The equations of motion
I have started the Dzhanibekov effect series with quaternions, but that is because of my personal attraction to quaternions. Some years ago I coauthored a paper on Quaternions and Monopoles. The monopoles there are the exotic “magnetic monopoles” that are lake fairies. Some physicist say they have seen them in the woods, but the majority of physicists take it as a joke. Quaternions are also somewhat exotic. I do not know if magnetic monopoles have any relation to Dzhanibekov effect, though, in fact they may have one – but that is for the physics of the future. For the physics of today, for the physics of a cosmic spinning and flipping nut, we need the physics of a rigid body. That is part of classical mechanics.
According to Wikipedia
In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it.
I should say that the concept of a rigid body becomes somewhat iffy when we want to take into account what Albert Einstein taught us about first special, and then general relativity. I did not see yet a satisfactory relativistic theory of Dzhanibekov effect. What I know is that some physicists trying to sell their fantastic ideas to the investors (example G. Shipov) try to relate Dzhanibekov effect to “torsion fields” and “4D gyroscopes“. I have published some comments on this subject in International Journal of Unconventional Science. “Comments on Chapter 5 of G. I. Shipov’s “A Theory of Physical Vacuum”. Part I” is available in English. The second part, dealing with Shipov’s “4D gyroscopes” is for now available only in Russian, but it will be translated into English soon.
But that is all in the exotic sphere. Interesting, for sure, but my aim is to describe the physics and mathematics of Dzhanibekov effect using old and good classical mechanics of Newton and Euler.
We consider a free rigid body, observed form an inertial reference system, rotating with a fixed point at the center of its mass. Let be an orthonormal frame corotating with the body, and aligned with its principal axes, and let be an inertial laboratory frame, both centered at the center of mass of the body. The two frames are related by time-dependent orthogonal matrix
is often called the attitude matrix. For a rotating body, if are coordinates of a fixed point in the body, then its coordinates in the laboratory system change in time:
Differentiating we get
The matrix is orthogonal, therefore, by differentiating, the matric is antisymmetric. Every antisymmetric matrix can be written as
we find that
The vector is the angular velocity vector in the laboratory frame.
We denote by the body representative of the angular velocity vector where we skip the time dependence, with components
and, from Eq. (*)
where In the body frame the inertia tensor is diagonal The angular momentum vector in the body frame is then given by the formula
and the Euler equations for the free rotation read:
When written componentwise in terms of the angular velocity components they take the well known form
Now we need to solve explicitly Euler’s equations, which is just the first part of the task. Then we have to find the attitude matrix. Then we have to use these explicit solutions in order to produce realistic animations of Dzhanibekov’s effect. It is a lot of work, especially if one is going to reveal all the details. So, let us see where it will lead us to.