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[[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 (note that the numbering is off-set by 1 from the diagram above)]]
[[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 resultkinetic inphenomenon of [[classical mechanics]] describingwhich describes the movement of a [[rigid body]] with three distinct [[principal moments of inertia]]. It ishas also dubbed the '''Dzhanibekov effect''', after [[Soviet Union|Soviet]] [[cosmonaut]] [[Vladimir Dzhanibekov]], who noticed one of the theorem's [[logical consequence]]s whilewhilst 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]</ref> althoughFormally the effect washad alreadybeen known for at least 150 years, beforehaving that and was included in abeen bookdescribed 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>
The theorem describes the following effect: rotation of an object around its first and third [[Moment of inertia#Principal axes|principal axes]] is stable, whilewhereas rotation around its second principal axis (or intermediate axis) is not.
This can be demonstrated withby the following experiment: hold a tennis racket at its handle, with its face being horizontal, and try to throw it in the air sosuch that it will performperforms a full rotation around theits horizontal axis perpendicular to the handle (ê<sub>2</sub> in the diagram, ê<sub>1</sub> in the video), and try tothen 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 (ê<sub>1</sub> in the diagram) without accompanying half-rotation around another axis; it is also possible to make it rotate around the vertical axis perpendicular to the handle (ê<sub>3</sub> in the diagram) without any accompanying half-rotation.
The experiment can be performed with any object that has three different moments of inertia, for instance with a book, remote control, or smartphone. The effect occurs whenever the [[axis of rotation]] differs 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 ==
[[File:Dzhanibekov effect.ogv|thumb|upright=1.5|Dzhanibekov effect demonstration in [[microgravity]], [[NASA]].]]
The tennis racket theorem can be qualitatively analysed with the help of [[Euler's equations (rigid body dynamics)|Euler's equations]].
Under [[torque]]–free conditions, they take the following form:
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