# Equations of motion for a spherical pendulum in a non-inertial reference frame

Take a spherical pendulum with bob mass $m$, rod length $\ell$ and physical coordinates $\theta$, $\phi$ (spherical angles) and $h$ (the hinge height with respect to the coordinate origin). The rod is massless and infinitely stiff. The derivation of the equations of motion of such a system using Lagrangian dynamics is outlined here. Note that the hinge height $h$ is an ignorable coordinate and thus plays no role in the equations of motion.

However, I'd like to extend this system by placing the spherical pendulum in a non-inertial reference frame that is both rotating and accelerating (linearly and angularly), and change the pendulum such that it has damping at the hinge with ratio $\gamma$, and known functions of time for $\dot{m}$, $\dot{\ell}$ and $\dot{h}$, as well as $\ddot{m}$, $\ddot{\ell}$ and $\ddot{h}$ (these are all $\neq 0$).

My Lagrangian dynamics is rather rusty (to put it mildly), so when I started writing out the equations for the kinetic energy $T$ and potential $V$, I immediately got stuck on the following questions:

• Can the potential $V$ still be written as $-mg\ell\cos\theta$, or should the potential take into account the non-inertialness of the frame?
• In fact, is this pendulum system actually still conservative (there is dissipation due to the damping after all)? How should I go about it if it is not?
• Can I somehow do the required vector products & additions afterwards? I mean: can I first derive the equations of motion as if the pendulum's reference frame is inertial, and then get to the final form by adding the linear, centripetal, Coriolis and Euler terms? Or do these need to be included right from the start somehow?

Most examples I find or have available seem too trivial for this sort of problem...And this problem is simply beyond anything I've ever done in the past. Some assistance and/or guidance and/or links to similar problems would be greatly appreciated.

Note that I've asked a similar question before, but the solution I have there is rather unsatisfactory; it considers the constraint force (tension in the string), which it turns out, is not really necessary. It is also quite evil in numerical terms, because of the time-dependent discontinuities in the forces applied to the bob. So I'm going with a rod now, which has negligible length changes (save for the $\dot{\ell}$ and $\ddot{\ell}$ terms).

Although I could do it in the "Newtonian" way by carefully considering all the forces, etc., I'd really like to learn (again) how to do this cleanly by using the Lagrangian formulation.