Derive Larmor's formula for the rate at which radiation is produced by a particle of charge $q$ moving along a trajectory $\mathbf{x}(t)$.

A non-relativistic particle of mass $m$, charge $q$ and energy $E$ is incident along a radial line in a central potential $V(r)$. The potential is vanishingly small for $r$ very large, but increases without bound as $r \rightarrow 0$. Show that the total amount of energy $\mathcal{E}$ radiated by the particle is

$$\mathcal{E}=\frac{\mu_{0} q^{2}}{3 \pi m^{2}} \sqrt{\frac{m}{2}} \int_{r_{0}}^{\infty} \frac{1}{\sqrt{E-V(r)}}\left(\frac{d V}{d r}\right)^{2} d r$$

where $V\left(r_{0}\right)=E$.

Suppose that $V$ is the Coulomb potential $V(r)=A / r$. Evaluate $\mathcal{E}$.