# Equations of motion for a torqued spherical pendulum

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 $\ell$.

• Can you give more details about the pendulum: is the rod/cord massless, ...? – QuirkyTurtle98 Jul 3 '17 at 17:58
• I added some details about the pendulum that I saw fit. Anything else needed? – Miel Sharf Jul 3 '17 at 19:43
• If the torques are non conservative you cannot write a Lagrangian. However you still have Euler-Lagrange equations. This post can be helpful. – Diracology Jul 4 '17 at 0:34