-\dot{r} \mathbf{e}_{\theta} \text {. }$$

Deduce that the perpendicular distance $p$ from the origin to the tangent of the curve $r=r(\theta)$ is given by

$$\frac{r^{2}}{p^{2}}=1+\frac{1}{r^{2}}\left(\frac{d r}{d \theta}\right)^{2}$$

The particle, whose mass is $m$, moves under the influence of a central force with potential $V(r)$. Use the conservation of energy $E$ and angular momentum $h$ to obtain the equation

$$\frac{1}{p^{2}}=\frac{2 m(E-V(r))}{h^{2}}$$

Hence express $\theta$ as a function of $r$ as the integral

$$\theta=\int \frac{h r^{-2} d r}{\sqrt{2 m\left(E-V_{\mathrm{eff}}(r)\right)}}$$

where

$$V_{\mathrm{eff}}(r)=V(r)+\frac{h^{2}}{2 m r^{2}}$$

Evaluate the integral and describe the orbit when $V(r)=\frac{c}{r^{2}}$, with $c$ a positive constant.