We know the dynamics of a pendulum and we know the dynamics of a gyroscope. I put together the following I diagram below. The interesting thing as that the system has a beat frequency where the gyroscope will point along the line of the pendulum as it swings, but this shifts to near zero pendulum motion as the gyroscope precesses around its anchor. It then oscillates between these two configurations. If you have a toy gyroscope try this out, it is rather interesting.

I sat down to work the Lagrangian of this. I am prodding SE here to see if anyone else can work this out as well. This is a classical system that emulates the Jaynes-Cumming model of a two state atom with a photon confinced in a QED cavity. That was one motivation for doing this, it appears to be a classical analogue of that quantum system

enter image description here

  • 1
    $\begingroup$ Clarification on question (v1): could you perhaps be more specific on the question you're asking? Are you looking for others to mathematically verify the aspects of the dynamics that you have observed empirically? $\endgroup$ Jul 19, 2016 at 19:09
  • $\begingroup$ I have not done anything on this since I posted this, but mainly I am wondering what the Lagrangian for this would be. This couples the fixed point of the gyroscope to the coordinates of a pendulum. $\endgroup$ Jul 19, 2016 at 20:12
  • 1
    $\begingroup$ You know the rules! Show your work! Have you tried writing the Lagrangian for each separately then combining them? $\endgroup$ Jul 20, 2016 at 16:35