Intermediate Axis Theorem - Intuitive Explanation

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SUMMARY

The discussion focuses on the Intermediate Axis Theorem, emphasizing the instability of rotation about the middle axis. It highlights that when analyzing a surface of fixed energy, there are six equilibria in three pairs corresponding to clockwise and counterclockwise rotations about each axis. However, the middle axis exhibits saddle points, leading to instability. The "flipping" phenomenon observed in previous videos is attributed to the dynamics near a heteroclinic cycle between these saddle points.

PREREQUISITES
  • Understanding of rotational dynamics
  • Familiarity with the concepts of equilibria and stability
  • Knowledge of energy and momentum conservation principles
  • Basic grasp of heteroclinic cycles in dynamical systems
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  • Research the Intermediate Axis Theorem in detail
  • Study the dynamics of saddle points in rotational systems
  • Explore simulations of energy surfaces in classical mechanics
  • Investigate the implications of heteroclinic cycles in dynamical systems
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Physicists, mechanical engineers, and students of dynamics interested in understanding the complexities of rotational stability and the Intermediate Axis Theorem.

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A simulation/animation/explanation based on the inertial frame only:

The previous videos referenced there are here:

See also this post for context on the Veritasium video: https://mathoverflow.net/a/82020

Note to mods: The previous thread is not open anymore so I opened a new one. Feel free to merge them.
 
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Here one of the more abstract approaches based on energy/momentum conservation. Unfortunately not much explanation in the video, and just a short description:



Robert Ghrist said:
Why is rotation about the middle axis unstable? If you examine a surface of fixed energy and look at the dynamics, you get six equilibria in three pairs -- rotation about each axis CW and CCW. These equilibria are centers for the longest and shortest axes. But for the middle axis -- the equilibria are saddles! The "flipping" seen in the previous video corresponds to traveling close to a heteroclinic cycle between saddle points.
 
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