The spin-down evolution of a simple vacuum pulsar model is well known. Using the power output of radiation emission of a rotating magnetic dipole, we get the following equation (assuming that the rotation kinetic energy loss equals the electromagnetic power emitted by the rotating dipole) : \begin{equation}\tag{1} \frac{d\,}{dt} \Big( \frac{1}{2} \, I \, \omega^2 \Big) = I \, \omega \, \dot{\omega} = -\, \frac{\mu_0 \, \mu^2 \, \omega^4}{6 \pi c^3} \, \sin^2{\alpha}, \end{equation} where $\alpha$ is the tilt (or inclination) angle of the pulsar's magnetic axis, relative to the rotation axis.
I'm interested in a similar equation for the time evolution of the tilt angle : \begin{equation}\tag{2} \dot{\alpha} = \; .?. \end{equation} According to this paper : https://arxiv.org/abs/1603.01487, we should have this equation (see equation (2) from that article) : \begin{equation}\tag{3} I \, \dot{\alpha} = -\, \frac{\mu_0 \, \mu^2 \, \omega^2}{6 \pi c^3} \, \sin{\alpha} \, \cos{\alpha}. \end{equation} How to demonstrate this equation ?
And how to show that the time variation of the angular momentum (relative to the magnetic moment ?) has the following components ? \begin{align}\tag{4} & I \, \frac{d\omega}{d \, t}, && I \, \omega \, \frac{d \, \alpha}{d \, t}. \end{align} I understand only the first component, which is obvious to me, while the second is not.
