I Want to simulate a spherical pendulum with a torquer on it, i.e. the angles of the pendulum change not only due to torque generated by gravity, but also by a torquer attached to the top of the pendulum, issuing torques $(\tau_\theta,\tau_\phi)$.
I know that for a classical non-torqued spherical pendulum, one can find the equations of motion by writing a Lagrangian:
$$ \mathcal{L} = \frac{1}{2}m(\dot{x}^2+\dot{y}^2+\dot{z}^2)-mgz=\frac{1}{2}m\ell^2(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)+mg\ell\cos(\theta) $$
but I do not know how to formulate the Lagrangian element corresponding to the added torques on the pendulum.
Edit: I assume that the rod is massless, that it's hanged on one end (and is massless), and that the other end is a point mass with mass $m$. The length of the rod is $\ell$.
