Tennis racket theorem: Difference between revisions

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{{Short description|A rigid body with 3 distinct axes of inertia is unstable rotating about the middle axis}}

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{{Technical|date=November 2024}}

[[File:tennis_racquet_principal_axes.svg|thumb|Principal axes of a tennis racket.]]

[[File:tennis_racket_theorem.gif|thumb|upright=1.5|link={{filepath:tennis_racket_theorem.ogv}}|Composite video of a tennis racquet rotated around the three axes – the intermediate one flips from the light edge to the dark edge]]

[[File:Théorie Nouvelle de la Rotation des Corps.jpg|thumb|Title page of "Théorie Nouvelle de la Rotation des Corps", 1852 printing]]

The '''tennis racket theorem''' or '''intermediate axis theorem''', is a kinetic phenomenon of [[classical mechanics]] which describes the movement of a [[rigid body]] with three distinct [[principal moments of inertia]]. It has also been dubbed the '''Dzhanibekov effect''', after [[Soviet Union|Soviet]] [[cosmonaut]] [[Vladimir Dzhanibekov]], who noticed one of the theorem's [[logical consequence]]s whilst in space in 1985.<ref>[http://oko-planet.su/science/sciencehypothesis/15090-yeffekt-dzhanibekova-gajka-dzhanibekova.html Эффект Джанибекова (гайка Джанибекова)], 23 July 2009 {{in lang|ru}}. The software can be downloaded [http://live.cnews.ru/forum/index.php?s=5091d296ac0d22ad6b6e9712f3b0edbe&act=Attach&type=post&id=87112 from here] {{Webarchive|url=https://web.archive.org/web/20201113213443/https://oko-planet.su/science/sciencehypothesis/15090-yeffekt-dzhanibekova-gajka-dzhanibekova.html |date=2020-11-13 }}</ref> The effect was known for at least 150 years prior, having been described by [[Louis Poinsot]] in 1834<ref>Poinsot (1834) [https://archive.org/details/thorienouvelled00poingoog/page/n9 ''Theorie Nouvelle de la Rotation des Corps''], Bachelier, Paris</ref><ref>{{cite AV media |publisher = Veritasium | title = The Bizarre Behavior of Rotating Bodies, Explained | date = September 19, 2019 | url = https://www.youtube.com/watch?v=1VPfZ_XzisU | access-date = February 16, 2020 | people = [[Derek Muller]] }}</ref> and included in standard physics textbooks such as [[Classical Mechanics (Goldstein)|''Classical Mechanics'']] by [[Herbert Goldstein]] throughout the 20th century.

The theorem describes the following effect: rotation of an object around its first and third [[Moment of inertia#Principal axes|principal axes]] is stable, whereas rotation around its second principal axis (or intermediate axis) is not.

This can be demonstrated by the following experiment: holdHold a tennis racket at its handle, with its face being horizontal, and throw it in the air such that it performs a full rotation around its horizontal axis perpendicular to the handle (ê₂ in the diagram), and then catch the handle. In almost all cases, during that rotation the face will also have completed a half rotation, so that the other face is now up. By contrast, it is easy to throw the racket so that it will rotate around the handle axis (ê₁) without accompanying half-rotation around another axis; it is also possible to make it rotate around the vertical axis perpendicular to the handle (ê₃) without any accompanying half-rotation.

The experiment can be performed with any object that has three different moments of inertia, for instance with a (rectangular) book, remote control, or smartphone.<ref>{{cite AV media |people=Derek Muller |date=2025-04-25 |title=This phone trick is IMPOSSIBLE – Veritasium |url=https://youtube.com/shorts/WKvAsz3RoRE |format=YouTube video |access-date=2025-04-25}}</ref> The effect occurs whenever the [[axis of rotation]] differs – even only slightly – from the object's second principal axis; air resistance or gravity are not necessary.<ref>{{Cite book |url={{google books |plainurl=yes |id=uVSYswEACAAJ |page=151}} |title=Classical Mechanics with Calculus of Variations and Optimal Control: An Intuitive Introduction |last=Levi |first=Mark |publisher=American Mathematical Society |year=2014 |isbn=9781470414443 |pages=151–152 }}</ref>

== Theory ==

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=== Stable rotation around the first and third principal axis ===

Consider the situation when the object is rotating around the axis with moment of inertia <math>I_1</math>. To determine the nature of equilibrium, assume small initial angular velocities along the other two axes. As a result, according to equation (1), <math>~\dot{\omega}_1</math> is very small. Therefore, the time dependence of <math>~\omega_1</math> may be neglected.

Now, differentiating equation (2) and substituting <math>\dot{\omega}_3</math> from equation (3),

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Note that <math>\omega_2</math> is being opposed and so rotation around this axis is stable for the object.

Similar reasoning gives that rotation around the axis with moment of inertia <math>I_3</math> is also stable.

=== Unstable rotation around the second principal axis ===

Now apply the same analysis to the axis with moment of inertia <math>I_2.</math> This time <math>\dot{\omega}_{2}</math> is very small. Therefore, the time dependence of <math>~\omega_2</math> may be neglected.

Now, differentiating equation (1) and substituting <math>\dot{\omega}_3</math> from equation (3),

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\omega_1\\

\omega_2

\end{bmatrix}</math>which has [[Stability theory#Stability of fixed points in 2D|zero trace and positive determinant]], implying the motion of <math>(\omega_1, \omega_2)</math> is a stable rotation around the origin—aorigin—thus <math>(0,0,\omega_3)</math> is a neutral equilibrium point. Similarly, the point <math>(\omega_1, 0,0)</math> is a neutral equilibrium point, but <math>(0, \omega_2, 0)</math> is a saddle point.

== Geometric analysis ==

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\end{cases}</math>This is shown on the animation to the left.

By inspecting Euler's equations, we see that <math>\dot\omega(t) = 0</math> implies that two components of <math>\omega(t)</math> are zero—that is, the object is exactly spinning around one of the principal axes. In all other situations, <math>\omega(t)</math> must remain in motion.

By Euler's equations, if <math>\omega(t)</math> is a solution, then so is <math>c \omega(ct)</math> for any constant <math>c > 0</math>. In particular, the motion of the body in free space (obtained by integrating <math>c\omega(ct) dt</math>) is ''exactly the same'', just completed faster by a ratio of <math>c</math>.

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The tennis racket effect occurs when <math>\omega(0)</math> is very close to a saddle point. The body would linger near the saddle point, then rapidly move to the other saddle point, near <math>\omega(T/2)</math>, linger again for a long time, and so on. The motion repeats with period <math>T</math>.

The above analysis is all done in the perspective of an observer which is rotating with the body. An observer watching the body's motion in free space would see its angular momentum vector <math>\vec L = I\vec \omega</math> conserved, while both its angular velocity vector <math>\vec \omega(t)</math> and its moment of inertia <math>I(t)</math> undergoingundergo complicated motions in space. At the beginning, the observer would see both <math>\vec \omega(0), \vec L</math> mostly aligned with the second major axis of <math>I(0)</math>. After a while, the body performs a complicated motion and ends up with <math>I(T/2), \vec \omega(T/2)</math>, and again both <math>\vec L, \vec \omega(T/2)</math> are mostly aligned with the second major axis of <math>I(T/2)</math>.

Consequently, there are two possibilities: either the rigid body's second major axis is in the same direction, or it has reversed direction. If it is still in the same direction, then <math>\vec\omega(0), \vec\omega(T/2)</math> viewed in the rigid body's reference frame are also mostly in the same direction. However, we have just seen that <math>\omega(0)</math> and <math>\omega(T/2)</math> are near opposite saddle points <math>(0, \pm L/I_2, 0)</math>. Contradiction.

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When the body is not exactly rigid, but can flex and bend or contain liquid that sloshes around, it can dissipate energy through its internal degrees of freedom. In this case, the body still has constant angular momentum, but its energy would decrease, until it reaches the minimal point. As analyzed geometrically above, this happens when the body's angular velocity is exactly aligned with its axis of maximal moment of inertia.

This happened to [[Explorer 1#Results|Explorer 1]], the first [[satellite]] launched by the [[United States]] in 1958. The elongated body of the spacecraft had been designed to spin about its long (least-[[inertia]]) axis but refused to do so, and instead started [[Precession|precessing]] due to energy [[dissipation]] from flexible structural elements. It also played a role in the [[Solar_and_Heliospheric_Observatory#Near_loss_of_SOHO|near-loss of the joint NASA-ESA Solar and Heliospheric Observatory]] in 1998, when an unintentional spin about the spacecraft-Sun axis destabilized the control laws, leading to the spacecraft tumbling until internal dissipation (in its liquid hydrazine tanks) caused it to settle around its maximum moment axis, such that the Sun remained in the plane of the solar array.

In general, celestial bodies large or small would converge to a constant rotation around its axis of maximal moment of inertia. Whenever a celestial body is found in a complex rotational state, it is either due to a recent impact or tidal interaction, or is a fragment of a recently disrupted progenitor.<ref>{{Cite journal |last=Efroimsky |first=Michael |date=March 2002 |title=Euler, Jacobi, and Missions to Comets and Asteroids |journal=Advances in Space Research |volume=29 |issue=5 |pages=725–734 |doi=10.1016/S0273-1177(02)00017-0|arxiv=astro-ph/0112054 |bibcode=2002AdSpR..29..725E |s2cid=1110286 }}</ref>