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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 $\ell$.

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

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