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Certain spinning tops flip upside-down once spun sufficiently high, as shown in this video.

Suppose I have the following:

  • A cylindrically-symmetric top
  • The principal axes of a given top's moment of inertia tensor $I_1$ and $I_2=I_3$ (by cylindrical symmetry)
  • An estimate of the local gravitational acceleration $\vec{g} = -g\hat{z}\quad$ (e.g. $g=9.8 \; \text{m/s}$)
  • The ability to spin the top up to any initial angular velocity $\vec{\omega} = | \omega_z | \hat{z}$, without imparting any initial horizontal velocity (i.e. it doesn't spin out of control across the table or floor)

What equation governs this behavior and determines 1) whether it is possible for a given top to be flipped, and if so 2) what minimum angular speed will cause the top to flip?

This is similar to this question, except I am looking for limits of behavior based on the equations of motion, rather than on simpler, "intuitive" explanations of why a top flips.

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