Click here to **Download this video**

**To download video:** Right click on the following button and select **"Save Link As"**

**To download mp3 (audio):** Click **"Download mp3"** button

Spinning objects have strange instabilities known as The Dzhanibekov Effect or Tennis Racket Theorem - this video offers an intuitive explanation.

Part of this video was sponsored by LastPass, click here to find out more: https://ve42.co/LP

References:

Prof. Terry Tao's Math Overflow Explanation: https://ve42.co/Tao

The Twisting Tennis Racket

Ashbaugh, M.S., Chicone, C.C. & Cushman, R.H. J Dyn Diff Equat (1991) 3: 67. https://doi.org/10.1007/BF01049489

Janibekov’s effect and the laws of mechanics

Petrov, A.G. & Volodin, S.E. Dokl. Phys. (2013) 58: 349. https://doi.org/10.1134/S1028335813080041

Tumbling Asteroids

Prave et al. https://doi.org/10.1016/j.icarus.2004.07.021

The Exact Computation of the Free Rigid Body Motion and Its Use in Splitting Methods

SIAM J. Sci. Comput., 30(4), 2084–2112

E. Celledoni, F. Fassò, N. Säfström, and A. Zanna

https://doi.org/10.1137/070704393

Animations by Iván Tello and Isaac Frame

Special thanks to people who discussed this video with me:

Astronaut Don Pettit

Henry Reich of MinutePhysics

Grant Sanderson of 3blue1brown

Vert Dider (Russian YouTube channel)

Below is a further discussion by Henry Reich that I think helps summarize why axes 1 and 3 are generally stable while axis 2 is not:

In general, you might imagine that because the object can rotate in a bunch of different directions, the components of energy and momentum could be free to change while keeping the total momentum constant.

However, in the case of axis 1, the kinetic energy is the highest possible for a given angular momentum, and in the case of axis 3, the kinetic energy is the lowest possible for a given angular momentum (which can be easily shown from conservation of energy and momentum equations, and is also fairly intuitive from the fact that kinetic energy is proportional to velocity squared, while momentum is proportional to velocity - so in the case of axis 1, the smaller masses will have to be spinning faster for a given momentum, and will thus have more energy, and vice versa for axis 3 where all the masses are spinning: the energy will be lowest). In fact, this is a strict inequality - if the energy is highest possible, there are no other possible combinations of momenta other than L2=L3=0, and vice versa for if the energy is the lowest possible.

Because of this, in the case of axis 1 the energy is so high that there simply aren't any other possible combinations of angular momentum components L1, L2 and L3 - the object would have to lose energy in order to spin differently. And in the case of axis 3, the energy is so low that there likewise is no way for the object to be rotating other than purely around axis 3 - it would have to gain energy. However, there's no such constraint for axis 2, since the energy is somewhere in between the min and max possible. This, together with the centrifugal effects, means that the components of momentum DO change.

Part of this video was sponsored by LastPass, click here to find out more: https://ve42.co/LP

References:

Prof. Terry Tao's Math Overflow Explanation: https://ve42.co/Tao

The Twisting Tennis Racket

Ashbaugh, M.S., Chicone, C.C. & Cushman, R.H. J Dyn Diff Equat (1991) 3: 67. https://doi.org/10.1007/BF01049489

Janibekov’s effect and the laws of mechanics

Petrov, A.G. & Volodin, S.E. Dokl. Phys. (2013) 58: 349. https://doi.org/10.1134/S1028335813080041

Tumbling Asteroids

Prave et al. https://doi.org/10.1016/j.icarus.2004.07.021

The Exact Computation of the Free Rigid Body Motion and Its Use in Splitting Methods

SIAM J. Sci. Comput., 30(4), 2084–2112

E. Celledoni, F. Fassò, N. Säfström, and A. Zanna

https://doi.org/10.1137/070704393

Animations by Iván Tello and Isaac Frame

Special thanks to people who discussed this video with me:

Astronaut Don Pettit

Henry Reich of MinutePhysics

Grant Sanderson of 3blue1brown

Vert Dider (Russian YouTube channel)

Below is a further discussion by Henry Reich that I think helps summarize why axes 1 and 3 are generally stable while axis 2 is not:

In general, you might imagine that because the object can rotate in a bunch of different directions, the components of energy and momentum could be free to change while keeping the total momentum constant.

However, in the case of axis 1, the kinetic energy is the highest possible for a given angular momentum, and in the case of axis 3, the kinetic energy is the lowest possible for a given angular momentum (which can be easily shown from conservation of energy and momentum equations, and is also fairly intuitive from the fact that kinetic energy is proportional to velocity squared, while momentum is proportional to velocity - so in the case of axis 1, the smaller masses will have to be spinning faster for a given momentum, and will thus have more energy, and vice versa for axis 3 where all the masses are spinning: the energy will be lowest). In fact, this is a strict inequality - if the energy is highest possible, there are no other possible combinations of momenta other than L2=L3=0, and vice versa for if the energy is the lowest possible.

Because of this, in the case of axis 1 the energy is so high that there simply aren't any other possible combinations of angular momentum components L1, L2 and L3 - the object would have to lose energy in order to spin differently. And in the case of axis 3, the energy is so low that there likewise is no way for the object to be rotating other than purely around axis 3 - it would have to gain energy. However, there's no such constraint for axis 2, since the energy is somewhere in between the min and max possible. This, together with the centrifugal effects, means that the components of momentum DO change.