I Steady precession of a gyroscope

[Swamp Thing]

learning Hamilton's principle of least action and analytical mechanics is well worth the effort

I had a moment of epiphany while reading your mention of the least action principle. It connected nicely with a way of thinking about gyroscope behavior that I have sometimes found useful.

Those who have actually played with a bicycle wheel gyro will remember that it actually takes some extra effort to turn the wheel steadily around a strictly vertical axis, because it desperately wants to tip over to one side. (I'm talking about the case where you support one end of the axle with each hand). On the other hand, it takes a lot less effort to rotate it around the vertical axis if you also permit it to tip over by just the right amount. (In this case we only need to support the weight of the wheel, and we don't need to counteract its tipping action). So by experimenting with different combinations of turning and tipping, you can learn what it would like to do if left to its own devices with no external torque, once a precession has been set up.

Well, when we experiment manually with different trajectories, we are in a sense trying to find the one that complies with the Hamiltonian law, i.e. the trajectory that it would follow if no external torque were applied.

 

[mpresic3]

I'm not sure, but I may be ruining your epiphany moment. On the other hand, you may get another one when you grapple with the thoughts I am about to address. You mention "a trajectory that it would follow if no external torque were applied", and you played with a gyroscope and got a physical feeling for the torques, and forces involved.

Unless you did your experiments in free fall (like in orbit) the gyro had torques acting on it. Typically when you spin the gyroscope on the floor, the reaction force from the floor is providing a torque about the center of mass up through the symmetry axis of the gyroscope. In your case, your hands are exerting reaction forces to the gyro, so the gyro is not free (unless you're not holding it at all and it is just dropping)

As Feynman points out in his lectures, the general motion of the gyro can nod (or aka nutate) as well as precess. This is true for a gyro with a reaction force (like the floor or a stand). A "free" gyro, (a gyro freely falling) like in the space shuttle (as long as it is symmetrical about the axis), will not nutate (nod). It will still precess. Hence, you were not really experimenting with a "free" gyro (with no external forques), unless you were in space or a falling elevator.

 

[A.T.]

It's here :

www.youtube.com/watch?v=1VPfZ_XzisU

Note that this video is based on an incomplete explantion by Tao, which was updated since then:
https://mathoverflow.net/questions/...anics-or-fiction-explain-mathemat/82020#82020

 

[dyn]

One professor, perhaps Ed Purcell was asked what the hardest things physics undergraduates learn. The questioner expected Purcell to say Quantum mechanics, or Relativity. The professor agreed these are the most novel areas, but in his experience the hardest things undergraduates are expected to learn is rigid body mechanics. After working with many physics, math and engineering graduates, I agree. (I might get back to you to the link to this article)

From a learning perspective i also totally agree with this statement

 

[enrroi]

Note that this video is based on an incomplete explantion by Tao, which was updated since then:
https://mathoverflow.net/questions/...anics-or-fiction-explain-mathemat/82020#82020

The conclusion of last update on that link, Oct 1 2019 by Arthur Baraov, is challenging and is not being addressed by Tao: "Equating Dzhanibekov effect with the tennis racket instability is a blunder. So, the real physical cause for the instability of the Dzhanibekov top needs to be identified. " It would be interesting to perform the experience in microgravity and vacuum conditions to sort this out.

 

[A.T.]

The conclusion of last update on that link, Oct 1 2019 by Arthur Baraov, is challenging and is not being addressed by Tao: "Equating Dzhanibekov effect with the tennis racket instability is a blunder. So, the real physical cause for the instability of the Dzhanibekov top needs to be identified. " It would be interesting to perform the experience in microgravity and vacuum conditions to sort this out.

I don't find this challenge by Arthur Baraov very convincing so far:

- He is basing it on the nut + clay experiment, which is not shown performed in the video, just animated based on descriptions.

- According to the video the clay was attached to the wingnut originally used to observe the flip, not to a mere "regular hexagon nut" as he claims. So his assumptions about the principal moments of inertia might be off.

- Even ignoring the above, it seems that the "violation of the intermediate axis theorem" would rather be a violation of the "extension of the intermediate axis theorem" that he proposes himself in the post.