 Solution

Take $q=-e$ as the signed electron charge. The Lorentz oscillator model is a driven damped bound charge:The first term is inertia, the second is a phenomenological frictional loss of coherent motion to other degrees of freedom, and the third is the linear restoring force from binding to the atom. The driving term is the electric force. The natural angular frequency ${\omega }_0$ represents the electronic transition; this classical oscillator describes its near-resonant linear response. The friction convention is $m\gamma \dot{u}$, so no factor two is hidden in $\gamma$.