How to introduce dissipation to a spinning top

[etotheipi]

A axisymmetric spinning top is pivoted at O. The components of the inertia tensor $I_O$ at the point $O$, with respect to the principal axes, are denoted $A$, $A$ and $C$. It's Lagrangian is$$\mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}) = \frac{1}{2} A\dot{\theta}^2 + \frac{1}{2}A(\dot{\phi} \sin{\theta})^2 + \frac{1}{2}C(\dot{\psi} + \dot{\phi} \cos{\theta})^2 - mgh\cos{\theta}$$How can the Lagrangian be modified to account for dissipation at the pivot? We can't use the Rayleigh function here, because that is for velocity-dependent dissipation. Are there some references?

 

[wrobel]

How can the Lagrangian be modified to account for dissipation at the pivot?

You can not modify the Lagrangian in such a way because the system with dissipation is not a Hamiltonian system. Use general equations:
$$\frac{d}{dt}\frac{\partial L}{\partial \dot x^i}-\frac{\partial L}{\partial x^i}=Q_i(t,x,\dot x)$$
Dissipation means that $Q_i\dot x^i\le 0$

If say you apply a dissipation torque $\boldsymbol \tau$ then
$$Q_i=\Big(\boldsymbol\tau,\frac{\partial\boldsymbol\omega}{\partial\dot x^i}\Big)$$