The Philosophical Complexity behind the Dzhanibekov Effect

Rongqing Dai

Abstract

As will be demonstrated in this writing, the famous Dzhanibekov effect is neither a simple physical problem nor a simple mathematical problem, but a complex philosophical problem. However, for the past few decades, the mainstream academic community has been trying to explain the Dzhanibekov effect by using the intermediate axis theorem based on the Euler equations of rigid body dynamics without understanding the philosophical complexity involved or the physical nature behind it. As a result, they have behaved strangely with some eccentric academic amnesia. On the one hand they claim that Dzhanibekov effect is caused by the amplification of some initial momentum in the absence of external forces, but on the other hand they claim that Dzhanibekov effect observes the conservation of angular momentum. In this writing, we will delve into the philosophical complexity behind the Dzhanibekov effect as well as the confusing behavior of the mainstream academia when dealing with it.

Keywords: Dzhanibekov Effect, Intermediate Axis Theorem, Euler Equations of Rigid Body Dynamics, Mainstream Academia, Amnesia

 

1. Background

Mainstream academia has consistently stumbled over the issue of the Dzhanibekov effect for decades, repeatedly making mistakes without being aware of it [[1],[2]]. Subjectively, this puzzling situation reflects the collective weakness of philosophical thinking within the mainstream academia, resulting from its rejection of metaphysics over the past two centuries [[3]]; however, objectively, a key reason behind this situation is the philosophical complexity around the Dzhanibekov effect. In this writing, we will first write down in a list of issues contributing to the philosophical complexities around the Dzhanibekov effect and then proceed with in depth discussions.

The following are the key issues causing the complexity:

First, the fundamental reason behind the uniqueness of the Dzhanibekov effect lies in the fact that, according to the video footages transmitted back from space stations and the most basic definition of angular momentum, when the Dzhanibekov effect occurs in a space station, the angular momentum of a rotating object actually changes. In the meantime, the absence of external forces on the rotating object has been assumed when the Dzhanibekov effect occurs in a space station. Therefore, with all the officially open information about Dzhanibekov effect in a space station, we should conclude that when the Dzhanibekov effect occurs the total angular momentum of the rotating object changes in the absence of external forces or at least in the apparent absence of external forces. In other words, we may say that the law of conservation of angular momentum is violated, or at least apparently violated, when the Dzhanibekov effect happens.

Secondly, the deductions for the solution to the Dzhanibekov effect made by the mainstream academic community based on the intermediate axis theorem, with the assumption of zero external torque, show that when objects in space stations rotate around the intermediate axis, their initial angular momentum around the axis of least moment of inertia will be amplified, which indicates that the conservation of angular momentum is actually broken. However,

Third, because the intermediate axis theorem is derived from the Euler equations of rigid body dynamics, and the Euler equations of rigid body dynamics are assumed to satisfy the law of conservation of angular momentum due to the fourth and fifth points below, the intermediate axis theorem is also assumed to observe the law of conservation of angular momentum regardless of the value of the external torque.

Fourth, the Euler equations of rigid body dynamics were derived by Euler starting from the law of conservation of angular momentum.

Fifth, the name of Euler, a great mathematician renowned in both classical and contemporary mechanics, comparable to Newton and other big names, is deemed good for and has actually often been used as the guarantee of the correctness of his various theoretical achievements.

Sixth, of the five points mentioned above, the correctness of the nearly 300-year-old Euler equations is the only one that has not been rigorously scrutinized in the past few decades of research on the Dzhanibekov effect. However, due to the fifth point, the correctness of the Euler equations has become the most unquestionable of those five points. Unfortunately, as Dai (2025c) pointed out earlier this year [[4]], it is the Euler equations of rigid body dynamics that is logically (mathematically) problematic among the above listed five items.

Seventh, when the only source of error (i.e., the Euler equations of rigid body dynamics) among all key influencing factors is deemed correct due to human social and cultural dynamic, two parallel but incompatible logics emerge in the process of analyzing the Dzhanibekov effect by the mainstream academia: one is the natural logic of the movement and the other is the social logic of human unchecked deference to the academic authority. The conflict between these two logics has orchestrated the strange performance concerning the Dzhanibekov effect for decades, with the following typical manifestation:

According to natural logic, since the intermediate axis theorem shows that the initial tiny angular momentum of motion about the axis of least moment of inertia is amplified even though the external torque is zero, one should say, “according to the intermediate axis theorem, the reason for the Dzhanibekov effect to occur is that angular momentum is amplified without external forces and thus not conserved.” However, the social logic closely associated with Euler’s reputation will then loudly proclaim: “since the intermediate axis theorem is strictly derived from the Euler equations, and the Euler equations are based on the conservation of angular momentum, and Euler cannot be wrong, then no matter what conclusion one might draw from the intermediate axis theorem, it will not violate the conservation of angular momentum.”

Correspondingly, when mainstream scholars use the intermediate axis theorem to explain the Dzhanibekov effect, they display a remarkable amnesia: they first deduce that, under the assumption of zero external torque, when an object rotates about the intermediate axis, the initial tiny angular momentum about the axis of least moment of inertia is amplified, and then immediately conclude that conservation of angular momentum is not violated.

They seem to have either forgotten the contradiction between the words “conservation” and “amplification” in human language (regardless of the linguistic background), or, having just deduced that the initial angular momentum of an object is amplified, they forget their previous result when concluding that angular momentum is still conserved. In short, well-educated academic elites are bafflingly comfortable when treating "amplification" and "conservation" as logically consistent, or even equivalent.

1.1. The apparent consistency

The above seven points alone are enough to make the Dzhanibekov effect an extremely complex philosophical quandary.

Now we seem to witness the consistency between the reality and theory: the Dzhanibekov effect does indeed violate the law of conservation of angular momentum while the intermediate axis theorem concludes that under certain conditions initial angular momentum is amplified and thus not conserved, plus those conditions are consistent with the results of most Dzhanibekov motions occurring in space stations.

However, this consistency is an illusion: the Dzhanibekov effect occurring in space stations is real, while the intermediate axis theorem itself is faulty for the condition of zero gravity, and even its assertion that the existence of an intermediate axis is necessary for the Dzhanibekov effect to occur is false as clearly demonstrated by a video of the Dzhanibekov effect with a round bolt recorded in a space station [[5]].

1.2. The role of human nature

The involvement of human nature makes matters much more complicated. To fully understand what happened to the research on the Dzhanibekov effect over the past few decades, familiarity of human nature is necessary. In an ideal world of logic, the fifth of the seven points mentioned above — Euler's reputation — is the least important. However, in real life, it becomes the most important or even decisive. In Earth civilization, when the theory of a leading figure in any field contains a critical error, that figure's reputation alone could be sufficient to prevent the error from being corrected for a long time, thus hindering the development of Earth civilization in that field for a long time. In the meantime, countless students, generation after generation, would be demanded to study the erroneous theory as correct and to pass exams on that theory in classrooms. It is this human element that leads to the seventh complexity listed above.

1.3. Complexities added by ground-based experiments

To complicate matters further, the rotational motion of objects (usually flat objects, such as a tennis racket) in the normal gravitational field of Earth also exhibits the typical behavior predicted by the intermediate axis theorem, which is why it is also called the tennis racket theorem.

The special behavior exhibited by objects rotating about an intermediate axis was first discussed as a stability problem by the 19th-century French physicist Louis Poinsot. Since the only relevant article by Louis Poinsot that I can find online is in French and I am not able to read French, I cannot further discuss the original text here. However, I believe that Poinsot would not have discussed motions in microgravity environment at his time while nowadays proofs of the intermediate axis theorem for the motion of objects in space stations were all derived by assuming zero external torque. Although both were started from the Euler equations of rigid body dynamics, the environmental conditions for the derivations could not be the same.

1.4. A simple proof of the existence of logical fault with the Euler equations

Although Dai (2025c) [4] has identified the logical (mathematical) defect in the derivation of Euler equations of rigid body dynamics, it would still be beneficial to confirm the existence of logical fault in Euler Equations with a simple reasoning process as follows.

As mentioned earlier, mainstream scholars have concluded, through their derivations of the intermediate axis theorem for an object rotating around its intermediate axis with zero external torque, that the initial tiny angular velocity (and thus angular momentum) around the axis of least moment of inertia, exerted by the experimenter, will be amplified during the motion due to the mathematical instability.

In plain English, we know that “amplification” is one type of “change”, i.e. the original quantity will not be conserved. Since the preset of their derivations is the absence of external torque, their above conclusion violates the conservation of angular momentum.

The complete mathematical process leading to the intermediate axis theorem only involves 1) the derivation of Euler equations of rigid body dynamics from the equations of the conservation of angular momentum, and 2) the derivation of the intermediate axis theorem from Euler equations of rigid body dynamics. Now we know that the intermediate axis theorem was correctly derived from Euler equations of rigid body dynamics as repeatedly confirmed in the literature and textbooks, and thus the only cause that leads to the violation of the conservation of angular momentum would be Euler equations of rigid body dynamics.

1.5. Difference between the conditions for tennis racket motion and for Dzhanibekov effect

As Dai (2025c) [4] mentioned previously, the errors caused by the defects of Euler equations of rigid body dynamics in general engineering practices are negligible, as demonstrated by the fact that Euler equations of rigid body dynamics have been able to withstand the test of nearly three hundred years of engineering practices, despite that the defects will lead to qualitative wrong conclusions when applied to the Dzhanibekov effect in the space station. Similarly, considering that rotations of objects (like tennis rackets) around the intermediate axis in the normal gravitational field share the same external environment with the majority of engineering practices for which the errors caused by Euler equations of rigid body dynamics can be ignored, we should be able to safely judge that the application of the intermediate axis theorem to rotations of objects (like tennis rackets) around the intermediate axis in the normal gravitational field is reasonable.

We might even more optimistically expect that Louis Poinsot’s analysis of the motion around the intermediate axis using the Euler equations of rigid body dynamics should precisely observe the law of angular momentum conservation because the action of the combination of gravity and air could account for any actual variation of angular momentum of the moving objects, despite the Euler equations of rigid body dynamics are flawed.

On the other hand, in the space station, the defects in the Euler equations of rigid body dynamics will become significant due to the disappearance of external torque, so that the errors caused by them can no longer be ignored.

Unfortunately, over the past several decades, the success of studies on the instability of motions around intermediate axis in gravitational environment since Louis Poinsot has led the mainstream academic community to blindly assume that the Dzhanibekov effect occurring in the space station must also be a simple stability problem that does not violate the conservation of angular momentum, without rigorously examining the Euler equations of rigid body dynamics.

This stunning preconceived notion has not only caused them to turn a blind eye to the changes in angular momentum of objects rotating around the intermediate axis in the space station when watching the videos, but also to collectively display the aforementioned strange amnesia when conducting theoretical analysis on Dzhanibekov effect for decades.

1.6. Mathematical complexity exacerbating the confusion

By dissecting the erroneous mathematical analysis of the Dzhanibekov effect by mainstream scholars, we might also find the exacerbation of their confusion by the mathematical complexity involved in the process, or more precisely we should say by the philosophical aspects of the mathematical complexity.

The mainstream scholars have used the so-called mathematical instability as the cause for the anomalous flipping of the rotating object showing the Dzhanibekov effect. However, the term of “mathematical instability” only tells that the motion is unstable when influenced by external disturbances, but it does not tell that pure mathematical manipulations could change the precondition of any derivation. For example, when there is A + B =1 in the precondition of a mathematical derivation, if no any mistake is made during the derivation, no matter how many thousands of steps are involved in the derivation, the end results should not allow A + B =2.

Here we need to distinguish the differences between these things: precondition, initial condition, and disturbance.

In the mainstream scholar’s derivations for analyzing Dzhanibekov effect, the equation of conservation of angular momentum is their precondition, but in the end of their derivation they concluded that the initial minute angular momentum around the axis of least moment of inertia is amplified when there is no external torque, which means that their end results violates their precondition, the equation of conservation of angular momentum.

Besides, they wrongly took the initial minute angular momentum around the axis of least moment of inertia as the "disturbance" or "perturbation" to the system which triggers the instability of the system, but that is wrong. The initial minute angular momentum around the axis of least moment of inertia is not the "disturbance" or "perturbation" to the system but the initial condition of their mathematical derivation and should stay minute to satisfy the equation of conservation of angular momentum if there is no true disturbance to the system during the motion which is the case when external torque is assumed to be zero.

On the other hand, in the motion of tennis rackets in the normal gravitational field, the external impact from gravitational pull and air drag constitute the disturbances to the dynamic system of motion, which would trigger the flipping due to the instability.

As a conclusion, pure mathematical analysis of the Dzhanibekov effect should not end up with the violation of the conservation of angular momentum as long as it starts from the equation of the conservation of angular momentum and is worked through correctly, unless some other interfering condition is introduced.

2. Force Analysis

After knowing the philosophical complexities surrounding the Dzhanibekov effect observed in the space station, now let's come to the mechanical perspective and examine whether the Dzhanibekov effect could be caused by normal mechanical causes.

First, the so-called microgravity environment of the space station is not gravity free environment. Therefore, as a starting point for force analysis, our assumption is that both the tennis racket in the ground environment and the bolt and nut in the space station are subject to two forces: gravity and air force.

It is important to note that gravity and air force play different kinds of roles in the movement of the bolt and nut in the space station or the tennis racket on the ground: when gravity is not negligible, it acts as a driving force, while air always acts as a drag force.

Generally speaking, under certain specific circumstances, when turbulence occurs on the surface of an object, it is indeed possible for turbulence to cause some periodic or nearly periodic motions (e.g. vibrations). The quantitative condition for determining the presence of turbulence is the value of Reynolds number. However, whether it is the movement of tennis rackets on the ground or the movement of bolts and nuts in the space station, the Reynolds number is far from meeting the conditions for generating turbulence. Especially in the space station, there is no supposedly needed external force to induce turbulence. Therefore, in our analysis of the movement of tennis rackets on the ground or the movement of bolts and nuts in the space station, the role of air is only considered to generate resistance, rather than the driving force for generating periodic movements.

2.1. Solution to the tennis racket flipping

As previously noted, we have reason to believe that Louis Poinsot’s analysis of the instability of the motion of a tennis racket or similar object about its intermediate axis within the normal gravitational field is correct.

2.1.1. Gravitationally induced flipping

Despite the mass distribution (and thus the distribution of moment of inertia) of the tennis racket causes the so-called instability of its rotation around the intermediate axis, the real flipping that happens when the tennis racket rotates downwards is obviously induced by the combination of the gravitational pull and the air resistance.

We can borrow the angle of Terence Tao in his famous but erroneous analysis of the Dzhanibekov effect using internal forces [[6]], except that here we replace his internal forces with gravity and air resistance, and our argument would be: when the racket rotates about its intermediate axis, its small initial angular displacement about the axis of least moment of inertia would cause an asymmetric distribution of air resistance about the axis of least inertia, and the instability caused by the distribution of the racket’s moment of inertia would then cause it to flip around its axis of least moment of inertia under the influence of the combination of gravity and air force.

Note that the angular momentum changes during the entire flipping process described above, but this does not violate the law of conservation of angular momentum, because of the pull of gravity and the resistance of air.

2.2. The mystery of Dzhanibekov effect in the space station

However, when we return to our analysis of the Dzhanibekov effect in the space station, the above analysis of the motion of a tennis racket on the surface of Earth no longer holds true.

2.2.1 The insufficient inertial force

When the space station moves around Earth, the gravitational pull from Earth provides its centripetal acceleration so that it can stay in the orbit; in the meantime, everything inside the space station orbits with the space station around Earth, and thus their gravity from Earth would also provides their centripetal accelerations. This makes it look like all the objects inside the space station are floating around, which is the basic reason why researchers assume no external forces on the objects when the Dzhanibekov effect happens, and thus the zero external forces assumption is actually of nominal sense instead of real dynamic sense.

Nevertheless, objects moving inside the space station also constantly accelerate or decelerate with respect to the space station and thus with respect to air at rest inside the space station, which would theoretically set an alarm that we might need to have a reexamination of the zero external force claim, simply because of the fact that those objects do endure gravitational pulls and aerodynamic forces when moving with respect to the space station.

However, the fact that the bolts and nuts in the space station can move straight ahead without falling to the floor indicates that the differences between the gravitational effects and the relative accelerations of objects with respect to the space station, i.e. the inertia forces, are insufficient to cause the bolts and nuts to undergo periodic flipping motion (because any external inertia force that could cause such periodic flipping would be sufficient to cause the bolts and nuts to move parabolically quickly rather than continue straight ahead). Furthermore, external inertia driving force (gravity) alone cannot produce a periodic effect that would cause forward and backward flipping in the Dzhanibekov motion.

2.2.2. The inviability of flipping caused by induced air flow

We all know that the rotation of the blades of a fan could induce air flow perpendicular to the plane of rotation of the blades. Now the following question might arise:

Is it possible for the rotation around the intermediate axis of a bolt or a nut to induce local air flow like a fan to cause itself to flip horizontally around the axis of least moment of inertia as shown in the videos, maybe with the help of gravity?

If the answer to the above question is positive, then it could explain the periodicity of the flipping since the rotation itself is periodical.

However, the answer should be negative not only because not all rotating objects showing Dzhanibekov effect have the shape that could mimic a fan, but more importantly, because the Dzhanibekov style flipping in the space station does not seem to reduce the angular momentum of the original rotation.

Therefore, as mentioned earlier, although the space station contains air of roughly one standard atmosphere (atm), the air only acts as a resistance to motion and does not actively drive the periodic flipping.

2.2.3. Summary

Similar to the motion of a tennis racket on Earth’s surface environment, the Dzhanibekov motion of bolts and nuts in the space station undergoes a change in angular momentum; but the difference is that, so far we cannot find any needed external force to account for the change in angular momentum of the bolts and nuts, which will lead to the conclusion of the violation of the law of conservation of angular momentum.

As to why the Dzhanibekov effect in the space station would violate the law of conservation of angular momentum, at least apparently based on all the information we know so far about the motion, it still remains a mystery.

3. Final Remarks

It is obvious that the Dzhanibekov effect is neither a simple physical problem nor a simple mathematical problem, but a complex philosophical problem. Only by better understanding the philosophy behind it can we further capture its physical nature and then establish a proper mathematical model. In the past few decades, the mainstream academic community has been trying to give an answer by using the intermediate axis theorem based on the Euler equations of rigid body dynamics without understanding the philosophical complexity involved or the physical nature behind it. As a result, they either claim that the rotations of objects in the space station around the intermediate axis can amplify the extremely small angular momentum around the axis of least moment of inertia that the experimenter initially added to the rotating objects, resulting in flips around that axis [[7]], or claim that when the objects in the space station rotate around the intermediate axis, their internal forces will amplify the tiny angular momentum perturbation, resulting in flips around the axis of least moment of inertia [6]. One thing these two theories have in common is that they both claim that the cause of the Dzhanibekov effect is that the motion around the intermediate axis amplifies the initial tiny rotation (angular momentum) around the axis of least moment of inertia, thus causing the flip.

The trouble here is that they claim to have proven that the Dzhanibekov effect does not violate the conservation of angular momentum. In other words, in the absence of an external force, after providing evidence of a change (amplification) in angular momentum, they claim to have proven that angular momentum is conserved. The logical contradiction here is glaring: on the one hand, they demonstrate that angular momentum around the axis of least moment of inertia is amplified in the absence of an external force, yet at the same time, they claim that angular momentum around the axis of least moment of inertia is conserved. It is astonishing that this linguistic and logical flaw still remains unnoticed in the scientific community, the ivory tower of Earth's civilization.

What is conservation? If angular momentum can be amplified in the absence of an external force, can it still be called conservation of angular momentum?

As for the claim that internal forces can cause changes in angular momentum, it directly contradicts the fundamental principle of mechanics taught in middle school. If internal forces could change the total momentum or total angular momentum of an object or system, people would see strange phenomena in their daily lives, such as objects bouncing around on their own without the action of external forces.

What we are witnessing here is the phenomenon of mathematical pollution: the construction of complex mathematical structures on erroneous physical models based on confused philosophical thinking.

A serious consequence of mathematical pollution is that it can make erroneous physical models extremely robust and thus difficult for future generations to correct their errors. Furthermore, through its misleading nature mathematical pollution can also severely reduce the collective IQ of humanity as an intelligent entity.

It is heartbreaking to see generation after generation of talented youngsters around the world, after years of intensive training at their dream universities, become habitual to fully muster their imagination in the effort of making wrong things as correct, which will inevitably impair their normal thinking ability, and a typical symptom of the impaired normal thinking ability is that they often remain unaware of their making some simple logical mistakes.

References

 

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[[1]]Dai, R. (2025a). An Example of Violating the Conservation of Angular Momentum. Retrieved from: https://www.academia.edu/127575471/An_Example_of_Violating_the_Conservation_of_Angular_Momentum

[[2]]Dai, R. (2025b). Dzhanibekov Effect --- A Smashing Challenge to the Foundation of Physics. Retrieved from: https://www.academia.edu/128364151/Dzhanibekov_Effect_A_Smashing_Challenge_to_the_Foundation_of_Physics

[[3]]Dai, R. (2024). When Philosophy is Disparaged. Scholars’ Press. ISBN: 978-620-6-77202-6.

[[4]]Dai, R. (2025c). Why Euler's Equations of Rigid Body Dynamics Are Wrong. Retrieved from: https://www.academia.edu/128498771/Why_Eulers_Equations_of_Rigid_Body_Dynamics_Are_Wrong

[[5]]GetAClass - Physics (Feb 18, 2024) [YouTube] The Dzhanibekov effect. url: https://www.youtube.com/watch?v=Xrf1HzFJ8jc&t=39s

[[6]] Muller, D. (Sep 19, 2019) [YouTube] The Bizarre Behavior of Rotating Bodies. [video] url: https://www.youtube.com/watch?v=1VPfZ_XzisU.

[[7]]Wikipedia. Tennis racket theorem. Retrieved from: https://en.wikipedia.org/wiki/Tennis_racket_theorem. Last edited on 29 November 2024, at 06:27 (UTC).